Waves and ElectromagnetismWavefunctions

Lecture notes

Alessandro De Angelis

November 21, 2013

Contents

1 Differential calculus applied to vector fields 41.1 Scalar and vector fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41.2 The gradient . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51.3 The divergence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61.4 The curl . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61.5 Second order derivatives, and the Laplacian . . . . . . . . . . . . . . . . . . 71.6 The differential equation of heat flow . . . . . . . . . . . . . . . . . . . . . . 91.7 Differential operators in polar coordinates . . . . . . . . . . . . . . . . . . . 91.8 The gradient theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101.9 The divergence theorem (Gauss’ theorem) . . . . . . . . . . . . . . . . . . . 12

1.9.1 Flux of a vector field . . . . . . . . . . . . . . . . . . . . . . . . . . . 121.9.2 Flux through a closed surface . . . . . . . . . . . . . . . . . . . . . . 131.9.3 The flux from a cube . . . . . . . . . . . . . . . . . . . . . . . . . . . 151.9.4 Heat conduction; the diffusion equation . . . . . . . . . . . . . . . . . 16

1.10 The curl theorem (Stokes’ theorem) . . . . . . . . . . . . . . . . . . . . . . . 171.10.1 Circulation of a vector field . . . . . . . . . . . . . . . . . . . . . . . 171.10.2 The circulation around a square . . . . . . . . . . . . . . . . . . . . . 18

1.11 Curl-free fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 201.12 Dirac’s delta function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

I Waves and electromagnetism 22

2 Waves 232.1 Translations and the wave equation . . . . . . . . . . . . . . . . . . . . . . . 24

2.1.1 An example: mechanical waves in one dimension . . . . . . . . . . . . 252.2 Energy transported by a wave . . . . . . . . . . . . . . . . . . . . . . . . . . 262.3 Sinusoidal waves and the Fourier theorem . . . . . . . . . . . . . . . . . . . . 262.4 Amplitude, wavelength, period . . . . . . . . . . . . . . . . . . . . . . . . . . 282.5 Transverse and longitudinal waves . . . . . . . . . . . . . . . . . . . . . . . . 282.6 An example: sound . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 292.7 An example: light . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 292.8 The classical Doppler effect . . . . . . . . . . . . . . . . . . . . . . . . . . . 31

2.8.1 Redshift of galaxies and the expansion of the Universe . . . . . . . . 322.8.2 The Vavilov-Cherenkov effect . . . . . . . . . . . . . . . . . . . . . . 35

2.9 Composition of waves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 352.9.1 Boundary conditions and steady waves . . . . . . . . . . . . . . . . . 36

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2.9.2 Beats . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 372.9.3 Group velocity and phase velocity . . . . . . . . . . . . . . . . . . . . 39

2.10 Waves in three dimensions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 412.10.1 Spherical waves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42

3 Electromagnetism and the Maxwell’s equations 433.1 Charge density and current density . . . . . . . . . . . . . . . . . . . . . . . 433.2 Maxwell’s equations in differential form . . . . . . . . . . . . . . . . . . . . . 443.3 Maxwell’s equations and continuity equation for charge . . . . . . . . . . . . 453.4 The potentials, vector and scalar . . . . . . . . . . . . . . . . . . . . . . . . 463.5 Maxwell’s equations, electrostatics and magnetostatics . . . . . . . . . . . . 47

4 Solutions of the Maxwell’s equations in vacuo 484.1 Maxwell’s waves and light . . . . . . . . . . . . . . . . . . . . . . . . . . . . 494.2 Properties of the electromagnetic waves . . . . . . . . . . . . . . . . . . . . . 51

4.2.1 Energy transported an the electromagnetic wave . . . . . . . . . . . . 534.2.2 The Poynting vector . . . . . . . . . . . . . . . . . . . . . . . . . . . 53

4.3 Photoelectric effect; the photon hypothesis . . . . . . . . . . . . . . . . . . . 544.4 Natural units . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56

5 Geometrical optics 595.1 Light propagation through different materials; transmission of electromagnetic

waves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 595.2 Huygens’ principle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 595.3 The laws of reflection and refraction . . . . . . . . . . . . . . . . . . . . . . . 60

5.3.1 The Fermat principle . . . . . . . . . . . . . . . . . . . . . . . . . . . 63

6 Interference and diffraction 646.1 Young’s interference experiment . . . . . . . . . . . . . . . . . . . . . . . . . 64

6.1.1 The two-slit case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 686.2 Diffraction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69

6.2.1 Diffraction through a single slit . . . . . . . . . . . . . . . . . . . . . 696.2.2 Diffraction grating . . . . . . . . . . . . . . . . . . . . . . . . . . . . 716.2.3 The diffraction limit . . . . . . . . . . . . . . . . . . . . . . . . . . . 71

7 Maxwell’s equations and Einstein’s special relativity 737.1 Classical electromagnetism is not a consistent theory . . . . . . . . . . . . . 737.2 Galilean transformations, relativity, and the ether . . . . . . . . . . . . . . . 74

7.2.1 Maxwell’s equations and the ether . . . . . . . . . . . . . . . . . . . . 757.3 The Michelson-Morley experiment . . . . . . . . . . . . . . . . . . . . . . . . 757.4 Einsteins’s postulates and relativity . . . . . . . . . . . . . . . . . . . . . . . 77

7.4.1 Relativity of simultaneity . . . . . . . . . . . . . . . . . . . . . . . . 777.4.2 Time dilation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 797.4.3 Length contraction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81

7.5 Invariance of the interval . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 827.6 Spacelike and timelike intervals; future and past . . . . . . . . . . . . . . . . 85

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8 Lorentz transformations and the formalism of special relativity 878.1 The Lorentz transformations . . . . . . . . . . . . . . . . . . . . . . . . . . . 87

8.1.1 A theorem by von Ignatowski . . . . . . . . . . . . . . . . . . . . . . 888.1.2 Transformation of velocities . . . . . . . . . . . . . . . . . . . . . . . 888.1.3 Time dilation and length contraction, again . . . . . . . . . . . . . . 898.1.4 The relativistic Doppler effect . . . . . . . . . . . . . . . . . . . . . . 90

8.2 4-vectors; covariant and controvariant representation . . . . . . . . . . . . . 928.2.1 Tensors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 938.2.2 Covariant derivatives . . . . . . . . . . . . . . . . . . . . . . . . . . . 948.2.3 Four-dimensional velocity . . . . . . . . . . . . . . . . . . . . . . . . 95

8.3 E=mc2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 958.4 4-momentum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 968.5 The photon . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99

8.5.1 Radiation pressure . . . . . . . . . . . . . . . . . . . . . . . . . . . . 998.6 Examples of relativistic dynamics . . . . . . . . . . . . . . . . . . . . . . . . 99

8.6.1 Decay . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 998.6.2 Elastic scattering . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100

8.7 Relativistic force . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102

9 Covariant formulation of electromagnetism 1049.0.1 The equations for the potentials . . . . . . . . . . . . . . . . . . . . . 1049.0.2 The electromagnetic tensor . . . . . . . . . . . . . . . . . . . . . . . . 1069.0.3 Covariant expression of Maxwell’s equations . . . . . . . . . . . . . . 107

9.1 Transformation of the fields (from Jarrell, electrodynamics) . . . . . . . . . . 1089.1.1 Invariants . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109

9.2 Covariant expression of the Lorentz force . . . . . . . . . . . . . . . . . . . . 1099.2.1 Motion of a particle under a constant force . . . . . . . . . . . . . . . 109

9.3 Radiation from an accelerated particle . . . . . . . . . . . . . . . . . . . . . 110

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Chapter 1

Differential calculus applied to vectorfields

1.1 Scalar and vector fields

The concept of field formalizes physical properties of space. To each point of space (let usstart with the usual 3-dimensional space) a physical quantity is associated. Mathematically,a physical field is a function whose domain is space.

The simplest possible physical field is a scalar field. By a scalar field we mean a fieldwhich is characterized at each point by a single number – a scalar. Of course the numbermay change in time; we shall talk for now about what the field looks like at a given instant.

As an example of a scalar field, let us consider a solid block of material which has beenheated at some places, so that the temperature of the body varies from point to point.Then the temperature will be a function of x, y, and z, the position in space measured in arectangular coordinate system. Temperature T (x, y, z) is a scalar field.

One way of picturing scalar fields is to draw contours, which are imaginary surfacesthrough points for which the field has the same value, just as contour lines on a map connectpoints with the same height. For a temperature field the contours are called “isothermalsurfaces” or isotherms. Figure 1.1 illustrates a temperature field and shows the dependenceof T on x and y when z = 0. Several isotherms are drawn.

There are also vector fields. In this case, a vector is given for each point in (a region of)space. As an example, consider a rotating body: the velocity of the material of the body atany point is a vector which is a function of position. As a second example, consider the flowof heat in a block of material. If the temperature in the block is high at one piece and lowat another, there will be a flow of heat from the hotter places to the colder ones. The heatwill be flowing in different directions in different parts of the block.

The heat flow is a directional quantity which we call ~h. Its magnitude is a measure ofhow much heat is flowing, and can be defined as the amount of thermal energy that passes,per unit time and per unit area, through an infinitesimal surface element, perpendicular tothe direction of flow. The vector points in the direction of flow. In symbols: if ∆J is thethermal energy that passes per unit time through the surface element ∆a, then

~h = lim∆a→0

∆J

∆a~ef (1.1)

where ~ef is the versor of the heat flow. Examples of the heat flow vector are also shown inFigure 1.1.

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Figure 1.1: The temperature field; isotherms.

1.2 The gradient

For a real-valued function T (x, y, z) on <3, the gradient ∇T (x, y, z) (or ~∇T (x, y, z)) is afunction on <3, that is, its value at a point (x, y, z) is the triple

∇T (x, y, z) =

(∂T

∂x,∂T

∂y,∂T

∂z

)=∂T

∂xi +

∂T

∂yj +

∂T

∂zk

in <3, where each of the partial derivatives is evaluated at the point (x, y, z). One can thinkof the symbol ∇ as being “applied” to a real-valued function T to produce a 3-dimensionalfunction ∇T .

Is ∇T (x, y, z) a vector? Of course it is not generally true that any three numbers form avector. It is true only if, when we rotate the coordinate system, the components of the vectortransform among themselves in the correct way for a vector. So it is necessary to analyzehow these derivatives are changed by a rotation of the coordinate system. We shall showthat ∇T (x, y, z) is indeed a vector: the derivatives do transform in the correct way whenthe coordinate system is rotated. We can observe that, by the total differential theorem,

dT =∂T

∂xdx+

∂T

∂ydy +

∂T

∂zdz (1.2)

for any d~r = (dx, dy, dz). And since df is a scalar and d~r is a vector, ∇T (we can now

call it ~∇T ) must be a vector (of course a demonstration based on the transformations ofcoordinates is also possible, but ennoying).

Is ~∇ a vector? Strictly speaking, no, since ∂∂x

, ∂∂y

and ∂∂z

are not numbers. But it helps to

think of it as a vector, as we shall see. The process of “applying” ∂∂x

, ∂∂y

, ∂∂z

to a real-valued

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function T (x, y, z) can be thought of as multiplying the quantities:(∂

∂x

)(T ) =

∂T

∂x,

(∂

∂y

)(T ) =

∂T

∂y,

(∂

∂z

)(T ) =

∂T

∂z

For this reason, ~∇ is often referred to as the “del operator”, since it “operates” on functions.We can write the expression (1.2) as

dT = (~∇f) · (d~r) . (1.3)

~∇T represents the spatial rate of change of T . The x−component of ~∇T shows how fastT changes in the x−direction, and in the same way for the other coordinates. What isthe direction of the vector ~∇T? Equation (1.3) shows that the rate of change of T in any

direction is the component of ~∇T in that direction. It follows that the direction of ~∇T isthat in which it has the largest possible component- in other words, the direction in whichT changes the fastest. The gradient of T has the direction of the steepest uphill slope (in

T ). ~∇T is perpendicular to the isotherms.

What would it mean for the gradient to vanish? If ~∇T = 0 at (x, y, z), then dT = 0 forsmall displacements about the point (x, y, z). This is, then, a stationary point of the functionT (x, y, z). It could be a maximum (a summit), a minimum (a valley), a saddle (a pass).

1.3 The divergence

What if we make the ~∇ operator to operate on a vector field (a function of <3 into <3)instead than on a scalar field? In this case we when two possible kinds of products, thescalar product and the vector product; we shall see that both correspond to interestingfunctions.

Giving a vector function ~F defined in a domain in <3, we define as the divergence of ~Fthe formal scalar product ~∇ and ~F :

~∇ · ~F =

(∂Fx∂x

+∂Fy∂y

+∂Fz∂z

), (1.4)

and it is clearly a scalar.What is its geometrical interpretation? ~∇ · ~F is a measure of how much the vector ~F

spreads out (diverges) from a point. For example, the vector function in Figure 1.2(a) hasa positive divergence (if the arrows pointed in, it would be a large negative divergence), thefunction in Figure 1.2(b) has zero divergence, and the function in Figure 1.2(c) again has apositive divergence.

Exercise: if the function pictured in Figure 1.2(a) is ~F(a) = x~i + y~j, and the function

pictured in Figure 1.2(b) is ~F(b) = ~j, compute their divergence.

1.4 The curl

In a similar way, we can define the vector product of ~∇ times a vector field ~F . This is calledthe curl of ~F :

~∇× ~F =~ux ~uy ~uz∂/∂x ∂/∂y ∂/∂zFx Fy Fz

. (1.5)

6

Figure 1.2: Divergence.

This can be demonstrated to be a vector function.Geometrical interpretation: ~∇× ~F is a measure of how much the vector ~F “curls around”

the point in question. Thus the three functions in Figure 1.2 all when zero curl (as you caneasily check for yourself), whereas the functions in Figure 1.3 when a nonzero curl, pointingin the z−direction, as the right-hand rule would suggest. A whirlpool would be a region oflarge curl.

Exercise: if the function pictured in Figure 1.3(a) is ~F(a) = −y~i + x~j, and the function

pictured in Figure 1.3(b) is ~F(b) = x~j, compute their curl.

1.5 Second order derivatives, and the Laplacian

So far we when had only first derivatives. Why not second derivatives? We can write severalcombinations:

1. ~∇× (~∇T )

2. ~∇ · (~∇× ~F )

3. ~∇ · (~∇T )

4. ~∇(~∇ · ~F )

5. ~∇× (~∇× ~F )

You can check that these are all the legal combinations.

7

Figure 1.3: Curl.

1. and 2. ~∇× (~∇T ) and ~∇ · (~∇× ~F ). The first two are identically zero. Let us check itfor the first one: one has, by the Schwartz’s lemma:

~~∇× ~∇T =~ux ~uy ~uz∂/∂x ∂/∂y ∂/∂z∂T/∂x ∂T/∂y ∂T/∂z

= 0.

We when thus demonstrated that the curl of a gradient is zero, which is easy to rememberbecause of the way the vectors work. It can be demonstrated that if the curl of a field ~F iszero, then this field is always the gradient of a scalar field: there is some scalar field φ suchthat ~F = ~∇φ.

Also ~∇ · (~∇× ~F ) = 0, and there is a similar theorem stating that if the divergence of ~F

is zero, ~F is the curl of some vector field ~A.

3. ~∇·(~∇T ). Let us examine the third expression now. For a real-valued function f(x, y, z),the Laplacian1 of T , denoted by ∇2T , is defined as

∇2T (x, y, z) = ~∇ · (~∇T ) =∂2T

∂x2+∂2T

∂y2+∂2T

∂z2. (1.6)

Sometimes the notation ∆T is used instead.The Laplacian can be thought as a scalar operator

∇2 =

(∂2

∂x2+

∂2

∂y2+

∂2

∂z2

)and as such it can be applied to vector functions as well, giving as output a vector:

∇2 ~A(x, y, z) =(∇2Ax(x, y, z),∇2Ay(x, y, z),∇2Az(x, y, z)

).

The explicit calculation confirms that this approach is justified.

1Pierre-Simon de Laplace (1749 - 1827) was a French mathematician and astronomer whose work waspivotal to the development of mathematical astronomy and statistics. He was one of the first scientists topostulate the existence of black holes and the notion of gravitational collapse.

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4. ~∇(~∇ · ~F ). It is a possible vector field, which may occasionally come up (for example,see next point).

5. ~∇× (~∇× ~F ). Let us compare this expression with the vector identity

~A× ( ~B × ~C) = ~B( ~A · ~C)− ( ~A · ~B)~C .

In order to use this formula, we should replace ~A and ~B by the operator ~∇ and put ~C = ~F .If we do that, we get

~∇× (~∇× ~F ) = ~∇(~∇ · ~F )−∇2 ~F . (1.7)

1.6 The differential equation of heat flow

Let us give an example of a law of physics written in vector notation. For heat conductors,the energy flows through the material from a surface at temperature T2 to a surface attemperature T2 < T1. The total energy flow is proportional to the area A of the faces, andto the temperature difference; it is also inversely proportional to d, the distance between theplates (for a given temperature difference, the thinner the slab the greater the heat flow).Letting J be the thermal energy that passes per unit time through the slab, we write

J = k(T2 − T1)A/d .

The constant of proportionality k is called the thermal conductivity of the material.For a small slab of area ∆a the heat flow per unit time is

∆J = k∆T∆a

∆d(1.8)

where ∆d is the thickness of the slab. ∆J/∆a is what we when defined earlier as the mag-

nitude of the vector ~h, whose direction is the heat flow. The heat flow will be perpendicularto the isotherms. Also, ∆T/∆d is just the rate of change of T with position. And since theposition change is perpendicular to the isotherms, ∆T/∆d is the maximum rate of change.

It is, therefore, in the limit for ∆d → 0, just the magnitude of ~∇T. Since the direction of~∇T is apposite to that of ~h, we can write the previous equation as a vector equation:

~h = −k ~∇T

(the minus sign is necessary because heat flows “downhill” in temperature.) This is thedifferential equation of heat conduction.

1.7 Differential operators in polar coordinates

Often – for example, in the case of central symmetries – it is convenient to use radialcoordinate systems when dealing with quantities such as the gradient, divergence, curl andLaplacian. We will present the expressions for these operators in spherical coordinates.

A point (x, y, z) can be represented in spherical (polar) coordinates (r, θ, φ), where x =r sin θ cosφ, y = r sin θ sinφ, z = r cos θ. θ (the angle down from the z axis) is called thepolar angle, and φ (the angle around from the x axis) is the azimuthal angle.

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Figure 1.4: Spherical coordinates.

At each point (r, θ, φ), ~er, ~eθ, ~eφ are unit vectors in the direction of increasing r, θ, φ,respectively (see Figure 1.4). Then the vectors ~er, ~eθ, ~eφ are orthonormal. By the right-handrule, we see that ~eθ × ~eφ = ~er.

We can summarize the expressions for the gradient and the Laplacian applied to a scalarfield T and to a vector field ~F in spherical coordinates in the following equations:

~∇T =∂T

∂r~er +

1

r

∂T

∂θ~eθ +

1

r sin θ

∂T

∂φ~eφ (1.9)

∇2T =1

r2

∂

∂r

(r2 ∂T

∂r

)+

1

r2 sin θ

∂

∂θ

(sin θ

∂T

∂θ

)+

1

r2 sin2 θ

∂2T

∂φ2. (1.10)

The derivation of the above formulas is conceptually straightforward but boring. Thebasic idea is to take the Cartesian equivalent of the quantity in question and to substituteinto that formula using the appropriate coordinate transformation.

1.8 The gradient theorem

We found previously that there were various ways of taking derivatives of fields. Somegave vector fields; some gave scalar fields. Although we developed many different formulas,everything could be summarized in one rule: the operators ∂/∂x, ∂/∂y, ∂/∂z are the three

components of a vector operator ~∇. We would now like to get some understanding of thesignificance of the derivatives of fields. We shall then when a better feeling for what a vectorfield equation means.

We when already discussed the meaning of the gradient operation (application of ~∇ ona scalar). We take up now an integral formula involving the gradient. The relation containsa very simple idea: since the gradient represents the rate of change of a field quantity, ifwe integrate that rate of change, we should get the total change (like in the fundamentaltheorem of calculus).

Let us first introduce the concept of line integral for a vector field. The line integral froma point a to a point b of the curve is nothing but the integral of the dot product of the valueof the function times the line element (i.e., at each point the value of the function is weighted

10

Figure 1.5: Line integral.

by the cosine of the angle formed by the vector itself and by the tangent to the line):∫ b

a

~E(x, y, z) · d~l

(see Figure 1.5). For example, the work done by the force ~F from a to b along a given path

is the line integral of ~F along that path.Suppose we have a scalar function of three variables T (x, y, z). Starting at point a, we

move by a small distance d~l1 (Figure 1.6). The function T will change by an amount

dT = ~∇T · d~l1 .

Now we move a little further, by an additional small displacement d~l2; the incremental changein T will be ~∇T · d~l2. In this way, proceeding by infinitesimal steps, we make a journey topoint b. At each step we compute the gradient of T (at that point) and dot it into the

displacement d~l: this gives us the change in T. Evidently the total change in T in going froma to b along the path selected is ∫ b

a

~∇T · d~l = T (~b)− T (~a) .

This is called the fundamental theorem for gradients; like the “ordinary” fundamental the-orem of calculus, it says that the integral (here a line integral) of a derivative (here thegradient) is given by the difference between the values of the function at the boundaries (aand b).

