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dipole in the same direction.) The quantity q is called the dipolemoment, notated D. (More complex dipoles can also be assigneda dipole moment — they are defined as having the same dipolemoment as the two-charge dipole that would experience the sametorque.)

Dipole moment of a molecule of NaCl gas example 5

In a molecule of NaCl gas, the center-to-center distance be-tween the two atoms is about 0.6 nm. Assuming that the chlo-

rine completely steals one of the sodium’s electrons, compute the

magnitude of this molecule’s dipole moment.

The total charge is zero, so it doesn’t matter where we choose

the origin of our coordinate system. For convenience, let’s choose

it to be at one of the atoms, so that the charge on that atom

doesn’t contribute to the dipole moment. The magnitude of the

dipole moment is then

D = (6

×10−10 m)(e )

= (6 × 10−10 m)(1.6 × 10−19 C)

= 1 × 10−28 C · m

Alternative definition of the electric field

The behavior of a dipole in an externally created field leads us toan alternative definition of the electric field:

alternative definition of the electric field

The electric field vector, E , at any location in space is definedby observing the torque exerted on a test dipole Dt placed

there. The direction of the field is the direction in which thefield tends to align a dipole (from − to +), and the field’smagnitude is |E| = τ/Dt sin θ.

The main reason for introducing a second definition for the sameconcept is that the magnetic field is most easily defined using asimilar approach.

Voltage related to electric field

Voltage is potential energy per unit charge, and electric field is forceper unit charge. We can therefore relate voltage and field if we startfrom the relationship between potential energy and force,

∆PE = −Fd , [assuming constant force and

motion parallel to the force]

and divide by charge,

∆PE

q= −F

qd ,

Section 22.3 The electric field 6

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o / Example 7.

giving

∆V = −Ed , [assuming constant force and

motion parallel to the force]

In other words, the difference in voltage between two points equalsthe electric field strength multiplied by the distance between them.The interpretation is that a strong electric field is a region of spacewhere the voltage is rapidly changing. By analogy, a steep hillsideis a place on the map where the altitude is rapidly changing.

Field generated by an electric eel example 6

Suppose an electric eel is 1 m long, and generates a voltage

difference of 1000 volts between its head and tail. What is the

electric field in the water around it?

We are only calculating the amount of field, not its direction, so

we ignore positive and negative signs. Subject to the possibly in-

accurate assumption of a constant field parallel to the eel’s body,

we have|E| =

∆V

∆x = 1000 V/m .

The hammerhead shark example 7

One of the reasons hammerhead sharks have their heads shaped

the way they do is that, like quite a few other fish, they can sense

electric fields as a way of finding prey, which may for example

be hidden in the sand. From the equation E = ∆V /∆x , we can

see that if the shark is sensing the voltage difference between

two points, it will be able to detect smaller electric fields if those

two points are farther apart. The shark has a network of sensory

organs, called the ampullae of Lorenzini, on the skin of its head.Since the network is spread over a wider head, the ∆x is larger.

Some sharks can detect electric fields as weak as 50 pico volts

per meter!

Relating the units of electric field and voltage example 8

From our original definition of the electric field, we expect it to

have units of newtons per coulomb, N/C. The example above,

however, came out in volts per meter, V/m. Are these inconsis-

tent? Let’s reassure ourselves that this all works. In this kind

of situation, the best strategy is usually to simplify the more com-

plex units so that they involve only mks units and coulombs. Since

voltage is defined as electrical energy per unit charge, it has units

of J/C:V

m=

J/C

m=

J

C · m.

To connect joules to newtons, we recall that work equals force

times distance, so J = N · m, so

V

m=

N · m

C · m=

N

C

602 Chapter 22 The nonmechanical universe

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p / Discussion question

As with other such difficulties with electrical units, one quickly

begins to recognize frequently occurring combinations.

Discussion questions

A In the definition of the electric field, does the test charge need to be1 coulomb? Does it need to be positive?

B Does a charged particle such as an electron or proton feel a forcefrom its own electric field?

C Is there an electric field surrounding a wall socket that has nothingplugged into it, or a battery that is just sitting on a table?

D In a flashlight powered by a battery, which way do the electric fieldspoint? What would the fields be like inside the wires? Inside the filamentof the bulb?

E Criticize the following statement: “An electric field can be representedby a sea of arrows showing how current is flowing.”

F The field of a point charge, |E| = k Q /r 2, was derived in the self-check above. How would the field pattern of a uniformly charged sphere

compare with the field of a point charge?G The interior of a perfect electrical conductor in equilibrium musthave zero electric field, since otherwise the free charges within it wouldbe drifting in response to the field, and it would not be in equilibrium. Whatabout the field right at the surface of a perfect conductor? Consider the

possibility of a field perpendicular to the surface or parallel to it.

H Compare the dipole moments of the molecules and molecular ionsshown in the figure.

I Small pieces of paper that have not been electrically prepared in

any way can be picked up with a charged object such as a charged pieceof tape. In our new terminology, we could describe the tape’s charge asinducing a dipole moment in the paper. Can a similar technique be usedto induce not just a dipole moment but a charge?

J The earth and moon are fairly uneven in size and far apart, like abaseball and a ping-pong ball held in your outstretched arms. Imagineinstead a planetary system with the character of a double planet: twoplanets of equal size, close together. Sketch a sea of arrows diagram oftheir gravitational field.