A geometrical interpretation will make use of an example. Suppose you wanted to deter-mine the height of the Eiffel Tower. You could climb the stairs, using a ruler to measure therise at each step, and adding them all up, or you could place altimeters at the top and thebottom, and subtract the two readings; you should get the same answer either way.

Line integrals ordinarily depend on the path taken from a to b. But the right side of inthe gradient theorem makes no reference to the path - only to the end points. Evidently,gradients when the special property that their line integrals are path independent:

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Figure 1.6: Line integral and gradient theorem.

Corollary 1:∫ ba(~∇T ) · d~l is independent of the path taken from a to b.

Corollary 2:∮

(~∇T ) · d~l = 0, since the beginning and end points are identical, and henceT (b)− T (a) = 0.

1.9 The divergence theorem (Gauss’ theorem)

1.9.1 Flux of a vector field

A surface integral is an expression of the form∫S

~h · ~n da

where ~h is a vector function, and da is an infinitesimal patch of area; ~n is a unit vector withdirection perpendicular to the surface. There are, of course, two directions perpendicularto any surface, so the sign of a surface integral is intrinsically ambiguous. If the surface isclosed (forming a “balloon”), one frequently puts a circle on the integral sign∮

S

~h · ~n da ;

in this case tradition dictates that “outward” is positive, but for open surfaces it is, again,arbitrary.

We will identify sometimes a flat surface with a vector perpendicular to the surface itself,and with intensity equal to the area of the surface. The expression of the surface integralbecomes then ∫

S

~h · d~a .

If ~h describes the flow of a fluid (mass per unit area per unit time), then∫S~h·d~a represents

the total mass per unit time passing through the surface - hence the alternative name, fluxΦ. In the case of heat flow, we may think: ~h is the “current density” of heat flow and the

12

Figure 1.7: The closed surface S defines the volume V. The unit vector ~n is the outwardnormal to the surface element da, and ~h is the heat-flow vector at the surface element.

surface integral of it is the total heat current directed out of the surface: that is, the thermalenergy per unit time (joule per second).

We can generalize this idea to the case to any vector field; for instance, it might be theelectric field. We can certainly still integrate the normal component of the electric field overan area if we wish. Although it does not appear to be the flow of anything, we still call itthe “flux”. We say flux of ~E through the surface S the quantity

Φ =

∫S

~E · d~a .

We thus generalize the word “flux” to mean the “surface integral of the normal component”of a vector.

1.9.2 Flux through a closed surface

We defined the vector ~h, which represents the heat through a unit area in a unit time.Suppose that inside a block of material we have some closed surface S which encloses thevolume V. We would like to find out how much heat is flowing out of this volume. We can,of course, find it by calculating the total heat flow out of the surface S (Figure 1.7).

We write da for the area of an element of the surface. The symbol stands for a two-dimensional differential. If, for instance, the area happened to be in the xy−plane, we wouldhave da = dxdy (later we shall have integrals over volume and for these it is convenient toconsider a differential volume that is a little cube; so when we write dV we mean dV =dxdydz)2.

The heat flow out through the surface element da is the area times the component of~h perpendicular to da. Returning to the special case of heat flow, let us take a situationin which heat is conserved. For example, imagine some material in which after an initialheating no further heat energy is generated or absorbed. Then, if there is a net heat flow out

2Some texts write d2a instead of da to remind that it is kind of a second-order quantity. They also writed3V instead of dV. We will use the simpler notation, and assume that you remember that an area has twodimensions and a volume has three.

13

Figure 1.8: A volume V contained inside the surface S is divided into two pieces by a “cut”at the surface Sab. We now when the volume V1 enclosed in the surface S1 = Sa + Sab and

the volume V2 enclosed in the surface S2 = Sb + Sab.

of a closed surface, the heat content of the volume inside must decrease. So, in circumstancesin which heat would be conserved, we say that∮

~h · d~a = −dQdt

, (1.11)

where Q is the heat inside the surface.We shall point out an interesting fact about the flux of any vector. Imagine that we have

a closed surface S that encloses the volume V. We now separate the volume into two partsby some kind of a “cut”, as in Figure 1.8. Now we have two closed surfaces and volumes:the volume V1 is enclosed in the surface S1, which is made up of part of the original surfaceSa and of the surface of the cut, Sab; the volume V2 is enclosed by S2, which is made up ofthe rest of the original surface S, and closed off by the cut Sab.

The sum of the fluxes through S1 and S2 equals the flux through the whole surface thatwe started with. The flux through the part of the surfaces Sab common to both S1 and S2

just exactly cancels out. For the flux of a generic vector ~C out of V1, we can write

ΦS1 =

∫Sa

~C · ~n da+

∫Sab

~C · ~n1da

and for the flux out of V2

ΦS2 =

∫Sb

~C · ~n da+

∫Sab

~C · ~n2da .

Since ~n1 = −~n2, the sum of the fluxes through S1 and S2 is just the sum of two integralswhich, taken together, give the flux through the original surface S.

We can similarly subdivide again the volume - say by cutting V1 into two pieces. Yousee that the same arguments apply. So for any way of dividing the original volume, it isgenerally true that the flux through the outer surface, which is the original integral, is equalto a sum of the fluxes out of all the little interior pieces. Then we can divide it by a largenumber of small cubes.

14

Figure 1.9: Computation of the flux of ~C out of a small cube.

1.9.3 The flux from a cube

We now take the special case of a small cube (or parallelepiped) and find an interestingformula for the flux out of it.

Consider a cube whose edges are lined up with the axes (Figure 1.9). Let us supposethat the coordinates of the corner nearest the origin are x, y, z. Let ∆x be the length of thecube in the x−direction, ∆y be the length in the y−direction, and ∆z be the length in thez−direction (they are small quantities). We wish to find the flux of a vector field ~C throughthe surface of the cube. We shall do this by making a sum of the fluxes through each of thesix faces.

First, consider the face marked 1 in the figure. The flux outward on this face is thenegative of the x−component of ~C, integrated over the area of the face. This flux is approx-imately

Φ1 = −Cx(1) ∆y∆z .

(since we are considering a small cube, we approximate the integral by the value of Cx atthe center of the face - which we call the point (1) - multiplied by the area of the face).Similarly, the flux out of face 2 is approximately

Φ2 = Cx(2) ∆y∆z .

Cx(1) and Cx(2) are, in general, slightly different. If ∆x is small enough, we can write

Cx(2) = Cx(1) +∂Cx∂x

∆x

(there are, of course, higher order terms, but we are considering the limit for ∆x → 0). Sothe flux through faces 1 and 2 is

Φ1 + Φ2 =∂Cx∂x

∆x∆y∆z .

15

The derivative should really be evaluated at the center of face 1; that is, at (x, y+ ∆y/2, z+∆z/2). But in the limit of an infinitesimal cube, we make a negligible error if we evaluate itat the corner (x, y, z).

Applying the same reasoning to each of the other pairs of faces, we find

Φ3 + Φ4 =∂Cy∂y

∆x∆y∆z ,

and

Φ5 + Φ6 =∂Cz∂z

∆x∆y∆z .

The total flux through all the faces is the sum of these terms. We find that∫cube

~C · d~a =

(∂Cx∂x

+∂Cy∂y

+∂Cz∂z

)∆x∆y∆z

and so we can say that for an infinitesimal cube

dΦ = (~∇ · ~C)dV .

We have shown that the outward flux from the surface of an infinitesimal cube is equal tothe divergence of the vector multiplied by the volume of the cube. We now see the meaningof the divergence of a vector. The divergence of a vector at the point P is the flux - theoutgoing “flow” of ~C per unit volume, in the neighborhood of a point P .

We have connected the divergence to the flux out of an infinitesimal volume. For anyfinite volume we can use the fact we proved above - that the total flux from a volume is thesum of the fluxes out of each part. We can, that is, integrate the divergence over the entirevolume. This gives us the theorem that the integral of the normal component of any vectorover any closed surface can also be written as the integral of the divergence of the vectorover the volume enclosed by the surface. This theorem is named after Gauss3.∮

S

~C · d~a =

∫V

(~∇ · ~C) dV

where S is any closed surface and V is the volume inside it.

1.9.4 Heat conduction; the diffusion equation

By the Gauss’ theorem, equation (1.11),∮~h · d~a = −dQ

dt, becomes, if we call q the amount

of heat per unit volume,

− d

dt(q∆V ) = (~∇ · ~h)∆V

and thus we can transform the integral equation (1.11) into a differential equation definedlocally, i.e., point by point:

−dqdt

= ~∇ · ~h .

This kind of law appears frequently in physics: it is a conservation law, or a continuityequation.

3Carl Friedrich Gauss (Brunswick, 1777 - Gottingen, 1855) was a German mathematician, generallyregarded as one of the greatest mathematicians of all time for his contributions to number theory, geometry,probability theory, geodesy, planetary astronomy, the theory of functions, and potential theory (includingelectromagnetism). Gauss was the only child of poor parents. He was rare among mathematicians in thathe was a calculating prodigy.

16

Figure 1.10: Left: the circulation of ~C around the curve Γ. Right: The circulation aroundthe whole loop is the sum of the circulations around the two loops Γ1 and Γ2.

1.10 The curl theorem (Stokes’ theorem)

1.10.1 Circulation of a vector field

The circulation of a vector field is the line integral around a closed loop.Playing the same kind of game we did with the flux, we can show that the circulation

around a loop is the sum of the circulations around two partial loops. Suppose we breakup our curve of Figure 1.10(left) into two loops, by joining two points (1) and (2) on theoriginal curve by some line that cuts across as shown in Figure 1.10(right).

There are now two loops, Γ1 and Γ2; Γ1 is made up of Γa, which is that part of theoriginal curve to the left of (1) and (2), plus Γab, the “short cut”; Γ2 is made up of the restof the original curve plus the short cut.

The integral along Γab will have, for the curve Γ2, the opposite sign compared to Γ1

because the direction of travel is opposite - we must take both our line integrals with thesame “sense” of rotation. Following the same kind of argument we used before, you can seethat the sum of the two circulations will give just the line integral around the original curveΓ: the parts due to Γab cancel. The circulation around the one part plus the circulationaround the second part equals the circulation about the outer line.

We can continue the process of cutting the original loop into any number of smallerloops. When we add the circulations of the smaller loops, there is always a cancellation ofthe parts on their adjacent portions, so that the sum is equivalent to the circulation aroundthe original single loop.

Now let us suppose that the original loop is the boundary of some surface. There are, ofcourse, an infinite number of surfaces which all have the original loop as the boundary. Ourresults will not, however, depend on which surface we choose. First, we break our originalloop into a number of small loops that all lie on the surface we have chosen. No matter whatthe shape of the surface, if we choose our small loops small enough, we can assume thateach of the small loops will enclose an area which is essentially flat. Also, we can chooseour small loops so that each is very nearly a square (Figure 1.11). Now we can calculate thecirculation around the big loop by finding the circulations around all of the little squaresand then taking their sum.

17

Figure 1.11: Some surface bounded by the loop Γ is chosen. The surface is divided into anumber of small areas, each approximately a square. The circulation around Γ is the sum

of the circulations around the little loops.

1.10.2 The circulation around a square

How shall we find the circulation for each little square? One question is, how is the squareoriented in space? We could easily make the calculation if it had a special orientation. Forexample, if it were in one of the coordinate planes. Since we have not assumed anythingas yet about the orientation of the coordinate axes, we can just as well choose the axes sothat the one little square we are concentrating on at the moment lies in the xy−plane, as inFigure 1.12. If our result is expressed in vector notation, we can say that it will be the sameno matter what the particular orientation of the plane.

We want now to find the circulation of the field ~C around our little square. It will be easyto do the line integral if we make the square small enough that the vector ~C does not changemuch along any one side of the square (the assumption is better the smaller the square, sowe are really talking about infinitesimal squares).

Starting at the point (x, y)−the lower left corner of the figure - we go around in the di-rection indicated by the arrows. Along the first side - marked (1) - the tangential componentis Cx(1) and the distance is ∆x. The first part of the integral is Cx(1)∆x. Along the secondleg, we get Cy(2)∆y. Along the third, we get −Cx(3)∆x, and along the fourth, −Cy(4)∆y.The minus signs are required because we want the tangential component in the direction oftravel. The whole line integral is then∮

~C · d~l = (Cx(1)− Cx(3))∆x+ (Cy(2)− Cy(4))∆y .

Since

Cx(3) ' Cx(1) +∂Cx∂y

∆y

and

Cy(4) ' Cy(2) +∂Cy∂x

∆x

one can write, at first order, ∮~C · d~l =

(∂Cy∂x− ∂Cx

∂y

)∆x∆y .

18

Figure 1.12: Computing the circulation of C around a small square.

Neglecting second order terms, the derivative can be evaluated at (x, y). The aboveexpression can be written in vector form:∮

~C · d~l =(~∇× ~C

)·∆~a .

The circulation around any loop Γ can now be easily related to the curl of the vector field.We fill in the loop with any convenient surface S, as in Figure 1.11, and add the circulationsaround a set of infinitesimal squares in this surface; the sum can be written as an integral.

Our result is a very useful theorem called Stokes’ theorem4∮Γ

~C · d~l =

∫S

(~∇× ~C) · d~a (1.12)

where S is any surface bounded by Γ.We must now speak about a convention of signs. The z−axis in Figure 1.12 would point

toward the reader in a “usual”-that is, “right-handed”-system of axes. When we took ourline integral with a “positive” sense of rotation, we found that the circulation was equal tothe z−component of ~∇× ~C. If we had gone around the other way, we would have gotten theapposite sign. Now how shall we know, in general, what direction to choose for the positivedirection of the “normal” component of ~∇ × ~C? The “positive” normal must always berelated to the sense of rotation by the “right-hand rule”.

4Sir George Gabriel Stokes (1819 - 1903) was a mathematician and physicist. Born in Ireland, he spentall of his career at University of Cambridge, where he served as the Lucasian Professor of Mathematics from1849 until his death. Stokes made seminal contributions to fluid dynamics (including the Navier-Stokesequations), optics, and mathematical physics.

19

1.11 Curl-free fields

We would like, now, to consider some consequences of our new theorems. Take first the caseof a vector whose curl is everywhere zero. Then Stokes’ theorem says that the circulationaround any loop is zero. Now if we choose two points (1) and (2) on a closed curve, itfollows that the line integral of the tangential component from (1) to (2) is independent ofwhich of the two possible paths is taken. We can conclude that the integral from (1) to(2) can depend only on the location of these points - that is to say, it is some function ofposition only. The same logic was used where we proved that if the integral around a closedloop of some quantity is always zero, then that integral can be represented as the differenceof a function of the position of the two ends. This fact allowed us to invent the idea of apotential. We proved, furthermore, that the vector field was the gradient of this potentialfunction. It follows that any vector field whose curl is zero is equal to the gradient of somescalar function. That is, if ~∇ × ~C = 0 everywhere, there is some scalar field ψ for which~C = ~∇ψ. We can, if we wish, describe this special kind of vector field by means of a scalarfield (this shows how restrictive is the class of the conservative fields).

Let’s show something else. Suppose we have any scalar field φ. If we take its gradient,~∇φ, the integral of this vector around any closed loop must be zero:∮

loop

~∇φ · d~l = 0 .

Using Stokes’ theorem, we conclude that∫surface

(~∇× (~∇φ)

)· d~a = 0

over any surface; but this means that the integrand is zero:

~∇× (~∇φ) = 0 .

We obtained in a different way a result we had obtained with vector algebra.We stated, without demonstration, that a field ~C with zero curl can be expressed as the

gradient of a scaler field. Now, thanks to the Stokes’ theorem, we can demonstrate it. If thecurl is zero, then the circulation along whatever loop is zero; thus the line integral from apoint A to a point B is independent of the path. I can thus define, choosing arbitrarily apoint A,

ϕ(B) =

∫ B

A

~C · d~l

and thus~C = ~∇ϕ .

Note that I can choose arbitrarily the point A; this reflects on the fact that, adding anarbitrary constant to ϕ, the above relation is still valid.

1.12 Dirac’s delta function

The Dirac5 delta function is used to define fields which are nonzero in a region of measurezero, but for which the integral over space is different from zero.

5Paul Dirac (1902 -1984) was an English theoretical physicist who made fundamental contributions tothe early development of both quantum mechanics and quantum electrodynamics. Among other discoveries,

20

Figure 1.13: A sketch of the function δ(x).

In one dimension, it can be defined as a real function δ(x) being zero in all points apartfrom the origin, with the constraint ∫ ∞

−∞δ(x) dx = 1

(i.e., its value must be infinite at the origin, see Figure 1.13) for a sketch.In the mathematical literature it is known as a generalized function, or distribution. It

can be obtained as the limit of a class of legitimate functions, for example, of the Gaussianfunctions:

δ(x) = limσ→0

1

σ√

2πe−

x2

2σ2 .

Integrals including δ(x) are perfectly legitimate, and in particular:∫ ∞−∞

δ(x− a)f(x) dx = f(a) .

It is easy to generalize the delta function to three dimensions:

δ3(~r) = δ(x)δ(y)δ(z) .

This three-dimensional delta function is zero everywhere except at (0, 0, 0), where it blowsup. Its volume integral is 1.

As in the one-dimensional case, integration with δ3(~r) picks out the value of a functionf at the location of the spike (if the spike is included in the domain of integration):∫

all space

δ(~r − ~a)f(~r) dV = f(~a) .

he formulated the Dirac equation, which describes the behaviour of fermions, and predicted the existence ofantimatter. Dirac shared the Nobel Prize in Physics for 1933 with Erwin Schrodinger, for his contributionsto Quantum Mechanics.

21

Part I

Waves and electromagnetism

22

Chapter 2

Waves

Until 1600, scientists were dreaming to give a complete description of nature through thereduction to a set of elementary particles. An elementary particle is ideally a point-likestructure; the density of its charges (mass, electrical charge, ...) can be represented by aDirac’s delta function. Isaac Newton1 in his studies related to optics used only the conceptof particle to treat light’s propagation.

In the XVII century, however, some particular phenomena were observed, which couldnot be described by such model: their description needed the displacement of a delocalizedperturbation. Phenomena like destructive interference (the sum of two perturbations couldcancel) needed an extension of the particle model. In order to explain such phenomena, theconcept of wave was formally introduced in 1600 in the Netherlands, where people had along tradition in the field of optics. Christiaan Huygens2 was one of the main actors of thisintellectual revolution.

The wave is a model that describes the general propagation of a perturbation. Typicalexamples of phenomena usually described as waves are:

• sound waves;

• elastic waves (local deformations);

• waves on the water surface (gravity waves);

• electromagnetic waves.

The representation of a generic wave is a function ξ(x, y, z, t).

1Sir Isaac Newton (1642 - 1727) was an English physicist, mathematician, astronomer, natural philoso-pher, alchemist and theologian, who has been considered by many to be the greatest and most influentialscientist who ever lived. His monograph Philosophiae Naturalis Principia Mathematica, published in 1687,laid the foundations for most of classical mechanics. Newton built the first practical reflecting telescope anddeveloped a theory of colour based on the observation that a prism decomposes white light into the manycolours that form the visible spectrum. He also formulated an empirical law of cooling and studied the speedof sound. In mathematics, Newton shares the credit with Leibniz for the development of differential andintegral calculus. Newton was also deeply involved in occult studies and interpretations of religion.

2Christiaan Huygens (1629 - 1695) was a prominent Dutch mathematician, astronomer, physicist andhorologist. His work included early telescopic studies elucidating the nature of the rings of Saturn and thediscovery of its moon Titan, the invention of the pendulum clock and other investigations in timekeeping,and studies of both optics and the centrifugal force. Huygens achieved note for his argument that lightconsists of waves.

23

Figure 2.1: A perturbation moving right with speed c.

2.1 Translations and the wave equation

If∂ξ/∂y = ∂ξ/∂z = 0 ,

then ξ(x, t) is called a plane wave; a function satisfying these conditions might propagatealong the x axis only.

Let us consider the function of the space coordinate ξ(x, 0) at a fixed time t = 0 (weset it arbitrarily as the origin of time). Suppose that we want that after a time t the samespace function is displaced by ∆x = vt (i.e., the perturbation represented by that functionis moving in the positive direction of the x axis at a speed v). The new function will beξ(x− vt).

In the same way, the functional form ξ(x+ vt) will represent a perturbation moving leftwith speed v.

If we set u± = (x± vt), then:∂ξ

∂x=

dξ

du±

∂u±∂x

=dξ

du±∂ξ

∂t=

dξ

du±

∂u±∂t

= ±v dξdu±

∂2ξ

∂x2=

d

du±

(dξ

du±

)∂u±∂x

=d2ξ

du2±

∂2ξ

∂t2=

d

du±

(dξ

du±

)∂u±∂t

(±v) = v2 d2ξ

du2±

24

Figure 2.2: Perturbation along a string.

⇒ ∂2ξ

∂x2=

1

v2

∂2ξ

∂t2(2.1)

This is called the d’Alembert’s3 equation, or wave equation, in 1 dimension. A plane waveis defined as a function satisfying the equation (2.1).

Since the wave equation is linear, the sum of two or more waves is still a wave. Thegeneral solution ξ(x, t), however, is not in general a function that moves left or right, but alinear combination of tho functions ξ+ and ξ− moving right and left respectively:

ξ(x, t) = ξ+(x− vt) + ξ−(x+ vt) . (2.2)

2.1.1 An example: mechanical waves in one dimension

Let us consider a string of length L with a tension T . In equilibrium the string is straight.Suppose that the string is perturbed from its rest state; a section is moved by a small distancecompared to its length, perpendicularly to the string itself.

Let AB be a piece of the string of length dx and mass dm, witch is moved to a distance ξfrom the equilibrium state. We have a force on each extreme of the element AB. Because ofthe bending of the string these two forces have the same intensity but not the same direction.Analyzing the forces acting on the element AB, we have:

Fx = T (cosα′ − cosα) = dmax

Fξ = T (sinα′ − sinα) = dmaξ.(2.3)

Because of the little bending, the two angles α and α′ are small, so we can write:

sinα′ − sinα ' α′ − α ' tanα′ − tanα

cosα′ − cosα ' 0 ,(2.4)

and then:

Fx = dmax ' 0

Fξ = dmaξ ' T (tanα′ − tanα).(2.5)

3Jean-Baptiste d’Alembert (1717 - 1783) was a French mathematician, physicist, philosopher, and music

theorist. He was also co-editor with Denis Diderot of the Encyclopdie.

25

We see that the horizontal net force is small compared to the vertical one (it is 0 at firstorder). We can associate tanα to the slope of the string, i.e., to the derivative with respectto x: ∂ξ/∂x, so we have:

Fξ = T

(∂ξ

∂x

∣∣x+dx− ∂ξ

∂x

∣∣x

)= T

∂

∂x

(∂ξ

∂x

)dx = T

∂2ξ

∂x2dx . (2.6)

Let µ = M/L be the linear density of the string. We have that the mass of element ABis µdx; moreover the vertical acceleration is ∂2ξ/∂t2. By the second law of motion we canwrite:

µdx∂2ξ

∂t2= T

∂2ξ

∂x2dx. (2.7)

Thus,

∂2ξ

∂t2=T

µ

∂2ξ

∂x2. (2.8)

As we can see this is the d’Alembert wave equation so we now know that the perturbabion

moves through the string with velocity v =√

Tµ

.

2.2 Energy transported by a wave

A wave transfers a perturbation, and thus, ultimately, energy. If u is the density of energyper unit volume associated to the perturbation, the the intensity of transmitted energy Uper units of area and time is:

I =dU

dSdt=

dU

dSdx

dx

dt= uv, (2.9)

where I is called intensity and is measured in Wm−2. In a sinusoidal mechanical waveξ0 sin(kx− ωt) (v = ω/k) a fixed point describes a simple harmonic oscillation, thus:

u =1

2kξ2

0 ∝1

2ω2ξ2

0 ⇒ I ∝ 1

2ω2ξ2

0v . (2.10)

2.3 Sinusoidal waves and the Fourier theorem

A periodical phenomenon is something that repeats itself equally on regular periods of time.More precisely in mathematics a function f(t) is said to be periodical of period T if:

f(t+ nT ) = f(t), ∀n ∈ N and ∀t. (2.11)

Well known examples of periodical functions are the trigonometric functions sin and cos.Sinusoidal (or, which is equivalent, cosinusoidal) waves are very important because of the

Fourier theorem.The Fourier theorem states that every periodical function f(t) of period T = 2π/ω which

is finite, continuos and differentiable can be expressed by the Fourier series:

f(t) = a0 +∞∑n=1

(an cosnωt+ bn sinnωt) = a0 +∞∑n=1

cn cos(nωt+ ϕn) (2.12)

26

Figure 2.3: Example of Fourier decomposition.

where:

cn =√a2n + b2

n (2.13)

tanϕn =anbn

(2.14)

a0 =1

T

∫ T

0

f(t)dt (2.15)

an =2

T

∫ T

0

f(t) cos(nωt)dt ; bn =2

T

∫ T

0

f(t) sin(nωt)dt . (2.16)

Thus a generic wave can be written as a linear combination of sine waves. Let us analyzethe proprieties of (co)sinusoidal waves:

ξ(x, t) = ξ0 cos[k(x− vt) + δ] . (2.17)

To represent a (co)sinusoidal wave we can use the complex representation:

ξ(x, t) = Re[ξ0ei[k(x−vt)+δ]] (2.18)

and so we can refer to the complex wave:

ξ(x, t) = ξ0eik(x−vt) (2.19)

where ξ0 = ξ0eiδ (the amplitude absorbs the phase). As we stated before, the real part of

the complex wave is the physical wave in this context. Since the wave equation is linear, wecan carry on our calculations using exponentials (the advantage of the complex notation isthat exponentials are much easier to manipulate than sines and cosines), and then go backto the cosine representation when we want.

27

2.4 Amplitude, wavelength, period

We introduce some important definitions related to of a harmonic (sinusoidal) wave ξ(x, t) =ξ0 cos[k(x− vt) + δ]:

• ξ0 = max{|ξ(x, t)|} is defined as the amplitude.

• T = 2π/ω is the period, i.e., the minimum time after which the function makes areplica of itself (the time that a point takes to do a complete oscillation).

• ν = 1/T is the frequency, i.e., the number of periods per second; the unit of thefrequency in the SI is the hertz Hz: [Hz]= [s−1].

• λ = 2π/k is the wavelength, the spatial period of the wave - i.e., the minimumdistance over which the wave’s shape repeats.

• v = λ/T = λν is the velocity of the wave.

• k = 2π/λ is defined as the wave number.

• ω = kv = 2π/T is the angular frequency.

• δ is the phase, and it depends on the choice of the starting time.

2.5 Transverse and longitudinal waves

Let us suppose that the quantity associated to a wave ξ is a vector, for example representingan oscillation, or the electric field. We define:

• Longitudinal waves the waves for which the direction of the perturbation is the sameas their direction of propagation.

• Transverse waves the waves for which the direction of the perturbation is perpendicularto the direction of propagation.

Figure 2.4: Transverse and longitudinal waves.

Transverse waves are said to be polarized if there is a simple law describing the directionof oscillation ξ in the plane of oscillation, say, yz. Particular cases of polarized waves are:

• Linearly polarized waves: waves in which the direction of oscillation ξ is fixed.

28

• Elliptically polarized waves. These are waves that satisfy the condition:

ξy = ξ0y sin(kx− ωt) (2.20)

ξz = ξ0z cos(kx− ωt) (2.21)

ξ2y

ξ20y

+ξ2z

ξ20z

= 1 (2.22)

In particular, if ξ20y = ξ2

0x the wave is said to be circularly polarized.

2.6 An example: sound

Sound is a sequence of (approximately longitudinal) waves of pressure that propagate throughcompressible media such as air, water, or solids. Sound that is perceptible by humans hasfrequencies from about 20 Hz to 20 000 Hz; in air at standard temperature and pressure, thecorresponding wavelengths of sound waves range from 17 m to 17 mm.

Pitch is an auditory sensation in which a listener assigns musical tones to relative positionson a musical scale based primarily on the frequency of vibration. Pitch is closely related tofrequency, but the two are not equivalent, apart from purely sinusoidal waves. Frequencyis an objective, scientific concept, whereas pitch is subjective, and related to the sector ofpsychoacoustics.

The speed of sound depends on the medium the waves pass through, and is a fundamentalproperty of the material. In general, the speed of sound is proportional to the square rootof the ratio of the elastic modulus K = −V dP/dV of the medium (where V is the volumeand P is the pressure) to its density.

v =

√K

ρ.

Physical properties (density and elastic modulus) and thus the speed of sound change withambient conditions. For example, the speed of sound in gases depends on temperature. Inair at NTP, the speed of sound is approximately 340 m/s. In fresh water at 20 ◦C, the speedof sound is approximately 1400 m/s. In steel, the speed of sound is about 6000 m/s.

2.7 An example: light

Light is a transverse electromagnetic wave (i.e., a wave propagating electromagnetic fields),traveling at speed c ' 3.00× 108 m/s in vacuo. We shall discuss in detail the properties oflight, which can be derived from the equations of electromagnetism, in the following chapters.

Visible light is the electromagnetic radiation that is visible to the human eye, and isresponsible for the sense of sight. It has a wavelength in the range from about 380 nm toabout 750 nm.

29

Color is the visual perceptual property corresponding in humans to the categories calledred, blue, yellow, green and others. It derives from the spectrum of light (distributionof light power versus wavelength) interacting in the eye with the spectral sensitivities ofthe light receptors. Color categories and physical specifications of color are also associatedwith objects, materials, light sources, etc., based on their physical properties such as lightabsorption, reflection, or emission spectra.

Color depends on wavelength, but the precise dependence of colour on wavelength is amatter of culture and historical contingency. A common list identifies six main bands: red,orange, yellow, green, blue, and purple; red has the smallest wavelength, and purple thehighest. Newton’s description in his Opticks included a seventh color, indigo, between blueand purple. However, colour is a perception of the brain: most humans are trichromatic, theretina containing three types of color receptor cells, called cones.

The ability of the human eye to distinguish colors is based upon the varying sensitivityof different cells in the retina to light of different wavelengths. One type, relatively distinctfrom the other two, is most responsive to light that we perceive as blue or blue-violet, withwavelengths around 450 nm; cones of this type are sometimes called short-wavelength cones,S cones, or blue cones. The other two types are closely related genetically and chemically:middle-wavelength cones, M cones, or green cones are most sensitive to light perceived asgreen, with wavelengths around 540 nm, while the long-wavelength cones, L cones, or redcones, are most sensitive to light we perceive as greenish yellow, with wavelengths around570 nm. The overall maximum perception per unit of energy for average humans is aroundgreen.

Beyond (above and below) the visible spectrum other bands of light are classified withspecial names: see the Figure for their definition.

30

Being (as we shall see) energy of light proportional to frequency, the most energeticcvisible light is blue, and, beyond the visible, in the gamma-ray band.

2.8 The classical Doppler effect

A peculiar effect of wave propagation is the so-called Doppler effect: the characteristics ofa wave depend on the motion of the emitter and of the receiver. It is a phenomenon thataffects all type of waves, and it is of primary importance in astrophysics, as we shall seenext.

The Doppler effect is the apparent change of frequency of the wave, when a source oran observer are in movement. We notice this effect, for example, when we are in relativemovement with respect to a source of sound waves: if we approach the source, the frequencyof the sound is higher; if we get far away, the frequency appears lower.

Let us consider a source, that emits a wave with frequency f , and an observer. Supposethat the source and the observer move with constant relative velocity, and that the directionof the velocity lies on the straight line joining the two objects. We want to analize how theDoppler effect manifests itself. We make also the semplification that one of the two objectsis stationary with respect to the medium in which the wave propagates (for example, in thecase of sound, the air of the atmosphere). We can immediately note that the phenomenondepends on who is moving and who is stationary with respect to the medium in which thewave propagates.

31

Figure 2.5: A stationary source of waves and a moving source.

Source moving, observer at rest. Let us consider first the case in which the sourcemoves with velocity vs < v in the direction of the observer. The length of the wave receivedλ′ changes. We have that

λ′ = λ− vsf.

Thus:

v

f ′=v

f− vsf

;

=⇒ f ′ = f · 1

1− vsv

;

These conditions hold only when the velocity of propagation v is larger than vs. In particularif vs � v, we can make the approximation

f ′ ' f(

1 +vsv

).

Source at rest, observer moving. Let us consider the case in which the source lies ona fixed point and the observer moves with velocity vo in the direction of the source. Thevelocity of the wave relative to the observer is v + vo, so we get

f ′ =v + voλ

=v + vov/f

= f(

1 +vov

).

We can notice that only in the first approximation the variation of frequency is the samein both cases. The case in which both the source and the observer move is more complicated,as the case in which the direction of the movement is not directed along the line joining them;however, there is nothing conceptually new.

2.8.1 Redshift of galaxies and the expansion of the Universe

The Doppler effect has important applications in astrophysics. Observing the Doppler effecton the spectrum of emission of galaxies and stars in the Universe, we can compute the relative

32

Figure 2.6: Experimental plot of the relative velocity (in km/s) of known astrophysicalobjects as a function of the distance from the Earth (in Mpc; 1 pc ' 3.3 ly). The line is a

fit to Hubble’s Law.

33

Figure 2.7: Redshift of emission spectrum of stars and galaxies at different distances.Using this shift we can calculate the relative velocity between these objects and the Earth.

velocity of these objects with respect to us, and then the distance, thanks to the so-calledHubble’s law.

In 1929 Edwin Hubble4, studying the emission of galaxies, observed from their Dopplerredshift that objects in Universe move away with us with velocity

v = H0d , (2.23)

where d is the distance between the objects, and H0 is a parameter called the Hubbleconstant (whose value is known today to be about 24km /s /Mly). The above relation iscalled Hubble’s law.

To give an idea of what H0 means, the speed of revolution of the Earth around the Sunis about 30 km/s; Andromeda, the large galaxy closest to the Milky Way, has a distance d ofabout 2.5 Mly. However, the Hubble’s law is just statistical and working for large distances,where gravitational attraction becomes negligible: Andromeda is indeed approaching us.

Dimensionally we note that H0 is a frequency: H0 ' (14 × 109 years)−1. A simpleinterpretation of this law is that, if the Universe has always been expanding at a constantrate, about 14 · 109 years ago its volume was zero. This result is consistent with presentestimates of the age of the Universe within the so-called big bang theory.

The redshift

z =λ′

λ− 1

is also used as a metric of distance of objects.

4Edwin Hubble (1889 - 1953) was an American astronomer who played a crucial role in establishingthe field of extragalactic astronomy and is generally regarded as one of the most important observationalcosmologists of the 20th century.

34

Figure 2.8: The Cherenkov cone.

2.8.2 The Vavilov-Cherenkov effect

The Vavilov-Cherenkov5 (commonly just called Cherenkov) effect occurs when a wave emittermoves through a medium faster than the speed of the wave in that medium.

In the case vs > v, as we can see in the Figure 2.8, the wave front is a cone, that takesthe name of Cherenkov cone. The following relation connects the angle θ with the velocitiesv and vs:

cos θ = v/vs. (2.24)

To find the value of cos θ let us consider two positions of the source S1 and S2, and thecorresponding points P and Q on the wave front: the wave emitted in S1 has, in P , the samephase of the one emitted in S2 in Q. For the same reason also the points S ′1 and S2 have thesame phase. The time that the source spends to go from S1 to S2 is equal to the time thatthe wave spends to go from S1 to S ′1. If we call a the distance S1S2 we have S1S

′1 = a cos θ.

Thus we get:a

vs=a cos θ

v=⇒ cos θ =

v

vs.

As an example, when cosmic rays interact with the atmosphere they generate showers ofparticles. The charged particles radiate light, and some of them are faster than light in theatmosphere, thus generating cones of collimated Cherenkov light. This light is detected byspecial-purpose telescopes.

2.9 Composition of waves

Since the wave equation is linear, when we want to sum two waves we can just sum thefunctions representing them. We should remember that energy is proportional to the squareof the amplitude: this can create effects of positive and negative interference that we shalldiscuss later.

5Pavel Alekseyevich Cherenkov (1904-1990) was a Soviet physicist who shared the Nobel Prize in physicsin 1958 with Ilya Frank and Igor Tamm for the discovery of Cherenkov radiation, made in 1934. Thediscovery was made during Cherenkov’s thesis, directed by the academician Nikolai Vavilov; when the Nobelprize was assigned, however, Vavilov was dead since 15 years.

35

Figure 2.9: Examples of algebraic sum of two waves.

Figure 2.10: Waves with opposite velocity.

2.9.1 Boundary conditions and steady waves

Let ξ(x, t) be the equation of a plane wave along the x axis direction. Generally we canwrite

ξ(x, t) = ξ+(x− vt) + ξ−(x+ vt) (2.25)

where ξ+(x − vt) propagates in the positive direction of the x coordinate, while ξ−(x + vt)propagates in the negative direction (Figure 2.10).

Consider now the case in which the wave is confined in a finite region, e.g., a wave on arope between two walls. When a wave ξ+ meets the wall, it changes verse of propagation,and “creates” a second wave ξ− with the same characteristics but opposite velocity. If wesum the two waves we obtain a wave with constant speed zero, also called stationary wave.

Let us analyze mathematically what happens. Since v = ω/k, and ω remains the same,the change of velocity from +v to −v corresponds to a change of wave number from +k to−k. So the two wave equations are:

ξ+(x, t) = ξ0 sin (kx− ωt) ;

ξ−(x, t) = ξ0 sin (−kx− ωt) .

Using the appropriate trigonometric formulae, the sum of the waves is:

ξ(x, t) = ξ+(x, t) + ξ−(x, t) =

= ξ0 sin (kx− ωt) + ξ0 sin (−kx− ωt) =

= 2ξ0 sin (−ωt) cos (kx) =

= −2ξ0 sin (ωt) cos (kx) .

36

The oscillation is maximum in the points x such that cos(kx) = 1, while is minimumin the points that satisfy the condition cos(kx) = 0. These points are called respectivelyantinodes and nodes.

Observe that only waves that have nodes on the extremities of the rope will survive:otherwise the amplitude of the sum of the waves in an extreme is not null, and so also theenergy (dispersed) is not null, but this contradicts the conservation of energy, because weare supposing the reflected wave to have the same amplitude ξ0.

Imagine you are perturbing a violin string: the extremities P and Q are fixed and yougenerate different waves displacing the string in different points. The wave you have gener-ated can be expressed as the sum of sinusoidal waves, from the Fourier’s theorem. All thesewaves have a same property: the points P and Q are nodes for them. After a transient,only the waves for which the points P and Q are nodes survive, otherwise the wave, whenreflected, generates a destructive interference. So only discrete values of λ are permitted,those for which the boundaries are nodal points. These are called the harmonics:

• the fundamental frequency with frequency f1 is the wave with maximum wavelenght:it is such that λ

2= L where λ is the wavelenght and L is the string lenght. Observe

that λ = 2L;

• the second harmonic f2 is a wave with wave lenght such that 2λ2

= L, that is λ = L;

• the third harmonic f3 is such that 3λ2

= L, that is λ = 23L;

• the fourth harmonic f4 has a wave lenght such that 4λ2

= L, i.e. λ = 12L;

• and so on . . .

The wave generated perturbing the violin string is, after a transient, the sum of thesewaves, as we can see in Figure 2.12. The fundamental frequency determines the pitch ofthe note, and together with the higher harmonics determine the timbre of the sound. Wecan obtain sounds with same pitch, but different timbre, just plucking the string in differentplaces.

2.9.2 Beats

We will now analyze another interesting example of wave composition. Let us consider twosinusoidal waves that propagate in the same direction, with the same amplitude ξ0 andvelocity v, but slightly different frequencies ωi.

The waves are defined by the following equations:

ξ1(x, t) = ξ0 cos (k1x− ω1t) ;

ξ2(x, t) = ξ0 cos (k2x− ω2t) .

What happens if we sum them? Let us put:

∆k = k1 − k2; 〈k〉 =k1 + k2

2;

∆ω = ω1 − ω2; 〈ω〉 =ω1 + ω2

2.

37

Figure 2.11: The first four harmonics.

38

Figure 2.12: A string is plucked in a certain point. This creates a wave that is sum of threewaves: the foundamental frequency, the second and the third harmonics. Their sum

produce a determined sound.

By trigonometric identities:

cos(α) + cos(β) = 2 cos

(α + β

2

)cos

(α− β

2

),

we obtain

f1(x, t) + f2(x, t) = ξ0 cos(k1x− ω1t) + ξ0 cos(k2x− ω2t) =

= 2ξ0 cos

(∆k

2x− ∆ω

2t

)cos(〈k〉x − 〈ω〉t). (2.26)

The resulting function is a product of two cosinusoidal functions. Figure 2.13 explains thebehaviour of the new wave. In this case the listener perceives an oscillating volume level;the frequency of the volume oscillation is much lower than the sound frequency.

This effect happens for example when two singers are not able to take the same tonality.

2.9.3 Group velocity and phase velocity

In this section we want to analize the behaviour of a wave packet, starting with the prerequi-site that the propagation velocity of a wave with equation ξ(x, t) = ξ0 cos(kx− ωt), is givenby v = ω/k.

A wave packet is a short envelope of waves that travels as a unit. If the wave packetpropagates in one direction (e.g. the x axis), using the Fourier’s theorem, its general formcan be written as:

ξ(x, t) =

∫Ξ(k)ei(kx−ω(k)t)dk,

39

Figure 2.13: Graphic representation of beats.

Figure 2.14: A wave packet.

where Ξ(k) is a function that takes a large value in a region of area ∆k around a certainpoint k, and goes to zero elsewhere. For example f could be a Gaussian function with verylow variance so that the funcion has a great peak near k. An example of wave packet isrepresented in Figure 2.14.

We want now to analize the speed of propagation of a wave packet. To do this we observethat beats are a simple wave packet, made by two waves. So we begin studying this case,that is simpler than the general one. We know that the speed of a wave whose equation isf(x, t) = ξ0 cos(kx− ωt) is given by:

v =ω

k. (2.27)

In the previous section we obtained that beats have an equation that is the product of twocosinusoidal functions. The speed of the envelope is given by the speed of the factor withlonger wavelenght, i.e., lower wavenumber. If we return to (2.26), we have to compare thenumbers 〈k〉 and ∆k

2, to understand which term has the lower wavenumber. It is easy to

prove that ∆k2≤ 〈k〉, so we have to consider the factor cos

(∆k2x− ∆ω

2t). Using the (2.27)

we find the velocity

venvelope =∆ω

∆k.