22.4 Calculating energy in fields

We found on p. 592 that fields have energy, and we now know aswell that fields act like vectors. Presumably there is a relationship

between the strength of a field and its energy density. Flipping thedirection of the field can’t change the density of energy, which is ascalar and therefore has no direction in space. We therefore expectthat the energy density to be proportional to the square of the field,so that changing E to −E has no effect on the result. This is exactlywhat we’ve already learned to expect for waves: the energy dependson the square of the amplitude. The relevant equations for the

Section 22.4 Calculating energy in fields 6

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gravitational and electric fields are as follows:

energy stored in the gravitational field per m3 = − 1

8πG|g|2

energy stored in the electric field per m3 =1

8πk|E2|

A similar expression is given on p. 651 for the magnetic field.

Although funny factors of 8π and the plus and minus signs mayhave initially caught your eye, they are not the main point. Theimportant idea is that the energy density is proportional to thesquare of the field strength in all cases. We first give a simplenumerical example and work a little on the concepts, and then turnour attention to the factors out in front.

In chapter 22 when we discussed the original reason for introducingthe concept of a field of force, a prime motivation was that otherwisethere was no way to account for the energy transfers involved whenforces were delayed by an intervening distance. We used to think of the universe’s energy as consisting of

kinetic energy

+gravitational potential energy based on the distances between

objects that interact gravitationally

+electric potential energy based on the distances between

objects that interact electrically

+magnetic potential energy based on the distances between

objects that interact magnetically ,

but in nonstatic situations we must use a different method:

kinetic energy

+gravitational potential energy stored in gravitational fields

+electric potential energy stored in electric fields

+magnetic potential stored in magnetic fields

Surprisingly, the new method still gives the same answers for the

static cases.Energy stored in a capacitor example 9

A pair of parallel metal plates, seen from the side in figure q,

can be used to store electrical energy by putting positive charge

on one side and negative charge on the other. Such a device is

called a capacitor. (We have encountered such an arrangement

previously, but there its purpose was to deflect a beam of elec-

trons, not to store energy.)


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q / Example 9.

r / Example 10.

In the old method of describing potential energy, 1, we think in

terms of the mechanical work that had to be done to separate

the positive and negative charges onto the two plates, working

against their electrical attraction. The new description, 2, at-

tributes the storage of energy to the newly created electric field

occupying the volume between the plates. Since this is a static

case, both methods give the same, correct answer.

Potential energy of a pair of opposite charges example 10

Imagine taking two opposite charges, r, that were initially far apart

and allowing them to come together under the influence of their

electrical attraction.

According to the old method, potential energy is lost because the

electric force did positive work as it brought the charges together.

(This makes sense because as they come together and acceler-

ate it is their potential energy that is being lost and converted to

kinetic energy.)

By the new method, we must ask how the energy stored in theelectric field has changed. In the region indicated approximately

by the shading in the figure, the superposing fields of the two

charges undergo partial cancellation because they are in oppos-

ing directions. The energy in the shaded region is reduced by

this effect. In the unshaded region, the fields reinforce, and the

energy is increased.

It would be quite a project to do an actual numerical calculation of

the energy gained and lost in the two regions (this is a case where

the old method of finding energy gives greater ease of computa-

tion), but it is fairly easy to convince oneself that the energy is

less when the charges are closer. This is because bringing thecharges together shrinks the high-energy unshaded region and

enlarges the low-energy shaded region.

Energy transmitted by ripples in the electric and magnetic fields

example 11

We’ll see in chapter 24 that phenomena like light, radio waves,

and x-rays are all ripples in the electric and magnetic fields. The

old method would give zero energy for a region of space contain-

ing a light wave but no charges. That would be wrong! We can

only use the old method in static cases.

Now let’s give at least some justification for the other features of the expressions for energy density, − 1

8πG |g|2 and 1

8πk |E2|, besidesthe proportionality to the square of the field strength.

First, why the different plus and minus signs? The basic idea is thatthe signs have to be opposite in the gravitational and electric casesbecause there is an attraction between two positive masses (whichare the only kind that exist), but two positive charges would repel.Since we’ve already seen examples where the positive sign in the

Section 22.4 Calculating energy in fields 6

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s / Discussion question A.

electric energy makes sense, the gravitational energy equation mustbe the one with the minus sign.

It may also seem strange that the constants G and k, and are inthe denominator. They tell us how strong the three different forcesare, so shouldn’t they be on top? No. Consider, for instance, analternative universe in which gravity is twice as strong as in ours.

The numerical value of G is doubled. Because G is doubled, all thegravitational field strengths are doubled as well, which quadruplesthe quantity |g|2. In the expression − 1

8πG |g|2, we have quadrupledsomething on top and doubled something on the bottom, whichmakes the energy twice as big. That makes perfect sense.


A The figure shows a positive charge in the gap between two capacitorplates. First make a large drawing of the field pattern that would be formedby the capacitor itself, without the extra charge in the middle. Next, show

how the field pattern changes when you add the particle at these two po-sitions. Compare the energy of the electric fields in the two cases. Does

this agree with what you would have expected based on your knowledgeof electrical forces?

B Criticize the following statement: “A solenoid makes a charge in thespace surrounding it, which dissipates when you release the energy.”

C In example 10, I argued that the fields surrounding a positiveand negative charge contain less energy when the charges are closertogether. Perhaps a simpler approach is to consider the two extreme pos-sibilities: the case where the charges are infinitely far apart, and the onein which they are at zero distance from each other, i.e., right on top ofeach other. Carry out this reasoning for the case of (1) a positive chargeand a negative charge of equal magnitude, (2) two positive charges ofequal magnitude, (3) the gravitational energy of two equal masses.

22.5

Voltage for nonuniform fields

The calculus-savvy reader will have no difficulty generalizing thefield-voltage relationship to the case of a varying field. The potentialenergy associated with a varying force is

∆PE = −

F dx , [one dimension]

so for electric fields we divide by q to find

∆V = −

E dx , [one dimension]

Applying the fundamental theorem of calculus yields

E = −dV

dx. [one dimension]


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Voltage associated with a point charge example 12

What is the voltage associated with a point charge?

As derived previously in self-check A on page 597, the field is

|E| =kQ

r 2

The difference in voltage between two points on the same radius

line is

∆V =

dV

= −

E x dx

In the general discussion above, x was just a generic name for

distance traveled along the line from one point to the other, so in

this case x really means r .

∆V = −

r 2

r 1

E r dr

= − r 2

r 1

kQ

r 2dr

=kQ

r

r 2

r 1

=kQ

r 2− kQ

r 1.

The standard convention is to use r 1 = ∞ as a reference point, so

that the voltage at any distance r from the charge is

V =kQ

r .

The interpretation is that if you bring a positive test charge closer

to a positive charge, its electrical energy is increased; if it was

released, it would spring away, releasing this as kinetic energy.

self-check B

Show that you can recover the expression for the field of a point charge

by evaluating the derivative E x = −dV /dx . Answer, p. 969

22.6 Two or three dimensions

The topographical map shown in figure t suggests a good way to vi-sualize the relationship between field and voltage in two dimensions.Each contour on the map is a line of constant height; some of theseare labeled with their elevations in units of feet. Height is related togravitational potential energy, so in a gravitational analogy, we can

Section 22.6 Two or three dimensions 6

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u / The constant-voltage curvessurrounding a point charge. Nearthe charge, the curves are soclosely spaced that they blendtogether on this drawing due to

the finite width with which theywere drawn. Some electric fieldsare shown as arrows.

t / Left: A topographical map of Stowe, Vermont. From one constant-height line to the next is a heightdifference of 200 feet. Lines far apart, as in the lower village, indicate relatively flat terrain, while linesclose together, like the ones to the west of the main town, represent a steep slope. Streams flow downhill,perpendicular to the constant-height lines. Right: The same map has been redrawn in perspective, withshading to suggest relief.

think of height as representing voltage. Where the contour lines arefar apart, as in the town, the slope is gentle. Lines close togetherindicate a steep slope.

If we walk along a straight line, say straight east from the town,then height (voltage) is a function of the east-west coordinate x.Using the usual mathematical definition of the slope, and writingV for the height in order to remind us of the electrical analogy, theslope along such a line is ∆V/∆x. If the slope isn’t constant, we

either need to use the slope of the V − x graph, or use calculus andtalk about the derivative dV/dx.

What if everything isn’t confined to a straight line? Water flowsdownhill. Notice how the streams on the map cut perpendicularlythrough the lines of constant height.

It is possible to map voltages in the same way, as shown in figure u.The electric field is strongest where the constant-voltage curves areclosest together, and the electric field vectors always point perpen-dicular to the constant-voltage curves.

Figure w shows some examples of ways to visualize field and voltage

patterns.Mathematically, the calculus of section 22.5 generalizes to three di-mensions as follows:

E x = −dV/dx

E y = −dV/dy

E z = −dV/dz


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v / Self-check C.

self-check C

Imagine that the topographical map in figure v represents voltage rather

than height. (a) Consider the stream that starts near the center of the

map. Determine the positive and negative signs of dV /dx and dV /dy ,

and relate these to the direction of the force that is pushing the current

forward against the resistance of friction. (b) If you wanted to find a lot

of electric charge on this map, where would you look? Answer, p.969

w / Two-dimensional field avoltage patterns. Top: A uformly charged rod. BottoA dipole. In each case, tdiagram on the left shows tfield vectors and constant-volta

curves, while the one on tright shows the voltage (up-do

coordinate) as a function of x ay. Interpreting the field diagramEach arrow represents the fiat the point where its tail hbeen positioned. For clarsome of the arrows in regionsvery strong field strength are shown they would be too long

show. Interpreting the constavoltage curves: In regions very strong fields, the curves anot shown because they womerge together to make soblack regions. Interpreting perspective plots: Keep in mthat even though we’re visualizthings in three dimensions, the

are really two-dimensional vage patterns being representThe third (up-down) dimensrepresents voltage, not position

Section 22.6 Two or three dimensions 6

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y / Example 13.

22.7 Field lines and Gauss’s law

When we look at the “sea of arrows” representation of a field, x/1,there is a natural visual tendency to imagine connecting the arrowsas in x/2. The curves formed in this way are called field lines, andthey have a direction, shown by the arrowheads.

x / Two different representations of an electric field.

An avalanche transceiver example 13

The dipole field pattern is universal: it always has the same math-

ematical shape, and this holds regardless of whether the field

is electric or magnetic, static or oscillating. One of its universal

properties is that the field lines always pass through the source,

and this is taken advantage of in a device called an avalanche

transceiver used by backcountry skiers and hikers in winter. The

device is worn on the body and is left in transmitting mode while

the user is traveling. It creates a dipole pattern of electric and

magnetic fields, oscillating at 457 kHz. If a member of the party isburied in an avalanche, his companions must find and locate him

within about 30-90 minutes, or he will suffocate. The rescuers

switch their own transceivers to receive mode, allowing them to

determine the direction of the field line; when the antenna is par-

allel to the field line, the oscillating electric field drives electrons

up and down it, creating the maximum possible current.