If we want to extend the result to general wave packets, we have to take the limit for ∆kthat goes to zero. So the velocity of the envelope, also called group velocity, is given by:

vg =d(ω(k))

dk. (2.28)

The phase velocity is the speed of propagation of a phase, for example of the point Prepresented in Figure 2.15. This velocity, in the case of beats, is equal to the propagationvelocity of the factor with shorter wavelenght and higher wavenumber, i.e., we have to look

40

Figure 2.15: In the example of beats the envelope wave (green) is given by the cosinus withlonger wavelenght.

at the term cos(〈k〉x − 〈ω〉t). Remembering that v = ω1/k1 = ω2/k2 the phase velocity ofbeats is:

v =〈ω〉〈k〉

=ω1 + ω2

k1 + k2

=vk1 + vk2

k1 + k2

=ω1

k1

=ω2

k2

= v.

Note that the phase velocity is equal to the speed of the two waves that generate beats. Ingeneral, in a dispersive medium (where v is a function of ω) the phase and group velocitiescan be different.

2.10 Waves in three dimensions

Every wave we have considered so far was a planar wave moving along the x axis, whoseequation is ξ(x, t) = ξ(x ± vt). A wave of this form can be decomposed in its harmonicswhich can be written as ξ(x, t) = ξ0 sin((kx± ωt) + φ).

To describe a wave moving in a general direction we define a new vector ~k =2π

λuv. So

we have that

ξ = ξ0 sin((~k · ~r ± ωt) + φ) = ξ0 sin((kxx+ kyy + kzz ± ωt) + φ)

knowing that |~k| = ω

vwe have

∂2ξ

∂x2+∂2ξ

∂y2+∂2ξ

∂z2= ∇2ξ =

1

v2

∂2ξ

∂t2.

This last equation also has nonplanar waves as solutions.We define wavefront a surface whose phase is constant in a given moment of time, and

ray the line orthogonal to the wavefront which represents in that point the direction of thewave and of the energy associated to it. If |v| is the same in every direction we have aspherical wavefront; if |v| is the same in every direction perpendicular to a given axis wehave a cylindrical wavefront.

41

2.10.1 Spherical waves

A small spherical object which pulsates periodically produces a sound wave, whose wavefrontsare spheres concentric to the object. These waves are an example of spherical waves.

The equation for a single harmonic component is ξ(r, t) = A(r) sin (kr − ωt). If themedium in which the wave is travelling in is motionless and has uniform density then thewaves propagate outwards with constant velocity, therefore the wavelength will not dependon the distance whereas, due to energy conservation, the amplitude will, decreasing as weget further from the source.

The energy flow per unit surface is I(r) = CA2(r), where C is a constant. If we have nodispersion, the power carried through a surface of radius r is constant, and equal to

I(r)S(r) = CA2(r) · 4πr2 = cost⇒ cost

r= A(r) =

ξ0

r

where S(r) = 4πr2 is the surface of the wavefront.Therefore the equation of a spherical wave in a nondispersive medium is

ξ(r, t) =ξ0

rsin(kr − ωt).

42

Chapter 3

Electromagnetism and the Maxwell’sequations

The Maxwell’s equations1∮S

~E · d~S =q

ε0(3.1)∮

C

~E · d~l = − d

dt

∫S(C)

~B · d~S (3.2)∮S

~B · d~S = 0 (3.3)∮C

~B · d~l = µ0I + ε0µ0d

dt

∫S(C)

~E · d~S , (3.4)

together with the equation describing the motion of a particle of electrical charge q in anelectromagnetic field

~F = q(~E + ~v × ~B) (3.5)

(Lorentz2 force), provide a complete description of electromagnetic field and of its dynamicaleffects.

We want to write Maxwell’s equations in a local form, and to transform the integro-differential equations above into purely differential equations.

3.1 Charge density and current density

We first write in terms of local variables the electric charge and the electric current.

1James Clerk Maxwell (1831 - 1879) was a Scottish physicist. His most prominent achievement wasformulating classical electromagnetic theory. Maxwell’s equations, published in 1865, demonstrate that elec-tricity, magnetism and light are all manifestations of the same phenomenon, namely the electromagnetic field.Maxwell also contributed to the Maxwell-Boltzmann distribution, which gives the statistical distribution ofvelocities in a classical perfect gas in equilibrium. Einstein kept a photograph of Maxwell on his study wall,alongside pictures of Faraday and Newton.

2Hendrik Lorentz (1853 - 1928) was a Dutch physicist who gave important contributions to electromag-netism. He also derived the equations subsequently used by Albert Einstein to describe the transformationof space and time coordinates in different inertial reference frames. He was awarded the 1902 Nobel Prizein Physics.

43

The charge density ρ(t, ~r) is defined as the charge per unit volume in a point ~r at a timet:

q =

∫V

ρ(t, ~r) dV . (3.6)

The current density~(t, ~r) is defined as the intensity of electrical current per unit surface:

I =

∫S

~(t, ~r) · d~S . (3.7)

3.2 Maxwell’s equations in differential form

Let us examine Equation (3.1). The charge q is contained in the volume volume V , thus wecan write

q =

∫V (S)

ρ dV .

By Gauss’ theorem, ∮S

~E · d~S =

∫V (S)

(~∇ · ~E) dV

and thus~∇ · ~E =

ρ

ε0. (3.8)

In the same way we get from Equation (3.3), by applying Gauss’ theorem,∮~B · d~S = 0 =⇒ ~∇ · ~B = 0 . (3.9)

Equations ∮C(S)

~E · d~l = − d

dt

∫S

~B · d~S∮C(S)

~B · d~l = µ0I + ε0µ0d

dt

∫S

~E · d~S

become respectively, by the application of Stokes’ theorem, and of Equation (3.7),∫S

(~∇× ~E) · d~S = − d

dt

∫S

~B · d~S∫S

(~∇× ~B) · d~S = µ0

∫S

~ · d~S + ε0µ0d

dt

∫S

~E · d~S

and thus

~∇× ~E = −∂~B

∂t

~∇× ~B = ε0µ0∂ ~E∂t

+ µ0~ .

44

These are called respectively the law of Faraday3-Lenz4 law and the Ampere5-Maxwell law.Thus Maxwell’s equations can be written in differential form as:

~∇ · ~E =ρ

ε0

~∇× ~E = −∂~B

∂t~∇ · ~B = 0

~∇× ~B = ε0µ0∂ ~E∂t

+ µ0~

These equations allow calculating the electric and magnetic fields from the charge distributionρ(t, ~r) and the current density ~(t, ~r).

3.3 Maxwell’s equations and continuity equation for

charge

If we take the divergence of both sides of the Ampere-Maxwell law

~∇× ~B = ε0µ0∂ ~E∂t

+ µ0~

we obtain

~∇ · (~∇× ~B) = ε0µ0~∇ · ∂

~E∂t

+ µ0~∇ ·~ .

But ~∇ · (~∇× ~B) = 0, and we can exchange the derivatives with respect to space and time:

ε0µ0∂

∂t(~∇ · ~E) + µ0

~∇ ·~ = 0 .

Gauss’ law tells us that ~∇~E = ρ/ε′; thus

~∇ ·~ +∂ρ

∂t= 0 . (3.10)

The equation above is a continuity equation for charge. If there is a net electric currentis flowing out of a region, then the charge in that region must be decreasing by the sameamount. Charge is conserved.

3Michael Faraday (1791 - 1867) was an English scientist who contributed to the fields of electromagnetismand electrochemistry. His main discoveries include those of electromagnetic induction, diamagnetism andelectrolysis. Although Faraday received little formal education he was one of the most influential scientistsin history.

4Heinrich Lenz (1804 -1865) was a Russian physicist of Baltic ethnicity. He is most noted for formulatingLenz’s law in electrodynamics in 1833.

5Andre-Marie Ampere (1775 - 1836) was a French physicist and mathematician who is generally regardedas one of the main founders of the science of electrodynamics.

45

3.4 The potentials, vector and scalar

The physics in the Maxwell equations depends only on the magnetic and electric fields. Weshall tray to express these fields in terms of some generating fields, called potentials, whichhave no immediate relation to physics.

The divergence of ~B is zero, and this means that we can always represent ~B as the curlof another vector field. Conversely, the divergence of a curl is always zero. Thus we canalways relate the magnetic field to a field we will call ~A by

~B = ~∇× ~A .

The field ~A is called the vector potential.We remind that the scalar potential φ, such that

~E = −~∇φ

in electrostatics, was not completely specified by its definition. If we have found φ, we canalways find another potential φ′ that is equivalent from the point of view of physics by addinga constant:

φ′ = φ+ C .

The new potential φ′ gives the same electric fields, since the gradient of a constant is zero;φ′ and φ represent the same physics.

Similarly, we can have different vector potentials ~A which give the same magnetic fields.Again, because ~B is obtained from ~A by differentiation, adding a constant to ~A does notchange anything physical. But we can add to ~A any field which is the gradient of some scalarfield, without changing the physics:

~B = ~∇× ~A =⇒ ~∇× ( ~A+ ~∇ψ) = ~∇× ~A+ ~∇× ~∇ψ = ~∇× ~A = ~B .

It is usually convenient to take some of the freedom out of ~A by arbitrarily placing someother condition on it (in much the same way that we found it convenient - often - to choose

to make the potential zero at large distances). We can, for instance, restrict ~A by choosing

arbitrarily what the divergence of ~A must be. We can always do that without affecting~B. This is because although ~A and ~A′ have the same curi, and give the same ~B, they donot need to have the same divergence. By a suitable choice of ψ we can make ~∇ × ~A′ anywell-behaved function we wish.

What should we choose for ~∇ · ~A? The choice should be made to get the greatestmathematical convenience and will depend on the problem we are doing. For magnetostatics,we make the simple choice

~∇ · ~A = 0 (3.11)

(later, when we take up electrodynamics, we shall make a different choice). Our completedefinition of A is then, for the moment,

~∇× ~A = ~B and ∇ · A = 0 .

46

3.5 Maxwell’s equations, electrostatics and magneto-

statics

Two of the Maxwell’s laws are the traditional Ampere-Maxwell and Faraday-Lenz laws.Can we obtain the two other laws of electrostatics and magnetostatics: the Coulomb’s

law, stating that the electric field by a point charge q is

~E =

(1

4πε0

)~urr2

(3.12)

and the Biot-Savart6 law, stating that the elementary magnetic field generated by a currentI over an element of conductor d~l at a radius ~r is

d ~B =(µ0

4π

)Id~l × ~urr2

. (3.13)

from the Maxwell equations? It turns out that this is indeed possible.We do the demonstration for the Coulomb’s law. The differential form of the Maxwell’s

law is equivalent to its integral form (3.1). Due to symmetry reasons, the field must bedirected radially. Then if we choose as a surface a sphere of radius r, equation (3.1) becomes

|~E| 4πr2 =q

ε0,

which demonstrates the statement.For the demonstration of the Biot-Savart law from Maxwell’s equations, see the Feyn-

man’s lectures on Physics, Volume 2, Chapter 14 (it is not part of the program of the exam).

6Jean-Baptiste Biot (1774 - 1862) was a French physicist, astronomer, and mathematician who establishedthe reality of meteorites, made an early balloon flight, and studied the polarization of light. Felix Savart(1791 - 1841), professor at College de France, was the co-originator of the Biot-Savart Law, along with Biot.Together, they worked on the theory of magnetism and electrical currents. Their law was developed about1820.

47

Chapter 4

Solutions of the Maxwell’s equationsin vacuo

Maxwell’s equations are four partial-derivatives first order equations; two of them are coupledin the unknown fields ~E and ~B. We shall see now that in regions of space and time whereno charges or currents are present, these can be decoupled in two equations, one for ~E andone for ~B. The resulting equations are wave equations; we shall discuss them, together withtheir implications.

Let us go back to the Maxwell’s equations in differential form:

• Coulomb’s law:~∇ · ~E =

ρ

ε0

• Faraday-Lenz’s law:

~∇× ~E = −∂~B

∂t

• the equation stating that there are no magnetic charges (monopoles):

~∇ · ~B = 0

• Ampere-Maxwell’s law:

~∇× ~B = ε0µ0∂ ~E∂t

+ µ0~ .

Now we consider a region without charges and we are going to analyze the situation. Wecan write the Maxwell’s equations in vacuo as:

~∇ · ~E = 0 (4.1)

~∇× ~E = −∂~B

∂t(4.2)

~∇ · ~B = 0 (4.3)

~∇× ~B = ε0µ0∂E

∂t(4.4)

A trivial solution would be that the electromagnetic field is everywhere zero. Let us examinethe characteristics of nontrivial solutions.

48

We suppose that there would be field generators outside the no-charges region. Conse-quently the region inherits information about the charges in the Universe. Equation 4.2 canbe written as:

~∇×(~∇× ~E

)= − ∂

∂t

(~∇× ~B

)= −ε0µ0

∂2~E∂t2

thanks to equation 4.4

On the other hand

−µ0ε0∂2~E∂t2

= ~∇×(~∇× ~E

)= ~∇

(~∇ · ~E

)︸ ︷︷ ︸

=0, eq. 4.1

−∇2~E = −∇2~E

and thus we obtain:∂2~E∂t2

=1

ε0µ0

∇2~E . (4.5)

We observe that the electric field in the empty space satisfies the d’Alembert’s equation:it is thus a wave.

Similarly to what was done for the electric field we can write

−µ0ε0∂2 ~B

∂t2= ε0µ0

∂

∂t

(~∇× ~E

)︸ ︷︷ ︸

eq.4.2

= ~∇×(~∇× ~B

)= ~∇

(~∇ · ~B

)︸ ︷︷ ︸

=0,eq. 4.3

−∇2 ~B = −∇2 ~B

and thus we obtain∂2 ~B

∂t2=

1

ε0µ0

∇2 ~B . (4.6)

This is the wave equation for the magnetic field in empty space; we note that it has the sameform as that of the electric field.

4.1 Maxwell’s waves and light

Since1

4πε0

' 9 · 109 N ·m2

C

µ0

4π= 10−7 T ·m

A

we can write the velocity of this wave:

v2 =1

ε0µ0

=1

4πε0µ04π

' 9 · 109

10−7' 9 · 1016 m2

s2.

Then:

v =

√1

ε0µ0

' 3 · 108 m/s (4.7)

49

that is consistent with the value of the speed of light. It becomes then natural to assumethat light is an electromagnetic wave.

Hertz1 presented the decisive experimental confirmation that light is a Maxwell’s wavein 1888, by generating light through an oscillating electromagnetic field. He wrote: “Theconnection between light and electricity is now established ... In every flame, in everyluminous particle, we see an electrical process ... Thus, the domain of electricity extendsover the whole of nature. It even affects ourselves intimately: we perceive that we possess... an electrical organ - the eye.” By 1900, then, three great branches of physics, electricity,magnetism, and optics, had merged into a single unified theory (and it was soon apparentthat visible light represents only a tiny window in the vast spectrum of electromagneticradiation, from radio though microwaves, infrared and ultraviolet, to X-rays and gammarays.)

The Ampere-Maxwell equation (4.4) becomes, if we do this assumption,

~∇× ~B = ε0µ0∂ ~E∂t

=1

c2

∂ ~E∂t

.

Thus the speed of light enters in the fondamental laws of nature.This fact has an important consequence: in classical mechanics the Maxwell’s equations

are not the same for different reference frames in relative motion with constant velocity.Indeed, in classical physics, all the laws depend on acceleration, which is invariant betweenthe inertial reference frames (the coordinates are transformed from one reference frame toanother by means of the Galilei2 transformations); velocities cannot enter in a fundamentallaw valid in all inertial frames, since they are not an invariant quantity.

Since the Maxwell’s equations violate the Galilean relativity, if we assume that they arecorrect, there could be two possibilities (not mutually exclusive):

• Relativity is not a legitimate assumption: there is a preferred reference frame, in whichelectromagnetism can be described (this has been called the aether). For example,this would be the case if electromagnetic fields would be the perturbation of a staticmedium.

• The relativistic transformations of Galilei do not work.

We shall see that the answer to this question brought to us one of the most fascinatingtheories of the XX century - Einstein’s3 special relativity.

1Heinrich Hertz (1857 - 1894) was a German physicist who clarified and expanded James Clerk Maxwell’selectromagnetic theory of light.

2Galileo Galilei (1564 - 1642) was an Italian physicist, mathematician, astronomer, and philosopher whoplayed a major role in the scientific revolution. His achievements include improvements to the telescope andconsequent astronomical observations and support for Copernicanism. His contributions to observationalastronomy include the discovery of the phases of Venus, of the four largest satellites of Jupiter (named theGalilean moons in his honour), and the observation and analysis of sunspots. Galilei also worked in militaryscience and technology.

3Albert Einstein (1879 - 1955) was a German-born physicist who deeply changed the representationof the Universe by the human species. While best known for his mass-energy equivalence formula E =mc2 (published in 1905), he received the 1921 Nobel Prize in Physics for his discovery of the law of thephotoelectric effect (also in 1905), which was pivotal in establishing quantum theory within physics. Nearthe beginning of his career, Einstein thought that Newtonian mechanics could not reconcile the laws dynamicswith the laws of the electromagnetic field. This led to the development of his special theory of relativity

50

4.2 Properties of the electromagnetic waves

Now we study the caracteristics of elettromagnetic waves in vacuo. We suppose to be veryfar away from the electromagnetic charges, so the equation of the wave depends on a singlevariable. Then ~E(x, y, z, t) ≡ ~E(z, t) and

∂ ~E∂x

=∂ ~E∂y

=∂ ~B

∂x=∂ ~B

∂y= 0

(the waves must be plane waves). Thus the Maxwell’s equations become:

∂Ez∂z

= 0 (4.8)

uz ×∂ ~E∂z

= −∂~B

∂t. (4.9)

Indeed

~∇× ~E = ux

(∂Ez∂y− ∂Ey

∂z

)− uy

(∂Ez∂x− ∂Ex

∂z

)+ uz

(∂Ey∂x− ∂Ex

∂y

)=

= uz ×∂ ~E∂z

= −∂~B

∂t.

Analogously for the magnetic field:

∂Bz

∂z= 0 (4.10)

uz ×∂ ~B

∂z=

1

c2

∂ ~E∂t

. (4.11)

We observe that

0 = uz ·

(uz ×

∂ ~B

∂z

)=

1

c2

∂Ez∂t

=⇒ ∂Ez∂t

= 0

0 = uz ·

(uz ×

∂ ~E∂z

)= −∂Bz

∂t=⇒ ∂Bz

∂t= 0

and thus we can conclude that Ez and Bz are constants, and in particular they do not dependon the variables x, y, z, t. If we postulate that the Universe has finite energy, we concludethat Ez = Bz = 0. Recall that the energy density associated with the presence of electricand magnetic field are respectively

u~E =ε0E2

2u ~B =

B2

2µ0

. (4.12)

(again in 1905). He realized, however, that the principle of relativity could also be extended to gravitationalfields, and with his subsequent theory of gravitation in 1916, he published a paper on the general theory ofrelativity. He was visiting the United States when Adolf Hitler came to power in 1933, and did not go backto Germany, where he was a professor in Berlin. He settled in the U.S., becoming a citizen in 1940. On theeve of World War II, he helped the set up of the Manhattan Project, and ultimately the construction of theatomic bomb. Later, however, he highlighted the danger of nuclear weapons. Einstein was affiliated withthe Institute for Advanced Study in Princeton until his death.

51

Unless the Universe has infinite energy, Ez = Bz = 0: the electromagnetic wave must betransverse.

We rewrite all the equations obtained to understand the relations between the electricfield and the magnetic field in an electromagnetic wave.

Ez = 0 (4.13)

Bz = 0 (4.14)

∂Ex∂z

= −∂By

∂t(4.15)

∂Ey∂z

=∂Bx

∂t(4.16)

∂Ex∂t

= c2∂By

∂z(4.17)

∂Ey∂t

= −c2∂Bx

∂z(4.18)

If we set υ = z − ct we have that ∂υ∂z

= 1 and ∂υ∂t

= −c. We can obtain from equation 4.15

∂By

∂t= −∂Ex

∂z= −∂Ex

∂υ

∂υ

∂z= −∂Ex

∂υ

⇒ By =

∫∂By

∂tdt = −

∫∂Ex∂υ

dt =1

c

∫∂Ex∂υ

dυ

⇒ By =Exc

+ constant

The constant must be equal to zero, not to have infinite energy; in the end we have then

By =Exc. (4.19)

Analogously from Equation (4.16)

Bx = −Eyc. (4.20)

In conclusion we have obtained that

| ~B| = |~E|c

and

~B · ~E =

(−EycEx +

ExcEy + 0

)= 0.

In other words, the electric wave is perpendicular and proportional to the magnetic wave.We also observe from (4.19) and (4.20) that

~E × ~B = EB ~uz : (4.21)

i.e., the vector product ~E × ~B gives the direction of propagation of the wave.

52

4.2.1 Energy transported an the electromagnetic wave

We found the following equations:

∂2~~E∂t2

= c2∂2~~E∂x2

c2 =1

ε0µ0

∂2 ~~B

∂t2= c2∂

2 ~~B

∂x2| ~~B| = |

~~E|c

The energy densities associated with the electric and magnetic fields are respectively

u~E =ε0E2

2u ~B =

B2

2µ0

so that the total energy density is

u =ε0E2

2+B2

2µ0

.

Considering that B = E/c and ε0µ0 = 1/c2,we have that in an electromagnetic wave themagnetic and electric components of the total energy are equal:

u ~B =B2

2µ0

=E2

2µ0c2= u~E

so we can write the total energy density as

u = 2u~E = ε0E2.