In the figure, the rescuer moves along the field line from A to B.

At B, she verifies that the strength of the signal has increased.

(If it hadn’t, she would have had to turn around and go in the

opposite direction.) She redetermines the direction of the field

and continues in the new direction. Continuing the process, sheproceeds along an approximation to the field line. At D she finds

that the field strength has fallen off again, so she knows that she

has just passed over the victim’s position.

Electric field lines originate from positive charges and terminate onnegative ones. We can choose a constant of proportionality that fixeshow coarse or fine the “grain of the wood” is, but once this choiceis made the strength of each charge is shown by the number of lines


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z / The number of field lincoming in and out of each regidepends on the total chargeencloses.

aa / Example 14.

that begin or end on it. For example, figure x/2 shows eight lines ateach charge, so we know that q1/q2 = (−8)/8 = −1. Because linesnever begin or end except on a charge, we can always find the totalcharge inside any given region by subtracting the number of linesthat go in from the number that come out and multiplying by theappropriate constant of proportionality. Ignoring the constant, we

can apply this technique to figure z to find qA = −8, qB = 2 −2 = 0,and qC = 5 − 5 = 0.

Figures x and z are drawn in two dimensions, but we should reallyimagine them in three dimensions, so that the regions A, B, andC in figure z are volumes bounded by surfaces. It is only becauseour universe happens to have three dimensions that the field lineconcept is as useful as it is. This is because the number of field linesper unit area perpendicularly piercing a surface can be interpretedas the strength of the electric field. To see this, consider an imag-inary spherical surface of radius r, with a positive charge q at itscenter. The field at the surface equals kq/r2. The number of field

lines piercing the surface is independent of r, but the surface’s areais proportional to r2, so the number of piercings per unit area isproportional to 1/r2, just like the electric field. With this interpre-tation, we arrive at Gauss’s law, which states that the field strengthon a surface multiplied by the surface area equals the total chargeenclosed by the surface. (This particular formulation only works inthe case where the field pierces the surface perpendicularly and isconstant in magnitude over the whole surface. It can be reformu-lated so as to eliminate these restrictions, but we won’t require thatreformulation for our present purposes.)

The field of a line of charge example 14

In example 4 on p. 599, we found by simple scaling argumentsthat the electric field of a line of charge is proportional to 1/r ,

where r is the distance from the line. Since electric fields are al-

ways proportional to the Coulomb constant k and to the amount of

charge, clearly we must have something of the form

E = (. . .)k (q /L)/r , where q /L is the charge per unit length and

(. . .) represents an unknown numerical constant (which you can

easily verify is unitless). Using Gauss’s law we can fill in the final

piece of the puzzle, which is the value of this constant.

Figure aa shows two surfaces, a cylindrical one A and a spherical

one B. Every field line that passes through A also passes through

B. By making A sufficiently small and B sufficiently large, we canmake the field on each surface nearly constant and perpendicular

to the surface, as required by our restricted form of Gauss’s law.

If radius r B is made large, the field at B made by the line of charge

becomes indistinguishable from that of a point charge, kq /r 2B . By

Gauss’s law, the electric field at each surface is proportional to

the number of field lines divided by its area. But the number of

Section 22.7 Field lines and Gauss’s law 6

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ab / Example 15.

field lines is the same in both cases, so each field is inversely pro-

portional to the corresponding surface area. We therefore have

E A = E B

AB

AA

=

kq

r 2B

4π r 2B

2π r AL

=2kq

Lr B .

The unknown numerical constant equals 2.

22.8

Electric

field of a continuous charge distribution

Charge really comes in discrete chunks, but often it is mathemat-ically convenient to treat a set of charges as if they were like acontinuous fluid spread throughout a region of space. For example,a charged metal ball will have charge spread nearly uniformly allover its surface, and in for most purposes it will make sense to ig-nore the fact that this uniformity is broken at the atomic level. Theelectric field made by such a continuous charge distribution is thesum of the fields created by every part of it. If we let the “parts”become infinitesimally small, we have a sum of an infinite number of infinitesimal numbers, which is an integral. If it was a discrete sum,we would have a total electric field in the x direction that was thesum of all the x components of the individual fields, and similarlywe’d have sums for the y and z components. In the continuous case,we have three integrals.

Field of a uniformly charged rod example 15

A rod of length L has charge Q spread uniformly along it. Find

the electric field at a point a distance d from the center of the rod,

along the rod’s axis. (This is different from examples 4 on p. 599

and 14 on p. 611, both because the point is on the axis of the rod

and because the rod is of finite length.)

This is a one-dimensional situation, so we really only need to

do a single integral representing the total field along the axis. Weimagine breaking the rod down into short pieces of length dz ,

each with charge dq . Since charge is uniformly spread along the

rod, we have dq = λdz , where λ = Q /L (Greek lambda) is the

charge per unit length, in units of coulombs per meter. Since

the pieces are infinitesimally short, we can treat them as point

charges and use the expression k dq /r 2 for their contributions to

the field, where r = d − z is the distance from the charge at z to


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the point in which we are interested.

E z =

k dq

r 2

=

+L/2

−L/2

k λdz

r 2

= k λ

+L/2

−L/2

dz (d − z )2

The integral can be looked up in a table, or reduced to an ele-

mentary form by substituting a new variable for d − z . The result

is

E z = k λ

1

d − z

+L/2

−L/2

=kQ

L 1

d

−L/2

− 1

d + L/2.

For large values of d , this expression gets smaller for two rea-

sons: (1) the denominators of the fractions become large, and

(2) the two fractions become nearly the same, and tend to cancel

out. This makes sense, since the field should get weaker as we

get farther away from the charge. In fact, the field at large dis-

tances must approach kQ /d 2, since from a great distance, the

rod looks like a point.

It’s also interesting to note that the field becomes infinite at the

ends of the rod, but is not infinite on the interior of the rod. Can

you explain physically why this happens?

Section 22.8 Electric field of a continuous charge distribution 6

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Summary

Selected vocabularyfield . . . . . . . . a property of a point in space describing the

forces that would be exerted on a particle if itwas there

sink . . . . . . . . a point at which field vectors converge

source . . . . . . . a point from which field vectors diverge; of-ten used more inclusively to refer to points of either convergence or divergence

electric field . . . the force per unit charge exerted on a testcharge at a given point in space

gravitational field the force per unit mass exerted on a test massat a given point in space

electric dipole . . an object that has an imbalance between pos-itive charge on one side and negative chargeon the other; an object that will experience atorque in an electric field

Notationg . . . . . . . . . . the gravitational fieldE . . . . . . . . . the electric fieldD . . . . . . . . . an electric dipole moment

Other terminology and notation

d, p, m . . . . . . other notations for the electric dipole moment

Summary

Experiments show that time is not absolute: it flows at differentrates depending on an observer’s state of motion. This is an exam-ple of the strange effects predicted by Einstein’s theory of relativ-

ity. All of these effects, however, are very small when the relativevelocities are small compared to c. This makes sense, because New-ton’s laws have already been thoroughly tested by experiments atsuch speeds, so a new theory like relativity must agree with the oldone in their realm of common applicability. This requirement of backwards-compatibility is known as the correspondence principle.

Since time is not absolute, simultaneity is not a well-defined con-cept, and therefore the universe cannot operate as Newton imagined,through instantaneous action at a distance. There is a delay in timebefore a change in the configuration of mass and charge in one cor-ner of the universe will make itself felt as a change in the forces

experienced far away. We imagine the outward spread of such achange as a ripple in an invisible universe-filling field of force.

We define the gravitational field at a given point as the force perunit mass exerted on ob jects inserted at that point, and likewise theelectric field is defined as the force per unit charge. These fields arevectors, and the fields generated by multiple sources add accordingto the rules of vector addition.


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When the electric field is constant, the voltage difference betweentwo points lying on a line parallel to the field is related to the fieldby the equation ∆V = −Ed, where d is the distance between thetwo points.

Fields of force contain energy. The density of energy is proportionalto the square of the magnitude of the field. In the case of static

fields, we can calculate potential energy either using the previousdefinition in terms of mechanical work or by calculating the energystored in the fields. If the fields are not static, the old method givesincorrect results and the new one must be used.

Summary 6

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Problem 1.

Problem 4.

Problems

Key√ A computerized answer check is available online. A problem that requires calculus.

A difficult problem.

1 In our by-now-familiar neuron, the voltage difference be-tween the inner and outer surfaces of the cell membrane is aboutV out − V in = −70 mV in the resting state, and the thickness of themembrane is about 6.0 nm (i.e., only about a hundred atoms thick).What is the electric field inside the membrane?

√

2 The gap between the electrodes in an automobile engine’sspark plug is 0.060 cm. To produce an electric spark in a gasoline-air mixture, an electric field of 3.0 × 106 V/m must be achieved.On starting a car, what minimum voltage must be supplied by theignition circuit? Assume the field is uniform.

√

(b) The small size of the gap between the electrodes is inconvenient

because it can get blocked easily, and special tools are needed tomeasure it. Why don’t they design spark plugs with a wider gap?

3 (a) At time t = 0, a positively charged particle is placed,at rest, in a vacuum, in which there is a uniform electric field of magnitude E . Write an equation giving the particle’s speed, v, interms of t, E , and its mass and charge m and q.

√

(b) If this is done with two different objects and they are observedto have the same motion, what can you conclude about their massesand charges? (For instance, when radioactivity was discovered, itwas found that one form of it had the same motion as an electron

in this type of experiment.)

4 Three charges are arranged on a square as shown. All threecharges are positive. What value of q2/q1 will produce zero electricfield at the center of the square? Solution, p. 965

5 Show that the magnitude of the electric field produced by asimple two-charge dipole, at a distant point along the dipole’s axis,is to a good approximation proportional to D/r3, where r is thedistance from the dipole. [Hint: Use the approximation (1 + ) p ≈1 + p, which is valid for small .]

6 Consider the electric field created by a uniform ring of total

charge q and radius b. (a) Show that the field at a point on the ring’saxis at a distance a from the plane of the ring is kqa(a2 + b2)−3/2.(b) Show that this expression has the right behavior for a = 0 andfor a much greater than b.


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7 Example 4 on p. 599 showed that the electric field of a long,uniform line of charge falls off with distance as 1/r. By a similartechnique, show that the electric field of a uniformly charged planehas no dependence on the distance from the plane.

8 Given that the field of a dipole is proportional to D/r3 (prob-lem 5), show that its voltage varies as D/r2. (Ignore positive and

negative signs and numerical constants of proportionality.)

9 A carbon dioxide molecule is structured like O-C-O, withall three atoms along a line. The oxygen atoms grab a little bit of extra negative charge, leaving the carbon positive. The molecule’ssymmetry, however, means that it has no overall dipole moment,unlike a V-shaped water molecule, for instance. Whereas the voltageof a dipole of magnitude D is proportional to D/r2 (problem 8), itturns out that the voltage of a carbon dioxide molecule along itsaxis equals k/r3, where r is the distance from the molecule and kis a constant. What would be the electric field of a carbon dioxidemolecule at a distance r?