4.2.2 The Poynting vector

Let u be the energy density per volume unit, the energy carried by a wave, perpendicularlyto its direction, per time and surface unit is

I =dU

dSdt=

dU

dSdx

dx

dt= uv.

We call I the intensity and it is measured in W/m2.So far we assumed that the origin of the energy flux is coincident to the wave source, ig-

noring possible dissipative effects. The energy carried by an electromagnetic wave in vacuumper time and surface unit is

S = uc = ε0E2c = ε0EBc2 =1

µ0

EB

By the properties of ~E and ~B in an electromagnetic wave we have that

~S =1

µ0

~E × ~B

has the same direction of the wave and its magnitude is equal to the power per surface unit.This vector is called the Poynting’s vector 4.

A sine wave of the form E = E0 cos(kz − ωt), B = B0 cos(kz − ωt) has an average powerof EB/2µ0.

The electromagnetic wave also carries momentum. Based on considerations supportedby Einstein’s relativity theory (see later), the momentum carried per surface unit is related

to energy as ~S/c.4John Poynting (1852 - 1914) was an English physicist, college professor in Birmingham.

53

Figure 4.1: Sketch of the apparatus used to study the photoelectric effect.

Figure 4.2: The photoelectric effect.

4.3 Photoelectric effect; the photon hypothesis

When light interacts with material objects, it often reveals a particle-like nature, with be-having like a group of point-like objects interacting with individual atoms or molecules.

In the phenomenon called photoelectric effect, light is able to knock electrons out ofthe surface of a metal, creating an electric current. The phenomenon was first observedby Hertz in 1887. The characteristics of such an emission cannot be explained unless oneassumes (Einstein suggested this explanation in one of his famous articles published in 1905,and he was awarded the Nobel prize for this) that light interacts like a group of wave packets(we shall call them photons) interacting as particles.

Let us analyse the photoelectric effect using the apparatus in figure 4.1, in which light offrequency ν is directed onto a target and ejects electrons from it.

A potential difference V is maintained between the target and the collector to sweepup these electrons, called photoelectrons. This collection produces a photoelectric current Ithat is measured with meter A. We adjust the potential difference V by moving the slidingcontact so that the collector is slightly negative with respect to the target. This potentialdifference acts to slow down the ejected electrons. We then vary V until it reaches a certain

54

value, called the stopping potential Vstop, at which point the reading of meter A has justdropped to zero. When V = Vstop, the most energetic ejected electrons are turned back justbefore reaching the collector.

ν0

dKmaxdν

= h

ν

Kmax

Figure 4.3: The kinetic energyKmax, as a function of the

frequency.

Then Kmax, the kinetic energy of these most energeticelectrons, is

Kmax = eVstop

where e is the elementary charge. Measurements show thatfor light of a given frequency, Kmax does not depend on theintensity of the light source.

Now let us vary the frequency ν of the incident light andmeasure the associated stopping potential Vstop. Figure 4.3is a plot of Vstop versus f . Note that the photoelectric effectdoes not occur if the frequency is smaller than a certain cut-off frequency ν0 or, equivalently, if the wavelength is greaterthan the corresponding cutoff wavelength λ0 = c

ν0. This is

true no matter how intense the incident light is.This observation is in sharp contrast with a classical view of reality, which would make

us expect to see electrons ejected for every frequency, given a bright enough light. This isnot what happens. For light below the cutoff frequency ν0, the photoelectric effect does notoccur, no matter how bright the light source is.

The existence of a cutoff frequency is, however, just what we should expect if the energyis transferred via photons. The electrons within the target are held there by electric forces.To just escape from the target, an electron must pick up a certain minimum energy Φ, whereΦ is a property of the target material called its work function. If the energy hν transferred toan electron by a photon exceeds the work function of the material (if hν > Φ), the electroncan escape the target. If the energy transferred does not exceed the work function (that is,if hν < Φ), the electron cannot escape. This is what figure 4.3 shows.

Einstein summed up the results of such photoelectric experiments in the equation

hν = Kmax + Φ (photoelectric equation) . (4.22)

This is a statement of the conservation of energy for a single photon absorption by a targetwith work function Φ. Energy equal to the photon’s energy hν is transferred to a singleelectron in the material of the target. If the electron is to escape from the target, it must pickup energy at least equal to Φ. Any additional energy (hν−Φ) that the electron acquires fromthe photon appears as kinetic energy K of the electron. In the most favourable circumstance,the electron can escape through the surface without losing any of this kinetic energy in theprocess; it then appears outside the target with the maximum possible kinetic energy Kmax.

Therefore we can conclude that the a light wave of frequency ν is made of quanta ofenergy

hν =h

2πω = ~ω

where h is the Planck’s constant, which has value

h ' 6.63× 10−34 J s = 4.14× 10−15 eV s .

As we have seen, a typical human eye will respond to wavelengths from about 750 nm(the wavelength of red) to 380 nm (the wavelength of purple). This range is similar to theSun’s radiation spectrum.

55

If we take a green laser (of wavelength 0.5 µm, that is 5.00× 10−7 m), we have

ν =3× 108 m/s

5.00× 10−7 m= 0.6× 1015 Hz.

Therefore the energy associated to the green light is

E = hν = 6× 10−34 · 0.6× 1015 ' 3× 10−19 J ' 2 eV.

4.4 Natural units

The International system of units (SI) can be constructed on the basis of four fundamentalunits: of length (the meter m), of time (the second s), of mass (the kilogram kg), of charge(the coulomb C)5.

These units are inappropriate for the world of fundamental physics: the radius of anucleus is of the order of 10−15 m, also called one femtometer (fm) or one fermi; the massme of an electron is of the order of 10−30 kg; the charge of an electron is (in absolute value)of the order of 10−19 C. By using such units we would carry along a lot of exponents! Thusin particle physics we better use units like the electron charge for the electrostatic charge,and the electronvolt eV and its multiples (keV, MeV, GeV, TeV) for the energy.

Length 1 fm 10−15 mMass 1 MeV/c2 1.78× 10−30 kgCharge |e| 1.60× 10−19 C

Note the unit of mass, in which the relation E = mc2 establishing equivalence of mass andenergy (we shall discuss this relationship later) is used implicitly: what one is doing here isto use 1 eV ' 1.60× 10−19 J as the new fundamental unit of energy.

With these new units, the mass of a proton is about 0.938 GeV/c2, and the mass of theelectron is about 0.511 MeV/c2. The fundamental energy level of Hydrogen is about −13.6eV.

In addition to this, nature is providing us with two constants which are particularlyappropriate for the world of fundamental physics: the speed of light c ' 3.00 × 108 m/s= 3.00 × 1023 fm/s, and the Planck’s constant ~ ' 1.05 × 10−34 J s ' 6.58 × 10−16 eV s.It seems then natural to express speeds in terms of c, and angular moments in terms of ~.When we do this we switch to the so-called Natural Units (NU). The minimal set of naturalunits (not including electromagnetism) could then be

Speed 1 c 3.00× 108 m/sAngular momentum 1 ~ 1.05× 1034 J sEnergy 1 eV 1.60× 10−19 J

In such a system, ~ = c = 1.After these conventions, just one unit can be used to describe the mechanical Universe:

we choose energy, and thus we can express al mechanical quantities in terms of eV and of

5For reasons related only to metrology, i.e., of reproducibility and accuracy of the definition, in thestandard SI the unit of electrical current, the ampere A, is used instead of the coulomb; the two definitionsare however conceptually equivalent.

56

its multiples. It is immediate to express momenta and masses directly in terms of energy;related to lengths and energies, we can use the fact that

~c ' 1.97× 10−13MeV m = 3.15× 10−26J m

~ ' 6.58× 10−22MeV s = 1.05× 10−34J s

(the first relation can also be written as ~c ' 0.197 GeV fm. By choosing natural units, allfactors of ~ and c may be omitted from equations, which leads to considerable simplifications.For example, the relativistic energy relation

E2 = p2c2 +m2c4

becomesE2 = p2 +m2 .

To express 1 m, 1 s, 1 kg in NU, we can just write

1m = 1m~c ' 5.10× 1012MeV−1

1s = 1s~ ' 1.52× 1021MeV−1

1kg = 1kg c2 ' 5.62× 1029MeV .

Both length and time are thus, in natural units, expressed as inverse of energy. The firstrelation can also be written as 1fm ' 5.10GeV−1: note that when you when a quantityexpressed in MeV−1, in order to express it in GeV−1, you must multiply (and not divide) bya factor of 1000.

Let us now find a general rule to transform quantities expressed in natural units into SIunits, and vice-versa.

To express a quantity in NU back to SI we first restore the ~ and c factors by dimensionalarguments and then use the conversion factors ~ and c (or ~c) to evaluate the result. Thedimension of c is [c] = [m/s]; the dimension of ~ is [kgm2s−1].

The vice-versa (from SI to NU) also easy. A quantity with metre-kilogram-second (mks)dimensions MpLqT r (where M represents the mass, L the length and T the time) has theNU dimensions [Ep−q−r], where E represents energy. Since ~ and c do not appear in NU,this is the only relevant dimension, and dimensional checks and estimates are very simple.The quantity Q in the SI can be expressed in NU as

QNU = QSI

(5.62× 1029 MeV

kg

)p(5.10× 1012 MeV−1

m

)q×

(1.52× 1021 MeV−1

s

)rMeVp−q−r

The NU and SI dimensions are listed for some important quantities in Table 4.1.Finally, let us discuss how to treat electromagnetism. To do so, we must introduce a new

unit, charge for example. We can redefine the unit charge by observing that

e2

4πε0

has the dimension of [J m], and thus is a pure number in NU. By dividing by ~c one has:

e2

4πε0~c' 1

137.

57

mks NUQuantity p q r nAction (~) 1 2 -1 0Velocity (c) 0 1 -1 0Mass 1 0 0 1Length 0 1 0 -1Time 0 0 1 -1Momentum 1 1 -1 1Energy 1 2 -2 1

Table 4.1: Dimensions of different physical quantities in the SI and NU.

Imposing that the electric permeability of vacuum ε0 = 1 (thus automatically µ0 = 1 for themagnetic permeability of vacuum, since from Maxwell’s equations ε0µ0 = 1/c2) we obtainthe new definition of charge, and with such a definition:

α =e2

4π' 1

137.

This is called the Lorentz-Heaviside convention. Elementary charge in NU becomes then apure number:

e ' 0.303 .

58

Chapter 5

Geometrical optics

In cases where the wavelength is small compared to other length scales in a physical system,light waves can be modeled by light rays, moving on straight-line trajectories representingthe direction of the propagation. This is the domain of the so-called geometrical optics.

5.1 Light propagation through different materials; trans-

mission of electromagnetic waves

If light travels through a transparent material, rather than in vacuum, the electromagneticequations still hold, substituting effective constants ε and µ for ε0 and µ0, where ε = εrε0 andµ = µrµ0 (µr and εr are respectively the relative magnetic constant and relative dielectricconstant). µr ' 1 for non-ferromagnetic materials, and εr ≥ 1; for example εr ' 1+6×10−4

in air at NTP, εr ' 80 in water (which has a highly polar molecule), and εr ' 5 to εr ' 10in common glass.

So, in transparent (non-polarizable) materials, light travels with speed:

v = c′ =1√εµ

=c

√εrµr

' c√εr

=:c

n

where n ' √εr ≥ 1 is the ratio between speed of light in vacuum and speed of light throughthe material, and it is called refractive index.

5.2 Huygens’ principle

In the XVII century Huygens formulated a conjecture about the propagation of waves thatonly later, around 1882, Kirchhoff1 demonstrated for electromagnetic waves on the basis ofMaxwell’s equations, introducing some improvements.

The Huygens’ principle states that all points on a given wave front can be taken as pointsources for the production of spherical secondary waves, called wavelets, which propagate inthe forward direction with the speed characteristic of waves in that medium.

1Gustav Kirchhoff (1824 - 1887) was a German physicist who contributed to the fundamental understand-ing of electrical circuits, spectroscopy, and the emission of black-body (he coined the term “black body”)radiation by heated objects.

59

Figure 5.1: Huygens’ principle.

The corrections introduced by Kirchhoff were that the amplitude varies according toa decreasing function of the angle θ with respect to the direction of propagation (f(θ) '(1 + cos θ)/2). In many problems, however, this can be neglected.

5.3 The laws of reflection and refraction

A Huygens’ construction can be used to derive the laws of reflection and refraction of lightbetween two optical media with different indices of refraction; these are called Snelll’s laws2.

Assume a wave with fronts separated by a wavelength λ1 traveling with speed v1 in anoptically clear medium incident on the boundary with a second optically clear medium.

Theorem 1. When a wave meets an obstacle on his path, the angle of incidence θ1 is equalto the angle of reflection θ2.

To demonstrate it, we can look at the Figure 5.3: when the wave front touches the surfaceof reflection, every point behaves like a source of a new wave, that propagates form the pointfrom which the principal wave came. The result of these propagations is a wave that appearslike the original wave, just reflected by an angle equal to the angle θ1.

A second application of the rule that Huygens postulated is the demonstration of thephenomenon of refraction.

Theorem 2. If a wave passes from a medium with rifraction index n1 to one with index n2,then the direction of the light obeys the following equation

n1 sin θ1 = n2 sin θ2, (5.1)

2Willebrord Snell, or Snellius (1580 - 1626), was a Dutch astronomer and mathematician. His name hasbeen attached to the laws of reflection and of refraction of light.

60

Figure 5.2: The phenomenon of reflection.

61

Figure 5.3: The phenomenon of refraction.

where θ1 and θ2 are the angles formed by the direction of the wave with the normal of thesurface, before and after the change of medium of propagation respectively (vi = c/ni is thespeed of light in the two media).

Let us look at Figure 5.3. If we call T the time at which the wavefront begins to crossthe second medium, and T + ∆t the time at which a second ray hits the second medium,then in a time of ∆t the wave formed by the wavelets has moved inside the second mediumof a distance of v2∆t. We obtain that AC sin θ1 = v1∆t, and AC sin θ2 = v2∆t. From thatwe obtain

sin θ1

sin θ2

=v1

v2

=n2

n1

. (5.2)

Note that if n1 > n2, an angle of incidency θc exists, called the critical angle, for whichthe wave does not propagate in the second medium. This is because the equation (5.2)becomes:

θ2 = arcsin

(n1

n2

sin θ1

).

Thus refraction (transmission) is possible only if

−1 <n1

n2

sin θ1 < 1;

arcsin

(−n2

n1

)< θ1 < arcsin

(n2

n1

).

This principle is used in optical fibres: the waves in a tiny tube cannot exit from the fibresbecause of the elevate refraction index.

We also observe that in the transmission between two media, frequency of waves does notchange, consistent with Huygens’ principle: in fact the wavelets have all the same frequencyas the frequency of the principal wave.

62

Figure 5.4: The Fermat’s Principle.

5.3.1 The Fermat principle

The Fermat3 principle is equivalent to the Huygens’ principle, and states that a wave thatpropagates in more than one medium, covers the path that request the minimum time.

To better understand this principle we can look at the following example. When we arein a point S on a beach and we want to reach a point P in the sea, the fastest path for us isnot along the straight line that connects S and P : it is better for us cover a longer distanceon the sands, where our speed is faster.

We try now to demonstrate the law of refraction using this principle. As we can see inFigure 5.4 we have to move from S to P . We call C the point along our path in which themedium changes. The total time is given by

t =

√a2 + x2

v1

+

√b2 + (c− x)2

v2

;

dt

dx=

x

v1

√a2 + x2

− c− xv2

√b2 + (c− x)2

=

=sin θ1

v1

− sin θ2

v2

.

Imposing the derivative of the function to equal 0, we get

sin θ1

sin θ2

=v1

v2

=n2

n1

.

that is the law of refraction.

3Pierre de Fermat (1607 - 1665) was a French lawyer in Toulouse, and an amateur mathematician whois given credit for early developments that led to infinitesimal calculus. He made notable contributions toanalytic geometry, probability, and optics, and in particular to number theory.

63

Chapter 6

Interference and diffraction

Light waves interfere with each other much like mechanical waves do: interference associatedwith light waves arises when the electromagnetic fields that constitute the individual wavescombine.

Since light frequency is very high compared to the sensitivity of the human eye (and ofmost instruments: the typical frequency of 1015 Hz is very difficult to reach), for interferencebetween two sources of light to be observed, there are two conditions which must be met:the sources must be coherent (they must maintain a constant phase with respect to eachother), and the waves must have identical wavelengths.

6.1 Young’s interference experiment

In 1801, Thomas Young1 experimentally proved that light is a wave, contrary to what mostother scientists then thought. He did so by demonstrating that light undergoes interference,as do water waves, sound waves, and waves of all other types. In addition, he was able tomeasure the average wavelength of sunlight; his value, 570 nm, is impressively close to themodern accepted value of 555 nm. We shall here examine Young’s experiment as an exampleof the interference of light waves.

Figure (6.1) shows the basic arrangement of Young’s experiment. Light from a distantmonochromatic source of wavelength λ illuminates slit S0 in screen A. The emerging lightthen spreads to illuminate two slits S1 and S2 in screen B (this is done to guarantee coherenceof the two sources).

The snapshot of Figure (6.1) depicts the interference of the overlapping waves. Weobserve the interference on a viewing screen C intercepting the light. Where it does so,points of interference maxima form visible bright rows-called bright bands, bright fringes,or (loosely speaking) maxima - that extend across the screen. Dark regions - called darkbands, dark fringes, or (loosely speaking) minima - result from fully destructive interferenceand are visible between adjacent pairs of bright fringes (maxima and minima more properlyrefer to the center of a band.) The pattern of bright and dark fringes on the screen is calledan interference pattern. Figure (6.2) is a photograph of part of the interference pattern thatwould be seen by an observer standing to the left of screen C at a distance L� d, where dis the distance between S1 and S2.

1Thomas Young (1773 - 1829) was an English medical doctor and polymath. He made notable scientificcontributions to the fields of vision, light, solid mechanics, energy, physiology, language, musical harmony,and in the decipherment of Egyptian hieroglyphs.

64

Figure 6.1: Arrangement of Young’s interference experiment.

Figure 6.2: Interference pattern.

65

Figure 6.3: Difference of path length.

Where are the dark fringes and the bright fringes located? Let us examine the differencein optical path between the two rays in Figure (6.3).

When one wave travels an integer number of wavelengths farther than the other, thewaves arrive in phase, and a bright fringe occurs. This condition is verified for

δ = r2 − r1 ' d sin θ = mλ

(this assumes the paths are parallel; they are not exactly parallel, but the above is a verygood approximation since L � d). The absolute value of the integer m is called the orderof the maximum.

When destructive interference occurs, a dark fringe is observed. This needs a path dif-ference of an odd half wavelength

δ = r2 − r1 ' d sin θ =

(m+

1

2

)λ .

Let us take a coordinate z on the screen, starting from z = 0 at θ = 0. Then, z = L tan θ,and, for small θ, z ' L sin θ. One has thus

zbright ' λL

dm (6.1)

zdark ' λL

d

(m+

1

2

)(6.2)

(with m = 0,±1,±2, ...). Young’s setup is thus effective to measure the wavelength of light,which is amplified by a factor L/d.

What happens in between the maxima and the minima? Let us solve the general problemof the interference of N equally spaced sources in phase, each of amplitude A (Figure 6.4).Let A(θ) be the common amplitude of each wave (which can be a function of θ due to theattenuation).

66

Figure 6.4: N-slits interference.

The total field Etot as a function of θ is

Etot =N∑n=1

En =N∑n=1

A(θ)ei(krn−ωt) = A(θ)ei(kr1−ωt)N∑n=1

eik(n−1)d sin θ . (6.3)

With ζ = eikd sin θ, the sum is a sum of the geometrical series 1 + ζ + ζ2 + ... + ζN−1 =(ζN − 1)/(ζ − 1). Thus

Etot = A(θ)ei(kr1−ωt)eikNd sin θ − 1

eikd sin θ − 1

= A(θ)ei(kr1−ωt)eik(N/2)d sin θ

eik(1/2)d sin θ

eik(N/2)d sin θ − e−ik(N/2)d sin θ

eik(1/2)d sin θ − e−ik(1/2)d sin θ

= A(θ)ei(kr1−ωt)eik((N−1)/2)d sin θ sin (N/2)kd sin θ

sin (1/2)kd sin θ.

The amplitude is thus

Atot(θ) = A(0)sin (N/2)kd sin θ

sin (1/2)kd sin θ≡ A(0)

sin (Nα/2)

sin (α/2)

with

α ≡ kd sin θ =2πd sin θ

λ.

67

Figure 6.5: 4-slits interference.

The amplitude at θ = 0 is

Atot(0) = limθ→0

Atot(θ) = NA(0) .

and thusAtot(θ)

Atot(0)=A(θ)

A(0)

sin (Nα/2)

N sin (α/2).

When we go to intensities,

Itot(θ)

Itot(0)=

(A(θ)

A(0)

sin (Nα/2)

N sin (α/2)

)2

.

Even for large angles, the effect of A(θ) is to simply act as an envelope function of theoscillating sine functions. We can always bring A(θ) back in if we want to, but the moreinteresting behavior of Atot(θ)) is the oscillatory part. We are generally concerned with thelocations of the maxima and minima of the oscillations and not with the actual value of theamplitude. We can pose, for the moment, Atot(θ) ' Atot(0).

What does the Itot(θ)/Itot(0) ratio look like as a function of θ? The plot for N = 4 isshown in Figure (6.5). If we are actually talking about small angles, then we have α =kd sin θ ' kdz/D.