√ 10 A proton is in a region in which the electric field is given byE = a + bx3. If the proton starts at rest at x1 = 0, find its speed,v, when it reaches position x2. Give your answer in terms of a, b,x2, and e and m, the charge and mass of the proton.

√ 11 Consider the electric field created by a uniformly chargedcylinder that extends to infinity in one direction. (a) Starting fromthe result of problem 8, show that the field at the center of thecylinder’s mouth is 2πkσ , where σ is the density of charge on thecylinder, in units of coulombs per square meter. [Hint: You can usea method similar to the one in problem 9.] (b) This expression is

independent of the radius of the cylinder. Explain why this shouldbe so. For example, what would happen if you doubled the cylinder’sradius?

12 In an electrical storm, the cloud and the ground act like aparallel-plate capacitor, which typically charges up due to frictionalelectricity in collisions of ice particles in the cold upper atmosphere.Lightning occurs when the magnitude of the electric field builds upto a critical value, E c, at which air is ionized.(a) Treat the cloud as a flat square with sides of length L. If it is ata height h above the ground, find the amount of energy released inthe lightning strike.

√

(b) Based on your answer from part a, which is more dangerous, alightning strike from a high-altitude cloud or a low-altitude one?(c) Make an order-of-magnitude estimate of the energy released bya typical lightning bolt, assuming reasonable values for its size andaltitude. E c is about 106 V/m.

See problem 18 for a note on how recent research affects this esti-mate.

Problems 6

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Problem 13.

13 The neuron in the figure has been drawn fairly short, butsome neurons in your spinal cord have tails (axons) up to a meterlong. The inner and outer surfaces of the membrane act as the“plates” of a capacitor. (The fact that it has been rolled up into acylinder has very little effect.) In order to function, the neuron mustcreate a voltage difference V between the inner and outer surfaces

of the membrane. Let the membrane’s thickness, radius, and lengthbe t, r, and L. (a) Calculate the energy that must be stored inthe electric field for the neuron to do its job. (In real life, themembrane is made out of a substance called a dielectric, whoseelectrical properties increase the amount of energy that must bestored. For the sake of this analysis, ignore this fact.) [Hint: Thevolume of the membrane is essentially the same as if it was unrolledand flattened out.]

√

(b) An organism’s evolutionary fitness should be better if it needsless energy to operate its nervous system. Based on your answer topart a, what would you expect evolution to do to the dimensions tand r? What other constraints would keep these evolutionary trendsfrom going too far?

14 To do this problem, you need to understand how to dovolume integrals in cylindrical and spherical coordinates. (a) Showthat if you try to integrate the energy stored in the field of a long,straight wire, the resulting energy per unit length diverges both atr → 0 and r → ∞. Taken at face value, this would imply that acertain real-life process, the initiation of a current in a wire, wouldbe impossible, because it would require changing from a state of zero magnetic energy to a state of infinite magnetic energy. (b)Explain why the infinities at r → 0 and r → ∞ don’t really happen

in a realistic situation. (c) Show that the electric energy of a pointcharge diverges at r → 0, but not at r → ∞.

A remark regarding part (c): Nature does seem to supply us withparticles that are charged and pointlike, e.g., the electron, but onecould argue that the infinite energy is not really a problem, becausean electron traveling around and doing things neither gains nor losesinfinite energy; only an infinite change in potential energy would bephysically troublesome. However, there are real-life processes thatcreate and destroy pointlike charged particles, e.g., the annihilationof an electron and antielectron with the emission of two gammarays. Physicists have, in fact, been struggling with infinities like

this since about 1950, and the issue is far from resolved. Sometheorists propose that apparently pointlike particles are actually notpointlike: close up, an electron might be like a little circular loop of string.


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Problem 15.

15 The figure shows cross-sectional views of two cubical ca-pacitors, and a cross-sectional view of the same two capacitors puttogether so that their interiors coincide. A capacitor with the platesclose together has a nearly uniform electric field between the plates,and almost zero field outside; these capacitors don’t have their platesvery close together compared to the dimensions of the plates, but

for the purposes of this problem, assume that they still have ap-proximately the kind of idealized field pattern shown in the figure.Each capacitor has an interior volume of 1.00 m3, and is charged upto the point where its internal field is 1.00 V/m. (a) Calculate theenergy stored in the electric field of each capacitor when they areseparate. (b) Calculate the magnitude of the interior field when thetwo capacitors are put together in the manner shown. Ignore effectsarising from the redistribution of each capacitor’s charge under theinfluence of the other capacitor. (c) Calculate the energy of theput-together configuration. Does assembling them like this releaseenergy, consume energy, or neither?

Problems 6

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Exercise 22: Field vectors

Apparatus:

3 solenoids

DC power supply

compass

ruler

cut-off plastic cup

At this point you’ve studied the gravitational field, g, and the electric field, E, but not themagnetic field, B. However, they all have some of the same mathematical behavior: they actlike vectors. Furthermore, magnetic fields are the easiest to manipulate in the lab. Manipulatinggravitational fields directly would require futuristic technology capable of moving planet-sizedmasses around! Playing with electric fields is not as ridiculously difficult, but static electriccharges tend to leak off through your body to ground, and static electricity effects are hard tomeasure numerically. Magnetic fields, on the other hand, are easy to make and control. Anymoving charge, i.e. any current, makes a magnetic field.

A practical device for making a strong magnetic field is simply a coil of wire, formally knownas a solenoid. The field pattern surrounding the solenoid gets stronger or weaker in proportionto the amount of current passing through the wire.

1. With a single solenoid connected to the power supply and laid with its axis horizontal, use amagnetic compass to explore the field pattern inside and outside it. The compass shows you thefield vector’s direction, but not its magnitude, at any point you choose. Note that the field thecompass experiences is a combination (vector sum) of the solenoid’s field and the earth’s field.

2. What happens when you bring the compass extremely far away from the solenoid?

What does this tell you about the way the solenoid’s field varies with distance?

Thus although the compass doesn’t tell you the field vector’s magnitude numerically, you canget at least some general feel for how it depends on distance.


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3. The figure below is a cross-section of the solenoid in the plane containing its axis. Make asea-of-arrows sketch of the magnetic field in this plane. The length of each arrow should at leastapproximately reflect the strength of the magnetic field at that point.

Does the field seem to have sources or sinks?

4. What do you think would happen to your sketch if you reversed the wires?

Try it.

Exercise 22: Field vectors 6

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5. Now hook up the two solenoids in parallel. You are going to measure what happens whentheir two fields combine in the at a certain point in space. As you’ve seen already, the solenoids’nearby fields are much stronger than the earth’s field; so although we now theoretically havethree fields involved (the earth’s plus the two solenoids’), it will be safe to ignore the earth’sfield. The basic idea here is to place the solenoids with their axes at some angle to each other,and put the compass at the intersection of their axes, so that it is the same distance from each

solenoid. Since the geometry doesn’t favor either solenoid, the only factor that would make onesolenoid influence the compass more than the other is current. You can use the cut-off plasticcup as a little platform to bring the compass up to the same level as the solenoids’ axes.

a)What do you think will happen with the solenoids’ axes at 90 degrees to each other, and equalcurrents? Try it. Now represent the vector addition of the two magnetic fields with a diagram.Check your diagram with your instructor to make sure you’re on the right track.

b) Now try to make a similar diagram of what would happen if you switched the wires on oneof the solenoids.

After predicting what the compass will do, try it and see if you were right.

c)Now suppose you were to go back to the arrangement you had in part a, but you changed oneof the currents to half its former value. Make a vector addition diagram, and use trig to predictthe angle.

Try it. To cut the current to one of the solenoids in half, an easy and accurate method issimply to put the third solenoid in series with it, and put that third solenoid so far away thatits magnetic field doesn’t have any significant effect on the compass.


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a / All three clocks are mo

ing to the east. Even though twest-going plane is moving to twest relative to the air, the

is moving to the east due to tearth’s rotation.

Chapter 23

Relativity and magnetismMany people imagine Einstein’s theory of relativity as somethingexotic and speculative. It’s certainly not speculative at this point inhistory, since you use it every time you use a GPS receiver. But it’seven less exotic than that. Every time you stick a magnet to yourrefrigerator, you’re making use of relativity. The title of this chapteris “Electromagnetism,” but we’ll start out by digging a little deeperinto relativity as preparation for understanding what magnetism isand where it comes from.

23.1 Relativistic distortion of space and time

Time distortion arising from motion and gravity

Let’s refer back to the results of the Hafele-Keating experiment de-scribed on p. 590. Hafele and Keating were testing specific quantita-tive predictions of relativity, and they verified them to within theirexperiment’s error bars. Let’s work backward instead, and inspectthe empirical results for clues as to how time works.

The east-going clock lost time, ending up off by −59 ± 10 nanosec-onds, while the west-going one gained 273 ± 7 ns. Since two trav-

eling clocks experienced effects in opposite directions, we can tellthat the rate at which time flows depends on the motion of the ob-server. The east-going clock was moving in the same direction asthe earth’s rotation, so its velocity relative to the earth’s center wasgreater than that of the clock that remained in Washington, whilethe west-going clock’s velocity was correspondingly reduced. Thefact that the east-going clock fell behind, and the west-going onegot ahead, shows that the effect of motion is to make time go moreslowly. This effect of motion on time was predicted by Einstein inhis original 1905 paper on relativity, written when he was 26.

If this had been the only effect in the Hafele-Keating experiment,

then we would have expected to see effects on the two flying clocksthat were equal in size. Making up some simple numbers to keep thearithmetic transparent, suppose that the earth rotates from west toeast at 1000 km/hr, and that the planes fly at 300 km/hr. Then thespeed of the clock on the ground is 1000 km/hr, the speed of theclock on the east-going plane is 1300 km/hr, and that of the west-going clock 700 km/hr. Since the speeds of 700, 1000, and 1300km/hr have equal spacing on either side of 1000, we would expect

6

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the discrepancies of the moving clocks relative to the one in the labto be equal in size but opposite in sign.