6.1.1 The two-slit case

In particular, for the two-slit case,

Itot(θ)

Itot(0)='

(sin (2α/2)

N sin (α/2)

)2

= cos2(α/2) . (6.4)

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Figure 6.6: Diffraction through a single slit.

6.2 Diffraction

Any wave passing through an opening experiences diffraction. Diffraction means that thewave spreads out on the other side of the opening rather than the opening casting a sharpshadow.

Diffraction is most noticeable when the opening is about the same size as the wavelengthof the wave. If light passes through a narrow slit, it produces a characteristic pattern oflight and dark areas called a diffraction pattern, which can be explained by the interferenceof light traveling different optical paths.

Light passing a sharp edge also exhibits a diffraction pattern. Huygens’ Principle de-scribes this spreading out, and a Huygens’ construction can be used to quantify the diffrac-tion phenomenon. For example, Figure (6.6) shows coherent light incident on an opening,which has dimensions comparable to the wavelength of the light. Rather than casting a sharpshadow, light spreads out on the other side of the opening. We can describe this spreadingout by using a Huygens’ construction and assuming that spherical wavelets are emitted atseveral points inside the opening. The resulting light waves on the right side of the openingundergo interference and produce a characteristic diffraction pattern.

Light waves can also go around the edges of barriers. In this case, the light far from theedge of the barrier continues to travel like the light waves shown in Figure (6.6). The lightnear the edge of the barrier seems to bend around the barrier and is described by the sourcesof wavelets near the edge.

6.2.1 Diffraction through a single slit

Huygens’ principle requires that the waves spread out after they pass through slits. Thisspreading out of light from its initial line of travel is called diffraction; in general, diffractionoccurs when waves pass through small openings, around obstacles or by sharp edges.

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Figure 6.7: Diffraction through a single slit.

Fraunhofer diffraction occurs when the rays leave the diffracting object in parallel direc-tions (screen very far from the slit).

We shall now discuss what happens when a plane wave impinges on just one wide slitwith width a instead of a number of infinitesimally thin ones (see Figure 6.7).

By Huygens’ principle, we can consider the wide slit to consist of an infinite number ofline sources (or point sources, if we ignore the direction perpendicular to the page) next toeach other, each creating a cylindrical wave. In other words, the diffraction pattern fromone continuous wide slit is equivalent to the large-N limit of the N−slit result.

We shall perform this calculation by an integral over all the phases from the possiblepaths from different parts of the wide slit. Let the slit run from y = −a/2 to y = a/2,and let B(θ)dy be the amplitude that would be present at a location θ on the screen if onlyan infinitesimal slit of width dy was open. So B(θ) is the amplitude (on the screen) perunit length (in the slit): B(θ)dy is the analog of the A(θ) in the case of interference. If wemeasure the pathlengths relative to the midpoint of the slit, then the path that starts atposition y is shorter by y sin θ. It therefore has a relative phase of e−ky sin θ.

Integrating over all the paths that emerge from the different values of y (through slits ofwidth dy) gives the total wave at position θ on the screen as (up to an overall phase fromthe y = 0 point, and ignoring the temporal part of the phase)

Etot(θ) =

∫ +a/2

−a/2dy B(θ) e−iky sin θ .

This is the continuous version of the discrete sum in the case of the multi-slit interference.B(θ) falls off like 1 =

√r; however, we shall assume that θ is small, which means that we

can assume cosθ ' 1.

Etot(θ) ' B(0)

∫ +a/2

−a/2dy e−iky sin θ =

B(0)

−ik sin θ

(e−ik(a/2) sin θ − eik(a/2) sin θ

)= B(0)

−2i sin(ka sin θ

2

)−ik sin θ

= B(0)asin(

12ka sin θ

)12ka sin θ

.

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Figure 6.8: The function sin2(β/2)/(β/2)2.

There is no phase here, so this itself is the amplitude Atot(θ). Taking the usual limitat θ = 0, we obtain Atot(0) = B(0)a. Therefore, Atot(θ)/Atot(0) = sin(β/2)/(β/2), whereβ = ka sin θ. Since the intensity is proportional to the square of the amplitude, we arrive atthe equation

Itot(θ)

Itot(0)=

(sin(β/2)

β/2

)2

with β = ka sin θ = 2πa sin θ/λ.From Figure (6.8), we see that most of the intensity of the diffraction pattern is contained

within the main bump where β < 2π. Numerically, you can shown that the fraction of thetotal area that lies under the main bump is about 90%. So it makes sense to say that theangular width of the pattern is given by

sin θ ' λ

a.

6.2.2 Diffraction grating

The diffracting grating consists of many equally spaced parallel slits. A typical gratingcontains several thousand lines per centimeter.

The intensity of the pattern on the screen is the result of the combined effects of inter-ference and diffraction.

6.2.3 The diffraction limit

Making a perfect lens that produces flawless images has been a dream of lens makers forcenturies. This is however impossible: whenever an object is imaged by an optical system,such as the lens of a camera, or our eyes, fine features are permanently lost in the image.

Roughly, the diffraction limit is given by the fact that a lens of aperture a has an in-trinsic diffraction of a light of wavelength λ such that the minimum uncertainty δθ on the

71

measurement of angles is, as seen in the previous Section,

δθ ' kλ

a

where k depends on the geometry of the lens and on the definition of uncertainty, and it isof the order of unity.

For example, the human eye has an opening of a diameter of about 1mm. Thus theminimum angle we can appreciate (let us assume a wavelength of 500 nm) is about 500nm/1mm = 0.5 mrad. The Moon is about d ' 4 × 108 km away from us: then, when welook to the Moon, one pixel of our view has – at best: the atmosphere introduces additionaldistortions – a typical size of 200 km. A perfect 10-m telescope could register pixels of 20m: size matters in astronomy and optics in general. In any case, we could not take from theEarth a picture of the astronauts on the Moon.

72

Chapter 7

Maxwell’s equations and Einstein’sspecial relativity

The principle of relativity is the requirement that the equations describing the laws of physicshave the same form in all admissible frames of reference.

Classical mechanics postulates with the law of inertia the existence of at least one admis-sible system; the second and the third law of mechanics have the same form in all referenceframes moving with uniform speed with respect to the first one. This defines the class of theinertial systems, which are the admissible frames for the theory.

The key is the fact that the only dynamical variable appearing in the laws of classicalmechanics is acceleration - and acceleration is invariant in the class of inertial systems, suchas mass, and thus force.

7.1 Classical electromagnetism is not a consistent the-

ory

The situation is different when we include electromagnetism. We demonstrated that a speed- the speed of light - enters in Maxwell’s laws, and speed is not invariant within the classicalprinciple of relativity.

One can thus easily construct paradoxes in which it appears that electromagnetism doesnot respect the principle of classical special relativity.

For example, let us consider two electrons at rest, and let r be the distance betweenthem. The (repulsive) force between the two electrons is the electrostatic force

F =1

4πε0

e2

r2,

where e is the charge of the electron; it is directed along the line joining the two charges.But an obsever is moving with a velocity v perpendicular to the line joining the two

charges will measure a force (still directed as F )

F ′ =1

4πε0

e2

r2− µ0

2πrv2e2 6= F .

The expression of the force is thus different in the two frames of reference. But masses,charges and accelerations are invariant: thus the second Newton’s law cannot hold in bothsystems.

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7.2 Galilean transformations, relativity, and the ether

Galilei first postulated that the laws of physics should be the same for all the observersmoving at constant relative velocity to one another.

For example (an example more modern than Galilei’s examples), let us consider a skaterA tossing a tennis ball up and down, while he is riding at a constant velocity V , relative tosome other skater B (acceleration is = 0).

In the coordinate frame S in which the skater A is at rest, the ball is going up and downalong the y axis, and with zero component along the x axis.So the Newtonian equations of motion are:

i) md2xdt2

= 0 ii) md2ydt2

= −mg

where m is the mass of the ball.The Galileian transformations of coordinates between frames S(the frame of skater B),

and S, are:

t = t (7.1)

x = x− V t (7.2)

y = y (7.3)

z = z (7.4)

and thusv = v − V .

Then in frame S the equations of motion are, again:

i) md2ydt2

= −mg ii) md2xdt2

= 0

This is an example of the principle of Galileian relativity: the equations of classicalmechanics are the same in different reference systems, when Galilei’s transformations areused to pass from one system of coordinates to the other. One cannot determine whichframe is picked out as special by the laws of physics, because Newton’s laws look the samein both frames, and that’s because there is no acceleration between the two frames: they areboth inertial frames.

An accelerated frame, instead, could be distinguished, since an extra acceleration wouldappear in the equations of motion.

74

7.2.1 Maxwell’s equations and the ether

Contrary to Newton’s laws, Maxwell’s equations do not depend on acceleration only, butthey also include a speed: the speed of light c. Hence they are not invariant when Galilei’stransformations of coordinates are applied.

At the end of the XIX century, scientists preferred give up relativity of the electromagneticfield since the success of the Maxwell’s equations was huge.

They explained the non-invariance of Maxwell’s equations by the existence of a preferredreference frame, the medium in which light propagates: this was called the luminiferousether.

Since Maxwell’s equations hold with good accuracy on the Earth, Earth should be inslow motion with respect to the luminiferous ether. However, the Earth orbits around theSun at a speed of around 30 km/s: there should be a difference of the order of 60 km/sbetween the speed of light measured in two opposite seasons.

Michelson and Morley performed in 1887 a notable experiment to measure the speed ofEarth with respect to the ether.

7.3 The Michelson-Morley experiment

The goal of Michelson-Morley experiment, is to find the ether by measuring the speed oflight in different directions, as the Earth travels around the Sun.

Let us illustrate the experiment by an example.

Let us consider the problem of two swimmers in a river of width L, where the current travelsat speed V , relative to the river bank. Each swimmer swims with a speed c, relative tothe rest frame of the water; Swimmer A swims to a place of the bank directly opposite thestarting point and back, and swimmer B swims a distance L down the bank, and then backto the starting point. We call S, the frame in which the water is at rest, and S the framein which the river bank is at rest. The coordinates in the two frames, applying Galileiantransformations, are:

t = tx = x+ V ty = y .

In the water (frame S), both swimmers travel with a speed c, but in the frame S (theriver bank), their speeds are not the same:

75

vx = vx − Vvy = vy

In the S coordinates, B swims a distance L up the river at speed vx = c − V , and then heswims back down the river a distance L, at speed vx = c+ V . The total elapsed time is:

tB =L

c+ V+

L

c− V=

2Lc

c2 − V 2

In the S coordinate system, swimmer A travels across the river in the y direction, witha velocity ±vy, but in the S frame, A travels a diagonal path across the water, with total

speed c =√vx

2 + vy2 , since vx = 0, vx = V , so vy = ±

√c2 − V 2.

So the total time elapsed by swimmer A, going across the river and back, is

tA =2L√c2 − V 2

.

The difference between the times needed by the swimmers A and B is:

∆t = tB − tA =2L

c

1

1− V 2

c2

− 1√1− V 2

c2

.

If V << c, we get that

∆t ' L

c

(V

c

)2

.

The Michelson-Morley experiment was set up to detect the presence of ether using twolight beams instead of two swimmers, and a set of mirror and a detector, instead of a riverbank. The instrument is called a Michelson interferometer.

The Michelson interferometer used in the experiment has a half-silvered mirror that splits abeam of monochromatic light from a source into two beams. One beam travels along arm Ato a mirror and back, and the other travels along arm B to a mirror and back, then the two

76

beams meet and they recombine to produce an interference pattern. Interference is sensitiveto differences of times smaller than the wavelength of light divided by the speed c.

If there is an ether, there must be some measurable relative velocity between the Earthand the ether, and that velocity should be comparable to the Earth’s orbital velocity aroundthe Sun: the minimum expected value of VE/c should be of about 10−4. Michelson andMorley wanted to measure that relative velocity; if the velocity of light changed in differentdirections, during the experiment the interference pattern should have shifted by a fractionof a fringe, but the pattern did not appear to shift at all, so they did not find the presence ofether: the velocity of light c was, within the experimental errors, the same in every direction.

7.4 Einsteins’s postulates and relativity

Since no dependence of the speed of light was detected, Einstein postulated they the speedof light should be the same in all inertial reference frames.

In Einstein’s 1905 paper On The Electrodynamics of Moving Bodies, Einstein postulatedthat:

0. Space is homogeneous and hisotropical; time is homogeneous.

1. All physical laws valid in one frame of reference are equally valid in any other frameof reference, moving uniformly relative to the first.

2. The speed of light (in vacuum) is the same in all inertial frames of reference, regardlessof motion of the light source.

The last postulate is not needed, if we include among the physicals laws the Maxwell’sequations in their expression including the speed of light .

These postulates are very important, because, if the speed of light is independent of themotion of the source, time and distance must be relative and not absolute: we must replacethe Galilei laws of transformations of coordinates with new laws.

Before computing the new laws for the transformations, let us examine some consequencesof Einstein’s postulates.

7.4.1 Relativity of simultaneity

Two events that happen at the same time in a frame of reference, can happen at differenttimes for another observer, moving at some uniform velocity relative to the first frame.

Let us make an example (a modern version of an example originally formulated by Ein-stein). A train is moving at velocity v past a platform. We will call the frame of reference ofthe observers in the train S, and the frame of reference of the observers on the platform S.

A person is standing in the middle of the train with a laser that fires two pulses at thesame time in opposite direction.

77

In the rest frame of the train, the distance pulse 1 travels to hit the rear end of the train,is equal to the distance pulse 2 travels to hit the front end of the train.

If both light pulses travels at speed c, the pulse 1 hits the rear end of the train at thesame time that pulse 2 hits the front of the train: so the two events 1 and 2 are simultaneousin the rest frame of the train.

Let us call L the length of the train, as measured by to observers on the platform. InGalileian mechanics, for pulse 1 we have: (c − v)t1 = L

2− vt1, and for pulse 2: (c + v)t2 =

L2

+vt2. And then we get that: t1 = L2c

, and t2 = L2c

. So, according to the Galileian relativity,the events are simultaneous in both frames.

But this is not happening if the speed of light is c for both the observers riding on thetrain, and those standing on the platform. For pulse 1 we get that:

ct1 =L

2− vt1 → t1 =

L

2(c+ v)

and for pulse 2 we get:

ct2 =L

2+ vt2 → t2 =

L

2(c− v).

78

The time difference between the two events according to the platform observers is:

∆t =vL

c2(1− v2

c2

) =γ2β

cL ,

where

β =v

c; γ =

1√1− β2

.

Thus, if the speed of light is the same in both frames, these two events that happen atthe same time according to observers in the train, do not happen at the same time accordingto observers on the platform; simultaneity is relative.

7.4.2 Time dilation

Our observer in the train builds a smart clock. A light clock measures time by sending abeam of light from the bottom to the top of the car where it is then reflected back to thebottom. Of course this clock is as good as any mechanical clock (otherwise one could tellthe absolute motion of a system by the difference between the times recorded by the twokinds of clocks).

79

A light pulse travels (in the y direction) a distance ∆y to the mirror on the top, andanother ∆y back to the starting point. The light travels at speed c, so the total travel timeis measured by S is

∆t =2∆y

c.

According to a observer on the platform, the train is moving at speed v, in the +xdirection:

80

The light travels a diagonal path in the x and y direction, for a distance

d = 2

√∆y2 + (

∆x

2)2

and since ∆x = v∆t, the distance is

d = 2

√∆y2 +

(v∆t

2

)2

.

We know that d = c∆t, so ∆t = 2∆y

c√

1− v2c2

.

Since ∆y = ∆y, we get

∆t =∆t√1− v2

c2

(∆t = γ∆t)

and since, for v 6= 0, γ > 1, one has that ∆t > ∆t.Thus the total time ∆t elapsed according to observers in frame S is greater than the total

time ∆t elapsed according to observers in frame S: this phenomenon is called relativistictime dilation.

This introduces the concept of proper time. Proper time is the elapsed time between twoevents as measured in the same place. Proper time intervals are the shortest possible. Thelight clock, as seen by S, measures a time interval in the same place, while for S it does not.

7.4.3 Length contraction

The observers on the car, whose frame of reference we have chosen to call S, measure thelength of their car to be L. They verify this by watching the front end of the car and then

81

the rear end of the car pass the center of the platform, and measuring the amount of time∆t′ that passes between those two events. According to the observers on the car, the car isat rest and the subway platform is moving at a velocity −V relative to the car.

To an observer standing at the center of the platform when the subway car goes by, thecar has a length L that can be calculated by measuring the time that passes between thetime when the front end of the car reaches the center of the platform and the time when therear end of the car reaches the center of the platform: L = V∆t.

The interval ∆t is a proper time interval for the observer standing on the platform,because the two events whose times are being measured both happen at the center of theplatform. Therefore the time intervals are related by

∆t′ = γ∆t =⇒ L = L′

γ.

The length at rest is the longest possible, and gets in general contracted for an observerobserving an object in motion.

Time dilation (the proper time is the shortest) and length contraction are closely related:see Figure 7.1.

7.5 Invariance of the interval

From the principle of relativity it follows that the velocity of propagation of electromagneticinteractions is the same in all inertial systems of reference. We assume that the velocity ofpropagation of interactions is a universal constant, and equal to c.

Three relevant consequences are:

• Relativity of simultaneity;

• Time dilation;

• Length contraction.

(time and space intervals are not absolute). If time and space intervals are no more invariant,what is invariant as a consequence of the Einstein’s postulates?

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Figure 7.1: The decay of the unstable particle called muon, hitting the Earth from theupper atmosphere: the point of view of the muon, and of a terrestrial observer.

Let us look at two inertial reference systems K and K ′ with coordinate axes XY Z andX ′Y ′Z ′, respectively, where the system K ′ moves relative to K along the X(X ′) axis.

Suppose signals start out from some point A on the X ′ axis in two opposite directions.Since the velocity of propagation of a signal in the K ′ system, as in all inertial systems, isequal (for both directions) to c, signals will reach points B and C, equidistant from A, atthe same time (in the K ′ system).

The same two events (arrival of the signal at B and C) can by no means be simultaneousfor an observer in the K system. In fact, the velocity of a signal relative to the K systemhas, according to the principle of relativity, the same value c, and since the point B moves(relative to the K system) toward the source of its signal, while the point C moves in thedirection away from the signal (sent from A to C), in the K system the signal will reachpoint B earlier than point C.

83

We shall frequently use the concept of event. An event is described by the place whereit occurred and the time when it occurred. Thus an event occurring in a certain materialparticle is defined by the three coordinates of that particle, and by the time the event occurs.We designate the time in the systems K and K ′ by t and t′ respectively.

Let a first event consist of sending out a signal, propagating with light velocity, from apoint having coordinates x1, y1, z1 in the K system, at time t1 in this system. We observe thepropagation of this signal in the K system. Let the second event consist of the arrival of thesignal at point x2, y2, z2 at the moment of time t2. The signal propagates with velocity c; thedistance covered by it is therefore c(t1 − t2). On the other hand, this same distance equals√

(x2 − x1)2 + (y2 − y1)2 + (z2 − z1)2. Thus we can write the following relation between thecoordinates of the two events in the K system:

(x2 − x1)2 + (y2 − y1)2 + (z2 − z1)2 − c2(t2 − t1)2 = 0 .

The same two events, i.e., the propagation of the signal, can be observed from the K’system. Since the velocity of light is the same in the K and K’ systems, we have, similarly, t

(x′2 − x′1)2 + (y′2 − y′1)2 + (z′2 − z′1)2 − c2(t′2 − t′1)2 = 0 .

If x1, y1, z1, t1 and x2, y2, z2, t2 are the coordinates of any two events, then the quantity

s12 =√c2(t2 − t1)2 − (x2 − x1)2 − (y2 − y1)2 − (z2 − z1)2

is called the interval between these two events.It follows from the principle of invariance of the velocity of light that if the interval

between two events is zero in one coordinate system, then it is equal to zero in all othersystems. If two events are infinitely close to each other, then the interval ds between themis

ds2 = c2dt2 − dx2 − dy2 − dz2 .

As already shown, if ds = 0 in one inertial system, then ds′ = 0 in any other system. Onthe other hand, ds and ds′ are infinitesimals of the same order. From these two conditionsit follows that ds2 and ds′2 must be proportional to each other:

ds2 = a ds′2

where the coefficient a can depend only on the absolute value of the relative velocity of thetwo inertial systems. It cannot depend on the coordinates or the time, since then differentpoints in space and different moments in time would not be equivalent, which would be incontradiction to the homogeneity of space and time. Similarly, it cannot depend on thedirection of the relative velocity, since that would contradict the isotropy of space. Let usconsider three reference systems K, K1, K2, and let V1 and V2 be the velocities of systemsK1 and K2 relative to K. We then have:

ds2 = a(V1)ds′21 ; ds2 = a(V2)ds22

andds2

1 = a(V12)ds′22

where V12 is the absolute value of the velocity of K2 relative to K1. Comparing these relationswith one another, we find that we must have

a(V2)/a(V1) = a(V12) .

84

But V12 depends not only on the absolute values of the vectors V1 and V2, but also on theangle between them. However, this angle does not appear on the left side of formula above.It is therefore clear that this formula can be correct only if the function a(V ) reduces to aconstant, which is equal to unity according to this same formula.

ds2 = ds′2

and from the equality of the infinitesimal intervals there follows the equality of finite intervals:s = s′. Thus we arrive at a very important result: the interval between two events is thesame in all inertial systems of reference, i.e., it is invariant under transformation from oneinertial system to any other. This invariance is the mathematical expression of the constancyof the velocity of light.

7.6 Spacelike and timelike intervals; future and past

Let us take some event O as our origin of time and space coordinates. Let us now considerwhat relation other events bear to the given event O. For visualization, we shall consideronly one space dimension and the time, marking them on two axes.

Uniform rectilinear motion of a particle, passing through x = 0 at t = 0, is representedby a straight line going through O and inclined to the t axis at an angle whose tangent isthe velocity of the particle. Since the maximum possible velocity is c, there is a maximumangle which this line can subtend with the t axis.

The two lines represent the propagation of signals (with the velocity of light) in oppositedirections passing through O (i.e. going through x = 0 at t = 0). All lines representing themotion of particles can lie only in the regions aOc and dOb. On the lines ab and cd, x = ±ct.

First consider events whose world points lie within the region aOc. For all the points ofthis region c2t2−x2 > 0. The interval between any event in this region and the event O is said

85

to be timelike. In this region t > 0, i.e. all the events in this region occur “after” the eventO. But two events which are separated by a timelike interval cannot occur simultaneouslyin any reference system. Consequently it is impossible to find a reference system in whichany of the events in region aOe occurred ”before” the event O, i.e., at time t < 0. Thus allthe events in region aOc are future events relative to O in all reference systems. This regioncan be called the absolute future relative to O.

In the same way, all events in the region bOd are in the absolute past relative to O; i.e.events in this region occur before the event O in all systems of reference.

Next consider the regions dOa and eOb. The interval between any event in this regionand the event O is smaller than 0, and it is said to be spacelike. These events occur atdifferent points in space in every reference system. Therefore these regions can be said to beabsolutely remote relative to O.

Note that if we consider all three space coordinates instead of just one, then instead ofthe two intersecting lines in the Figure we would have a “hypercone” x2 + y2 + z2− c2t2 = 0in the 4-dimensional coordinate system x, y, z, t, the axis of the cone coinciding with the taxis. (This is called the light cone.) The regions of absolute future and absolute past arethen represented by the two interior portions of this cone.

Two events can be related causally to each other only if the interval between them istimelike; this follows immediately from the fact that no interaction can propagate with avelocity greater than the velocity of light. As we have just seen, it is precisely for these eventsthat the concepts “earlier” and “later” have an absolute significance, which is a necessarycondition for the concepts of cause and effect to have meaning.

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Chapter 8

Lorentz transformations and theformalism of special relativity

8.1 The Lorentz transformations

Nel lavoro di Einstein vengono ricavate leggi di trasformazione tra due riferimenti inerziali chesostituiscono le trasformazioni di Galilei; Einstein ricava tali leggi, che in seguito prenderannoil nome di trasformazioni di Lorentz in quanto ricavate gia da Lorentz in un contesto diversouna decina di anni prima, richiedendo la validita del principio di relativita discusso sopraper le equazioni di Maxwell e che la velocita della luce sia invariante per tali trasformazioni,oltre che a richiedere omogeneita e isotropia per un riferimento inerziale.

Consideriamo ancora un sistema di riferimento inerziale K e sia K′ un sistema di rifer-imento in moto rettilineo uniforme con velocita V rispetto a K. Possiamo supporre cheall’istante t = 0 le origini dei due riferimenti coincidano, sia spazialmente sia temporal-mente. Consideriamo inoltre, per semplificare, che il moto di K′ rispetto a K avvenga lungol’asse positivo delle x (il requisito di isotropia dello spazio fa sı che quest’ipotesi non leda lageneralita).

Quali sono le trasformazioni piu generali che legano le coordinate spaziali e temporalimisurate da K e sia K′?

Se vogliamo che un moto rettilineo uniforme resti tale in ogni sistema di riferimento(covarianza della prima legge di Newton), la trasformazione dovra essere lineare. D’altraparte, per l’omogeneita e l’isotropia dello spazio dovra essere

y′ = y ; z′ = z .

Quindi la trasformazione piu generale e della coordinata x e del tipo

x′ = γx+ ρt .

Va detto innanzitutto che il fatto che il tempo scorra nello stesso modo nei due sistemidi riferimento e solo un’ipotesi ad hoc formulata da Newton: in generale anche la misura deltempo potra essere diversa. Avremo quindi in generale una legge di trasformazione anchetra i tempi:

t′ = σx+ τt

o, per affrontare lo studio di coordinate omogenee,

ct′ = σx+ τct .

87

Sappiamo che, per l’invarianza dell’intervallo,

x2 − (ct)2 = x′2 − (ct′)2 . (8.1)

E conveniente parametrizzare la trasformazione mediante le trasformazioni trigonometricheiperboliche, che garantiscono a vista l’invarianza della differenza fra i quadrati: cosh2 ψ −sinh2 ψ = 1. Si ha

x = x′ coshψ + ct′ sinhψ

ct = x′ sinhψ + ct′ coshψ

Applicando le equazioni precedenti all’origine degli assi coordinati in K′ si ha

x

ct=V

c= tanhψ

da cui

sinhψ =V/c√

1− V 2/c2; coshψ =

1√1− V 2/c2

.

Per riassumere, le coordinate di un evento date nei due riferimenti inerziali K e K′considerati fino ad ora sono legate dalla seguente legge di trasformazione:

x′ =x− V

c(ct)√

1− V 2

c2

y′ = y

z′ = z

ct′ =(ct)− V

cx√

1− V 2

c2

8.1.1 A theorem by von Ignatowski

Una quindicina di anni dopo il lavoro di Einstein, von Ignatowsky dimostra elegantementepartendo dalle ipotesi che lo spazio-tempo sia omogeneo e isotropo, che valga il principio direlativita e la e che le trasformazioni fra i sistemi di riferimento formino un gruppo, che letrasformazioni sopra ricavate, con una velocita limite invariante. C. Rimane il problemadella determinazione di C; da un punto di vista matematico non esiste alcun vincolo, eneppure la meccanica ne fornisce uno. Nel limite C →∞ si riottengono le trasformazioni diGalilei.

8.1.2 Transformation of velocities

Let us now put

γ =1√

1− V 2/c2; β =

V

c

so that

ct′ = γ((ct)− βx)

x′ = γ(x− β(ct))

y′ = y

z′ = z

88

and

ct = γ((ct′) + βx′)

x = γ(x′ + β(ct′))

y = y′

z = z′

What is the transformation of velocities between the reference frames? One has

dt = γ

(dt′ +

V

c2dx′)

dx = γ(dx′ + V dt′)

dy = dy′

dz = dz′

and thus

vx =v′x + V

1 + v′xVc2

vy =1

γ

v′y

1 + v′xVc2

vz =1

γ

v′z

1 + v′xVc2

.

They correspond to Galilei’s transformations in the limit c→∞.The speed of light is invariant, and the sum of two whatever velocities smaller than c is

again < c. This is easy to verify for a particle moving along x:

v =v′ + V

1 + v′Vc2

.

8.1.3 Time dilation and length contraction, again

We can reexamine time dilation and length contraction as a consequence of the Lorentztransformations.

Time dilation

We consider measuring the time duration of a process in an object which is stationary in amoving frame S’, but we wish to transform the starting and ending times of the process toa stationary frame. Since the object is stationary in the moving frame, x′1 = x′2.

Applying the Lorentz transformations to the start and finish times of our process, wehave:

t1 = γ(t′1 + V x′1/c2) ; t2 = γ(t′2 + V x′2/c

2) .

Subtracting the first equation from the second to get the time duration of the process, wehave:

∆t = t2 − t1 = γ(t′2 − t′1) = γ∆t′ .

89

Length contraction

We consider measuring the length of our object as taking measurements of the positions ofthe two ends x1 and x2 at the same time in the stationary reference frame. That is to say,t1 = t2.

An observer which sees the object in movement measures that

x′1 = γ(x1 − V t1) ; x′2 = γ(x2 − V t2) .

Subtracting the first equation from the second to get the length, we have:

∆x′ = x′2 − x′1 = γ(x2 − x1) = γ∆x

where we have used the fact that t1 = t2 makes the time difference factor zero.

8.1.4 The relativistic Doppler effect

In classical physics, the Doppler Effect does not depend only on relative velocity betweenemitter and receiver of the wave, but also on who is moving. This cannot be the case forlight, which does not propagate in any medium.

Let us now consider in relativistic physics, the Doppler Effect of electromagnetic radia-tion. In this case, the signal transmitted by one observer and received by another is a wavetraveling at speed c. It makes a significant difference compared to the situation related tothe Doppler Effect whit sound waves: there is no medium, and the velocity of the wave (thespeed of light) is the same for both the observers.

Let us start reminding the main result about the classical Doppler effect. We have:

In case of approaching observer f ′ = f

(1 +

vrv0

)In case of approaching source f ′ = f/

(1− vr

v0

)' f

(1 +

vrv0

)for

vrv0

� 1

where vr is the relative velocity between source and observer and v0 is the velocity of thewave (the case of a source and of an observer getting away from each other can be obtainedsimply by changing the sign of the speed). The equality of these two cases (moving sourceand moving observer) holds only in the limit in which vr is very small compared with thevelocity v0 of the wave.

In the relativistic case, the speed c of the wave is invariant compared with the referencesystem.

We will consider the following situation: the transmitter is in the frame O moves awaywith speed −u from the frame O′ in which the receiver is located - by relativity this has tobe equivalent to the case in which the transmitted is at rest, and the receiver moves withspeed u. At some moment the transmitter starts to broadcast a light wave; let us considerthe emission and the observation of N consecutive wavefronts.

Let us evaluate the time 4t′ that elapsed between the beginning of transmission andthe moment the wave-front reached the observer O, measured in the O′ frame. For the O′

observer, the N waves sent out from the O source are stretched over the distance u×4t′ +c×4t′ = 4t′ (u+ c).

90

Figure 8.1: The relativistic Doppler effect.

Hence, the wavelength λ′ according to the O′ observer is:

λ′ =4t′ (u+ c)

N.

Denote the time registered in the O frame between the beginning of transmission and themoment the wave-front reached O′ as 4t0 and the frequency of the signal for the O observeras f . The frequency is the number of waves sent out in a time unit. Therefore, the totalnumber of waves emitted is

N = f ×4t0.

By combining the two equations, we obtain:

λ′ =4t′ (u+ c)

f · 4t0The general relation between the wavelength and frequency of a light wave is: wavelength =(speed of light)/(frequency). Therefore, the wavelength and the frequency the O′ observerregisters are related as:

λ′ =c

f ′

(=4t′ (u+ c)

f · 4t0

)⇒ f ′ = f · 4t0

4t′· 1

1 +u

c

Now, we can use the time dilation formula: 4t′ = γ · 4t0. We obtain:

f ′ = f

√1− u2/c2

1 + u/c= f

√(1− u/c) (1 + u/c)√(1 + u/c) (1 + u/c)

= f

√(1− u/c)√(1 + u/c)

. (8.2)

If the source speed u is small compared with the speed of light, we can approximate:

f ′ = f

√1− u/c√1 + u/c

' f√

1− 2u/c ' f(

1− u

c

).

91

It is the same formula that we obtained for sound waves. However, for source or observerspeeds comparable with the speed of light one can no longer use this approximation. But arethere any such situations that we can observe? Yes, in fact due to the expansion of Universe,distant galaxies are moving away from our galaxy with such speeds that using the correctformula makes an observable difference.

8.2 4-vectors; covariant and controvariant representa-

tion

Lorentz’s transformations of coordinates guarantee automatically the invariance of

c2dt2 − dx2 − dy2 − dz2

and now we want to extend the properties to 4-ples behaving like (cdt, dx, dy, dz).Let us introduce a simple convention: in the 4-ple (cdt, dx, dy, dz) the elements will be

numbered from 0 to 3. Greek indices like µ will run from 0 to 4, and Roman symbols willrun from 1 to 3 (i = (1, 2, 3)) as in the usual 3-dimensional case.

We define 4-vector a quadruple

Aµ =(A0, A1, A2, A3

)=(A0, ~A

)which transforms like (cdt, dx, dy, dz) for changes of reference systems. The Aµ (with highindices) is called controvariant representation of the 4-vector.

Correspondingly, we define the 4-ple

Aµ = (A0, A1, A2, A3) =(A0,−A1,−A2,−A3

)=(A0,− ~A

)that is called covariant representation.

The coordinates of an event (ct, x, y, z) can be considered as the components of a four-dimensional radius vector in a four-dimensional space. So we shall denote its componentsby xµ, where the index µ takes the values 0, 1, 2, 3 and

x0 = ct x1 = x x2 = y x3 = z .

By our definition, the quantity ≡∑

µAµAµ ≡ AµA

µ is invariant. Omitting the sum signwhen an index is repeated once in controvariant position and once in covariant position iscalled the Einstein’s sum convention. Sometimes, when there is no ambiguity, this quantityis also indicated as A2.

In analogy to the square of a four-vector, one forms the scalar product of two differentfour-vectors:

AµBµ = A0B0 + A1B1 + A2B2 + A3B3 = A0B0 − A1B1 − A2B2 − A3B3

It is clear that it can be written either as AµBµ or AµBµ - the result is the same.

The product AµBµ is a four-scalar : it is invariant under rotations of the four-dimensionalcoordinate system.

The component A0 is called the time component and (A1, A2, A3) the space componentsof the four-vector. Under purely spatial rotations the three space components of the four-vector Ai form a three dimensional vector ~A.

92

The square of a four-vector can be positive, negative or zero; accordingly, the 4-vector iscalled timelike-, spacelike- and null-vector respectively.

We can write Aµ = gµνAν , where

gµν =

1 0 0 00 −1 0 00 0 −1 00 0 0 −1

is called metric tensor, a symmetric matrix which transforms the controvariant Aµ in thecovariant Aµ and vice-versa.

In fact we can also write Aµ = gµνAµ, where

gµν =

1 0 0 00 −1 0 00 0 −1 00 0 0 −1

is the covariant representation of the same metric tensor.

gµν is the completely controvariant metric tensor, gµν is the completely covariant metrictensor. The scalar product of two vectors can therefore be written in the form

AµAµ = gµνAµAν = gµνAµAν .

Besides we have that gµνgµρ = δρµ = 1. In this way we have enlarged the space adding a

fourth dimension A0: the time dimension.The generic transformation between reference frames can be written expressing the

Lorentz transformations by means of a 4 matrix Λ:

A′µ = ΛνµAν

where

Λνµ =

γ −γ V/c 0 0

−γ V/c γ 0 00 0 1 00 0 0 1

. (8.3)

8.2.1 Tensors

A four-dimensional tensor of the second rank is a set of 16 quantities Aµν , which transformslike products of components of two four-vectors (i.e., they enter in covariant equations). Wecould similarly define four-tensor of higher rank.

For example we could have the expression AµBµσ ≡ Cσ where Aµ transforms as a vector,Bµσ transforms as a product of vectors and Cσ is a four-vector.

A second-rank tensor can be written in three ways: covariant Aµν , controvariant Aµν andmixed Aµν . The connection between different type of components is determined from thisgeneral rule: raising or lowering a space index (1, 2, 3) changes the sign of the component,while raising or lowering the time index (0) does not.

The quantity Aµµ = tr(Aνµ)

is the trace of the tensor.

93

Our aim is now to rewrite the physical laws using these four-vectorial entities. To do thatwe introduce the completely antisymmetric tensor of rank 4. Like the tensors gµν , g

µν whichare special because their components are the same in all coordinate systems, εµνρσ has thesame property.

The definition isε0123 = 1 .

The components change sign under interchange of any pair of indices and clearly the nonzerocomponents are those for which all four indices are different. Every permutation with anodd rank changes the sign. The number of component with nonzero value is 4! = 24.

We have

εµνρσ = gαµgβνgγρgδσεαβγδ = −εµνρσ.

Thus, εαβγδεαβγδ = −24 (number of nonzero elements changed of sign).In fact with respect to rotations of the coordinate system, the quantities εαβγδ behave likethe components of a tensor, but if we change the sign of one or three of the coordinates thecomponents αβγδ, being defined as the same in all coordinate systems, do not change, whereas some of the components of a tensor should change sign.

An example of tensor: the metric tensor

The invariant interval can be written as

ds2 = gµνdxµdxν .

Under a Lorentz transformation, we get

ds2 = gµνdx′µdx′ν = gµνΛ

µρΛ

νσdx

ρdxσ .

Since the interval is invariant, we get

gµνΛµρΛ

νσdx

ρdxσ = gρσdxρdxσ

=⇒(gµνΛ

µρΛ

νσ − gρσ

)dxρdxσ = 0 .

Since the last equation must be true for any infinitesimal interval, the quantity in parenthesesmust be zero, so

gρσ = gµνΛµρΛ

νσ .

8.2.2 Covariant derivatives

As a consequence of the total differential theorem,∂s

∂xµdxµ is equal to the scalar ds. Thus

∂s

∂xµis a four-vector and since xµ is controvariant,

∂s

∂xµis covariant because ds is scalar. We

call the operator ∂µ =∂

∂xµfour-gradient.

We can write∂φ

∂xµ=

(1

c

∂φ

∂t, ~∇φ

). In general, the operators of differentiation whit

respect to the coordinates xµ ≡ (ct, x, y, z), should be regarded as the covariant componentsof the operator four-gradient. For example:

∂µ =∂

∂xµ=

(1

c

∂

∂t, ~∇φ

)∂µ =

∂

∂xµ=

(1

c

∂

∂t,−~∇

).

94

We can build from covariant quantities the operator

∂µ∂µ =

1

c2

∂2

∂t2− ~∇2 ≡ � ;

this is called the D’Alembert operator.

8.2.3 Four-dimensional velocity

The ordinary three-dimensional velocity vector ~v =d~r

dtcannot be the spatial part of a four-

vector: indeed the denominator, dt, is not invariant - while d~r could be the component of a4-vector. We can however obtain a four-vector from it, if we replace dt with ds/c, which isequal to dt in the limit v → 0. In the rest frame of the particle d~r = 0 and the time is the

proper time: |dt| = |dsc|. This four-dimensional velocity of a particle has a space component

the vector ui = cdxi

ds.

We can write

ui = cdxi

ds=

dxi

dt

√1− v2

c2

=vi√

1− v2

c2

.

This is a good three-vector and it is equivalent to the classical velocity when (v/c)→ 0.We can define as four-velocity the quadruple:

uµ =

(γc,

d~r

ds

)=

c√1− v2

c2

,~v√

1− v2

c2

.

To verify that it is a four-vector the product uµ · uµ = u2 must be invariant:

u2 =c2

1− v2

c2

− v2(1− u2

c2

) = c2

So the 4-velocity is a good four-vector.

8.3 E=mc2

La relativita speciale implica che alla massa di una particella e associato un contenuto dienergia.

Sia K un riferimento inerziale; K′ sia in moto rispetto al primo con velocita V direttalungo x.

Un corpo in quiete rispetto a K emetta a un istante t due onde luminose di energia ε/2ciascuna in versi opposti lungo x. Per simmetria il corpo continuera ad essere in quiete.

Se indichiamo con Ei l’energia del corpo prima dell’emissione e con Ef l’energia finale,

Ei = Ef + ε/2 + ε/2 = Ef + ε .

95

V"

ε/2" ε/2"

γ (1&V/c)"ε/2" γ (1+V/c)"ε/2"

In K′ prima e dopo l’emissione dei fronti d’onda luminosi il corpo si muove con velocitacostante di modulo V . Tenendo conto dell’effetto Doppler, (l’energia e’ legata alla frequenzadalla relazione E = hf), si avra

E ′i = E ′f +ε

2γ

(1− V

c

)+ε

2γ

(1 +

V

c

)= E ′f + γε .

La differenza di energia del corpo misurata nel riferimento in cui questo e in moto ed ilriferimento in cui e in quiete e pari alla sua energia cinetica K; quindi

E ′i − Ei = E ′f − Ef + ε(γ − 1) =⇒ Ki = Kf + ε(γ − 1) .

Dunque l’energia cinetica del corpo diminuisce a causa dell’emissione dei fronti d’ondaelettromagnetici; ma la velocita del corpo non cambia, di conseguenza deve essere la suamassa che e diminuita.

Per V/c→ 0 si ha

(γ − 1)→ 1

2

V 2

c2=⇒ Ki −Kf →

1

2εV 2

c2

che se confrontata con la relazione

K =1

2mV 2

implica una dimunuzione di massa del corpo pari a ε/c2. Dunque la massa di un corpodipende dalla sua quantita di energia:

∆K =1

2∆E

V 2

c2=

1

2∆mV 2 =⇒ ∆E = (∆m)c2 =⇒ E = mc2 . (8.4)

8.4 4-momentum

La conservazione della quantita di moto classica, m~v, non e compatibile con le trasformazionidi Lorentz.

96

In riferimento alla figura, nel sistema c.m., si ha, prima dell’urto (a):

px = mv +m(−v) = 0 ; py = 0 .

Dopo la collisione (b):px = 0 ; = 0 .

Mettiamoci ora in un sistema di riferimento K′ in moto con velocita v diretta lungol’asse x rispetto al primo. Il sistema vede la particella di sinistra ferma prima dell’urto. Letrasformazioni delle velocita danno, prima dell’urto:

px = m−2v

1 + v2/c2+ 0m =

−2mv

1 + v2/c2; py = 0 ,

e dopo l’urto:p′x = −2mv ; p′y = 0 .

Dunque la quantita di moto non e conservata.Vogliamo definire la quantita di moto relativistica in modo tale che:

1. costituisca un vettore nello spazio tridimensionale;

2. coincida con quella classica per v � c;

3. la sua conservazione sia compatibile con le trasformazioni di Lorentz.

97

E evidente che le prime due condizioni sono verificate dalla scelta

~p(r) = m~vγ ; (8.5)

si verifica che anche la terza lo e.Riferiamoci alla figura precedente, e poniamo per semplicita β = v/c. Trascuriamo la

componente y, per la quale la quantita di moto e, evidentemente, ancora conservata. Si ha,dopo l’urto:

p(r)x = −2mv1

(1− β2)1/2.

Prima dell’urto

p′(r)x = −2mv1

(1 + β2)

1(1−

(2β

1+β2

)2)1/2

= −2mv1

(1 + β2)

1(1− 4β2

(1+β2)2

)1/2=

= −2mv1

(1 + β2)

(1 + β2)

((1 + β2)2 − 4β2)1/2= −2mv

1

(1− β2)1/2= p(r)x .

Vogliamo ora costruire un quadrivettore formato dall’energia e dalla quantita di moto diuna particella. Nel sistema di quiete della particella questo quadrivettore dovra scriversi

pquiete = (mc2, 0, 0, 0) .

D’altra parte:

• la norma quadrata del quadrivettore, m2c4, deve essere invariante;

• la parte spaziale del quadrivettore dovra essere data da

pi = mcviγ

(il fattore c e inserito per avere le dimensioni di un’energia, come la componente diindice 0).

Definiamo quindi il quadrivettore energia-momento1

p = (√m2c4 +m2c2~v2γ2, γmc~v) = (E = γmc2, γmc~v) .

Le componenti del quadrivettore energia-momento rappresentano quantita conservate.Si noti, sviluppando in serie il termite 0 (l’energia) che

p0 = mc2 +1

2mv2 + ...

e quindi l’energia meccanica, a meno di una costante (che e ininfluente per le equazioni delmoto) coincide fino a termini del quarto ordine in v/c esclusi con l’energia meccanica classica.

1Einstein’s and de Broglie’s relations are thus part of a covariant equation

(E, c~p) = ~(ω, c~k) .

98

8.5 The photon

Dalla forma del quadrivettore energia-momento

p = (E = γmc2, γmc~v)

si ricava la~v

c=

~p

E.

Mentre in meccanica classica non si puo avere una particella di massa nulla, in meccanicarelativistica cio e possibile. La particella avra quadrimomento

pµ = (E, ~pc)

conE2 − p2c2 = 0

e dunque si muovera alla velocita della luce.Vale anche il fatto che se una particella si muove alla velocita della luce deve avere massa

nulla.Un esempio di particella a massa nulla e dunque la particella di luce (il fotone).

8.5.1 Radiation pressure

The energy carried by an electromagnetic wave in vacuum per time and surface unit is givenby the Poynting vector

~S =1

µ0

~E × ~B .

The electromagnetic wave also carries momentum E/c. Thus, the momentum carried per

surface unit per unit time is ~S/c.Since classically F = dp/dt, the quantity in the above equation is a pressure: it is called

radiation pressure. For a totally absorbing surface, the force will be S/c, while for a perfectlyreflecting surface, it will be 2S/c.

8.6 Examples of relativistic dynamics

8.6.1 Decay

Let us consider the spontaneous decay of a body of mass M into two parts with masses m1

and m2. The law of conservation of energy in the decay, applied in the system of referencein which the body is at rest, give

Mc2 = E10 + E20 .

where E10 and E20 are the energies of the emerging particles. Since for a particle of mass mE ≥ m, this requires that M ≥ (m1 + m2), i.e., a body can disintegrate spontaneously intoparts the sum of whose masses is less or equal than the mass of the body. On the other hand,if M < (m1 + m2) the body is stable (with respect to the particular decay) and does not

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decay spontaneously. To cause the decay in this case, we would have to supply to the bodyfrom outside an amount of energy at least equal to its “binding energy” (m1 +m2 −M)c2.

Momentum as well as energy must be conserved in the decay process. Since the initialmomentum of the body was zero (in its rest frame), the sum of the momenta of the emergingparticles must be zero: p10 + p20 = 0. Consequently p2

10 = p220 or

E210 −m2

1c2 = E2

20 −m22c

2 .

Solving the two equations above, one gets

E10 =M2 +m2

1 −m22

2Mc2 ; E20 =

M2 +m22 −m2

1

2Mc2 .

8.6.2 Elastic scattering

Let us consider, from the point of view of relativistic mechanics, the elastic collision ofparticles. We denote the momenta and energies of the two colliding particles (with massesm1 and m2 by pi1 and pi2 respectively; we use primes for the corresponding quantities aftercollision. The laws of conservation of momentum and energy in the collision can be writtentogether as the equation for conservation of the four-momentum:

pµ1 + pµ2 = p′µ1 + p

′µ2 .

Let us rewrite as pµ1 + pµ2 − p′µ1 = p

′µ2 and square:

pµ1 + pµ2 − p′µ1 = p

′µ2 =⇒ m2

1c4 + pµ1p2µ − p1µp

′µ1 − p2µp

′µ1 = 0 . (8.6)

Similarly,pµ1 + pµ2 − p

′µ2 = p

′µ1 =⇒ m2

2c4 + pµ1p2µ − p2µp

′µ2 − p1µp

′µ2 = 0 . (8.7)

Let us consider the collision in a reference system in which one of the particles (m2)

was at rest before the collision. Then ~p2 = 0, and pµ1p2µ = E1m2c2, p2µp

′µ1 = m2E

′1c

2,

p1µp′µ1 = E1E

′1 − p1p

′1c

2 cos θ1 where cos θ1 is the angle of scattering of the incident particlem1. Substituting these expressions in Eq. (8.6) we get:

cos θ1 =E ′1(E1 +m2c

2)− E1m2c2 −m1c

4

p1p′1c2

.

We note that ifm1 > m2, i.e. if the incident particle is heavier than the target particle, thescattering angle θ1 cannot exceed a certain maximum value. It is easy to find by elementarycomputations that this value is given by the equation

sin θ1max = m2/m1

which coincides with the familiar classical result.

Compton effect

The so-called Compton scattering is the scattering of a photon hitting an electron at rest.Il fotone ha massa m = 0; sia hf la sua energia prima dell’urto, hf ′ la sua energia dopo

l’urto.

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pfotone, prima + pelettrone, prima = pfotone, dopo + pelettrone, dopo

=⇒ (hf, hf, 0, 0) + (mec2, 0, 0, 0) = (hf ′, hf ′ cosφ, hf ′ sinφ, 0)

+ (γmec2, γme|~v| cos θ, γme|~v| sin θ, 0)

=⇒

mec

2 + hf = hf ′ + γmec2

hf = hf ′ cosφ+ γme|~v| cos θ0 = hf ′ sinφ− γme|~v| sin θ

=⇒ ∆λ =

(c

f ′− c

f

)=

h

mec(1− cosφ) .

∆λ =

(c

f ′− c

f

)=

h

mec(1− cosφ) ,

Si noti che nel caso di urti del fotone con elettroni legati la massa effettiva dell’elettronee maggiore di me: questo spiega il fatto che Compton non ottenne una riga, ma una dis-tribuzione di risultati come nella figura.

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8.7 Relativistic force

Fields are required to find the forces on charges, and forces determine the motion of thecharge. In the case of electromagnetic fields and charges, for example,

~F = q( ~E + ~v × ~B) .

The dynamical effect of forces in classical mechanics is

~F = d~p/dt .

We now discuss this equation from the point of view of relativity: can rewrite these equationsin a four-vector notation? We know that the momentum is part of a four-vector whose timecomponent is the energy, divided by c. So we might think to replace the right-hand side ofthe last equation by dpi/dt. But dt is not relativistically invariant.

The time derivative of a four-vector is no longer a four-vector, because d/dt requires thechoice of some special frame for measuring t. We got into that trouble before when we triedto make the velocity into a four-vector. However, ds/c = dt/γ (where γ = (1 − v2/c2)1/2,and v is the speed of the particle) is independent of the reference frame, and it is equal todt (the proper time) in the rest frame of a particle.

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We can thus writedpi

ds= f i (f i = γF i) .

Then we need only find a time component to go with the f i. Since p0 is the energy, theobvious choice is

f 0 = γ(~F · ~v) .

One can thus write

fµ =dpµ

ds.

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Chapter 9

Covariant formulation ofelectromagnetism

We shall now write the equations of electromagnetism in a completely covariant form. Firstlet us express the electric and magnetic fields through the vector and scalar potential.

9.0.1 The equations for the potentials

We want to express Maxwell’s equations as a function of the potentials, vector and scalar.We shall give a definition of the scalar potential valid for electrodynamics.

We begin with ~∇ · ~B = 0 - the simplest of the equations. We know that it implies that~B can be expressed as the curl of a vector field. So, if we write

~B = ~∇× ~A . (9.1)

We take next the Faraday law, ~∇ × ~E = −∂ ~B/∂t. If we express ~B as a function of thevector potential, and differentiate with respect to it, we can write Faraday’s law in the form~∇× ~E + ∂(~∇× ~A)/∂t = 0. Since we can differentiate either with respect to time or to spacefirst, we can also write this equation as

~∇×

(~E +

∂ ~A

∂t

)= 0 .

We see that ~E + ∂ ~A/∂t is a vector whose curl is equal to zero. Therefore that vector can be

expressed as the gradient of a scalar field. When studying electrostatics, we had ~∇× ~E = 0,and thus ~E was the gradient of something. We took it to be the gradient of −φ, (the minus

for technical convenience). We do the same thing for ~E + ∂ ~A/∂t; we set

~E +∂ ~A

∂t= −~∇φ .

We use the same symbol φ, so that, in the electrostatic case where nothing changes, therelation ~E = −~∇φ still holds. Thus Faraday’s law can be put in the form

~E = −~∇φ− ∂ ~A

∂t. (9.2)

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We have solved two of Maxwell’s equations already, and we have found that to describethe electromagnetic fields ~E and ~B, we need four potential functions: a scalar potential φ,and a vector potential ~A, which is, of course, three functions.

Now that ~A determines part of ~E , as well as ~B, what happens when we change ~A to~A′ = ~A+ ~∇ψ? Although, as we have seen, ~B does not change since ~∇× ~∇ψ = 0, in general~E would change. We can, however, still allow ~A to be changed without affecting the electricand magnetic fields - that is, without changing the physics - if we always change ~A and φtogether by the rules

~A′ = ~A+ ~∇ψ ; φ′ = φ− ∂ψ

∂t.

Previously, we chose to set ~∇ · ~A = 0, to make the equations of statics simpler. We are notgoing to do that now; we are going to make a different choice - we shall justify it soon.

Now we return to the two remaining Maxwell equations which will give us relationsbetween the potentials and the sources. Once we can determine ~A and φ from the currentsand charges, we can always get ~E and ~B from Equations (9.1) and (9.2), so we will haveanother form of Maxwell’s equations.

We begin by substituting Equation (9.2) into ~∇ · ~E = ρ/ε0; we get

~∇ ·

(−~∇φ− ∂ ~A

∂t

)=

ρ

ε0

=⇒ −∇2φ− ∂

∂t~∇ · ~A =

ρ

ε0. (9.3)

This is one equation relating ρ and ~A to the sources.Our final equation will be the most complicated. Thanks to Equations (9.1) and (9.2),

the 4th Maxwell equation

~∇× ~B = µ0~ +1

c2

∂ ~E∂t

can be written as

~∇× (~∇× ~A) = µ0~ +1

c2

∂

∂t

(−~∇φ− ∂ ~A

∂t

)and since ~∇× (~∇× ~A) = ~∇(~∇ · ~A)−∇2 ~A we can write

−∇2 ~A+

[~∇(~∇ · ~A) +

1

c2

∂

∂t~∇φ]

+1

c2

∂2 ~A

∂t2= µ0~

=⇒ −∇2 ~A+

[~∇(~∇ · ~A+

1

c2

∂φ

∂t

)]+

1

c2

∂2 ~A

∂t2= µ0~ . (9.4)

Fortunately, we can now make use of our freedom to choose arbitrarily the divergence of ~A.What we are going to do is to use our choice to fix things so that the equations for ~A andfor φ are separated but have the same form. We can do this by taking (this is called theLorenz1 gauge:

~∇ · ~A = − 1

c2

∂φ

∂t. (9.5)

1Ludvig Lorenz (1829 - 1891), not to be confused with Hendrik Lorentz, was a Danish mathematicianand physicist, professor at the Military Academy in Copenhagen.

105

When we do that, the two terms in square brackets in ~A and φ in Equation (9.4) cancel, andthat equation becomes much simpler:

1

c2

∂2 ~A

∂t2− ~∇2 ~A = µ0~ =⇒ � ~A = µ0~ (9.6)

and also the equation for φ takes a similar form:

1

c2

∂2φ

∂t2− ~∇2φ =

ρ

ε0=⇒ �φ =

ρ

ε0. (9.7)

These equations are particularly fascinating. We remind that we obtained before formthe Maxwell’s equations the continuity equation for charge:

~∇ ·~ +∂ρ

∂t= 0 .

If there is a net electric current is flowing out of a region, then the charge in that regionmust be decreasing by the same amount. Charge is conserved. This provides a proof thatµ = (ρ/c,~) is a 4-vector, since it can be written

∂µ µ = 0 .

If we define the 4-ple Aµ = (φ/c, ~A), the equations (9.6),(9.7) can be written all together as

�Aµ = µ0µ . (9.8)

Thus the 4-ple Aµ is also a 4-vector; we call it the 4-potential of the electromagnetic field.Note that, considering this fact, it appears clear that the Lorenz gauge (9.5) is covariant,

and can be written∂µA

µ = 0 .

In regions where there are no longer any charges and currents, the solution of Equation(9.8) is a 4-potential which is changing in time but always moving out at the speed c. This4-field travel onward through free space.

9.0.2 The electromagnetic tensor

Let us move to natural units from now on (c = ε0 = 1); we must just remember in the endthat the dimension of the magnetic field B is [B] = [E/c], and that the dimension of the

scalar potential φ is such that [φ/c] = [ ~A].Since Aµ is a 4-vector, it is thus a 4-tensor the antisymmetrix matrix

F µν = ∂µAν − ∂νAµ .

Obviously the diagonal elements of this tensor are null. The 0-th row and column arerespectively:

F 0i = ∂0Ai − ∂iA0 =∂Ai

∂t+∂φ

∂xi= −E i

F i0 = −F 0i = E i .

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The 1...3 part or the matrix is

F 12 = ∂1A2 − ∂2A1 = −(~∇× ~A)z = −Bz

F 13 = ∂1A3 − ∂3A1 = (~∇× ~A)y = By

F 23 = ∂2A3 − ∂3A2 = −(~∇× ~A)x = −Bx

and correspondingly for the symmetric components.Finally, the electromagnetic tensor is

F µν =

0 −Ex −Ey −EzEx 0 −Bz By

Ey Bz 0 −Bx

Ez −By Bx 0

(9.9)

The components of the electromagnetic field are thus elements of a tensor, the electro-magnetic tensor.

9.0.3 Covariant expression of Maxwell’s equations

The non-homogeneous Maxwell equations have been written as (Equation 9.8):

�Aµ = (∂ν∂ν)Aµ = µ .

We can write∂νF

νµ = ∂ν(∂νAµ − ∂µAν) = (∂ν∂

ν)Aµ − ∂µ(∂νAν)

and since ∂νAν = 0,

∂νFνµ = (∂ν∂

ν)Aµ = �Aµ = µ .

The covariant equation∂νF

νµ = µ

is equivalent to the non-homogeneous Maxwell equations.Analogamente si ha per le equazioni omogenee

~∇× ~E +∂ ~B

∂t= 0 ; ~∇ · ~B = 0

il seguente risultato (4 equazioni):

∂2F 03 + ∂3F 20 + ∂0F 32 = 0 ... ∂1F 23 + ∂2F 31 + ∂3F 12 = 0

Quindi (~∇× ~E = −∂

~B

∂t& ~∇ · ~B = 0

)⇐⇒ εαβγδ∂

βF γδ = 0 (α = 0, 1, 2, 3) .

107

9.1 Transformation of the fields (from Jarrell, electro-

dynamics)

The transformation properties of ~E and ~B are easily worked out by making use of ourknowledge of how a rank-two tensor must transform:

(F ′)αβ = ΛαγΛβ

δFγδ, (9.10)

or, in matrix notation,F ′ = ΛF Λ (9.11)

where Λ is the transpose of the matrix representing the Lorentz transformation Λ. If we picka frame K ′ which is moving at velocity V directed along the x axis, then

Λ =

γ −V γ 0 0−V γ γ 0 0

0 0 1 00 0 0 1

≡ Λ. (9.12)

Given the field tensor from (9.9), we have

F Λ =

V γEx −γEx −Ey −EzγEx −V γEx −Bz By

γEy − V γBz −V γEy + γBz 0 −Bx

γEz + V γBy −V γEz − γBy Bx 0

(9.13)

and

F ′ =

0 −Ex −γEy + V γBy −γEz − V γBy

Ex 0 V γEy − γBz V γEz + γBy

γEy − V γBz −V γEy + γBz 0 −Bx

γEz + V γBy −V γEz − γBy Bx 0

. (9.14)

This is an antisymmetric tensor - as it should be - and we can equate individual elements tothe appropriate components of ~B′ and ~E ′. One finds

B′x = Bx B′y = γ(By + V Ez) B′z = γ(Bz − V Ey)E ′x = Ex E ′y = γ(Ey − V Bz) E ′z = γ(Ez + V By). (9.15)

By examining these relations one can see that

~E ′‖ = ~E‖ ~E ′⊥ = γ[~E⊥ + (~V × ~B)]~B′‖ = ~B‖ ~B′⊥ = γ[ ~B⊥ − (~V × ~E)]

(9.16)

where the subscripts refer to components of the fields parallel or perpendicular to ~V .

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9.1.1 Invariants

Due to the tensor nature of Fµν , the two following quantities are invariant for transformationsbetween inertial frames:

1

2FµνF

µν = B2 − E2

1

8εαβγδF

αβF γδ = ~B · ~E

where εαβγδ is the completely antisymmetric unit tensor of rank 4.We could derive explicitly the above invariants from the transformation properties of the

fields.

9.2 Covariant expression of the Lorentz force

Newton’s law can be written in covariant form. Let us analyse just the x component for themoment:

f i = γ(q~E + ~v × ~B)x = qγ(Ex + v1Bz − v3By) .

Now we must put all quantities m their relativistic notation. First, γvi are the space com-ponents of the 4-velocity vµ; Ei and Bi are components of the tensor F µν - let us call now,for clarity, the 4-vector components txyz instead of 0123. The γvi are components of the4-velocity ui, and thus

fx = q(utFxt − uyFxy − uzFxz) .Every term has the subscript x, which is reasonable, since we’re finding an x−component.Then all the others appear in pairs, except that the xx−term is missing. So we just add it(is is anyway zero, since F is antysymmetruc)). and write

fx = q(utFxt − uxFxx − uyFxy − uzFxz) .

The reason for putting in the xx−term is so that we can write the equation as

f 1 = quµFµ1 .

You can easily demonstrate that this works equally well for y and z, but what about thetime component? When we translate back to the electric and magnetic fields we get

ft = qγ(0 + vxEx + vyEy + vzEz) = γq(~E · ~u) .

But this is just γ ~F · ~v, since the magnetic field makes 0 work.Summarizing, our equation of motion can be written in the elegant form

f ν = quµFµν .

Although it Is nice to see that the equations can be written that way, this form is notparticularly useful: it is usually more convenient to solve for particle motions by using theoriginal equations.

9.2.1 Motion of a particle under a constant force

??

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9.3 Radiation from an accelerated particle

The classical formula for the radiated power from an accelerated particle is, in the limitv � c:

P =2

3

(1

4πε0

)q2

c3a2 (9.17)

Any charged particle which moves in a curved path or is accelerated in a straight-linepath will emit electromagnetic radiation. Various names are given to this radiation in dif-ferent contexts. For example, when is occurs upon electron impact with a solid metal targetin an x-ray tube, it is called “bremsstrahlung” (from German: braking radiation). Syn-chrotron radiation is the name given to the radiation which occurs when charged particlesare accelerated in a curved path or orbit.

Particularly in the application to circular particle accelerators like synchrotrons, wherecharged particles are accelerated to very high speeds, the effect can be important. Sincein the nonrelativistic limit a = v2/R, the radiated energy is proportional to the fourthpower of the particle speed and is inversely proportional to the square of the radius of thepath. It becomes the limiting factor on the final energy of particles accelerated in electronsynchrotrons like the LEP at CERN. In the ultrarelativistic limit (v ' c), we just replace vwith the relativistic speed u = γv, obtaining

Psync '2

3

(1

4πε0

)q2γ

4c

R2. (9.18)

The LEP electron synchrotron has a rated energy of 50 GeV and a radius of 4300 meters.This gives a relativistic gamma of about 98 000 compared to a gamma of 54 for a 50 GeVproton. Using the above formula, one obtains a loss of two-tenths of a microwatt per second.

Two-tenths of a microwatt may not sound like much loss, but per electron it is enor-mous! At this energy the proton velocity would also be essentially c; thus the synchrotronradiation loss for the two particles scales like their gammas. So the loss rate for the elec-tron is (97833/54)4 or over 1013 times the loss for a proton of the same energy in the samesynchrotron.

110