In fact, the two effects are unequal in size: −59 ns and 273 ns.This implies that there is a second effect involved, simply due tothe planes’ being up in the air. This was verified more directlyin a 1978 experiment by Iijima and Fujiwara, figure b, in which

identical atomic clocks were kept at rest at the top and bottom of amountain near Tokyo. This experiment, unlike the Hafele-Keatingone, isolates one effect on time, the gravitational one: time’s rateof flow increases with height in a gravitational field. Einstein didn’tfigure out how to incorporate gravity into relativity until 1915, aftermuch frustration and many false starts. The simpler version of thetheory without gravity is known as special relativity, the full versionas general relativity. We’ll restrict ourselves to special relativityuntil chapter 27, and that means that what we want to focus onright now is the distortion of time due to motion, not gravity.

b / A graph showing the time difference between two atomic clocks.One clock was kept at Mitaka Observatory, at 58 m above sea level. Theother was moved back and forth between a second observatory, Norikura

Corona Station, at the peak of the Norikura volcano, 2876 m above sealevel. The plateaus on the graph are data from the periods when theclocks were compared side by side at Mitaka. The difference between

one plateau and the next is the gravitational effect on the rate of flow oftime, accumulated during the period when the mobile clock was at the topof Norikura. (This is the original graph showing the actual data, exceptfor some clarification of the labels on the axes.)

We can now see in more detail how to apply the correspondenceprinciple. The behavior of the three clocks in the Hafele-Keatingexperiment shows that the amount of time distortion increases as

624 Chapter 23 Relativity and magnetism

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c / The correspondence prciple requires that the relativis

distortion of time become smfor small velocities.

the speed of the clock’s motion increases. Newton lived in an erawhen the fastest mode of transportation was a galloping horse, andthe best pendulum clocks would accumulate errors of perhaps aminute over the course of several days. A horse is much slowerthan a jet plane, so the distortion of time would have had a relativesize of only ∼ 10−15 — much smaller than the clocks were capable

of detecting. At the speed of a passenger jet, the effect is about10−12, and state-of-the-art atomic clocks in 1971 were capable of measuring that. A GPS satellite travels much faster than a jet air-plane, and the effect on the satellite turns out to be ∼ 10−10. Thegeneral idea here is that all physical laws are approximations, andapproximations aren’t simply right or wrong in different situations.Approximations are better or worse in different situations, and thequestion is whether a particular approximation is good enough in agiven situation to serve a particular purpose. The faster the motion,the worse the Newtonian approximation of absolute time. Whetherthe approximation is good enough depends on what you’re tryingto accomplish. The correspondence principle says that the approxi-mation must have been good enough to explain all the experimentsdone in the centuries before Einstein came up with relativity.

By the way, don’t get an inflated idea of the importance of theseatomic clock experiments. Special relativity had already been con-firmed by a vast and varied body of experiments decades beforethe 1970’s. The only reason I’m giving such a prominent role tothese experiments, which were actually more important as tests of general relativity, is that they were conceptually very direct. Sincewe’re only studying special relativity, it would be nice to have anequally simple and transparent atomic clock experiment in which

only the effect of motion was singled out, with no gravitational ef-fect. This does not ever seem to have been done, probably partlybecause testing special relativity was no longer considered cutting-edge science by the 1970’s and partly because it is difficult to movea clock at a high enough velocity without putting it in some kind of aircraft, which then has to fly at some altitude.

The Lorentz transformation

Relativity says that when two observers are in different frames of reference, each observer considers the other one’s perception of timeto be distorted. We’ll also see that something similar happens totheir observations of distances, so both space and time are distorted.

What exactly is this distortion? How do we even conceptualize it?

The idea isn’t really as radical as it might seem at first. We can vi-sualize the structure of space and time using a graph with positionand time on its axes. These graphs are familiar by now, but we’regoing to look at them in a slightly different way. Before, we usedthem to describe the motion of objects. The grid underlying thegraph was merely the stage on which the actors played their parts.

Section 23.1 Relativistic distortion of space and time 6

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d / Two events are given aspoints on a graph of positionversus time. Joan of Arc helps torestore Charles VII to the throne.

At a later time and a differentposition, Joan of Arc is sentencedto death.

e / A change of units distortsan x -t graph. This graph depictsexactly the same events as figure

d. The only change is that the x and t coordinates are measured

using different units, so the grid iscompressed in t and expanded inx .

f / A convention we’ll use torepresent a distortion of time andspace.

Now the background comes to the foreground: it’s time and spacethemselves that we’re studying. We don’t necessarily need to havea line or a curve drawn on top of the grid to represent a particu-lar object. We may, for example, just want to talk about events,depicted as points on the graph as in figure d. A distortion of theCartesian grid underlying the graph can arise for perfectly ordinary

reasons that Newton would have readily accepted. For example, wecan simply change the units used to measure time and position, asin figure e.

We’re going to have quite a few examples of this type, so I’ll adoptthe convention shown in figure f for depicting them. Figure f sum-marizes the relationship between figures d and e in a more compactform. The gray rectangle represents the original coordinate grid of figure d, while the grid of black lines represents the new versionfrom figure e. Omitting the grid from the gray rectangle makes thediagram easier to decode visually.

Our goal of unraveling the mysteries of special relativity amountsto nothing more than finding out how to draw a diagram like f in the case where the two different sets of coordinates representmeasurements of time and space made by two different observers,each in motion relative to the other. Galileo and Newton thoughtthey knew the answer to this question, but their answer turnedout to be only approximately right. To avoid repeating the samemistakes, we need to clearly spell out what we think are the basicproperties of time and space that will be a reliable foundation forour reasoning. I want to emphasize that there is no purely logicalway of deciding on this list of properties. The ones I’ll list are simplya summary of the patterns observed in the results from a large body

of experiments. Furthermore, some of them are only approximate.For example, property 1 below is only a good approximation whenthe gravitational field is weak, so it is a property that applies tospecial relativity, not to general relativity.

Experiments show that:

1. No point in time or space has properties that make it differentfrom any other point.

2. Likewise, all directions in space have the same properties.

3. Motion is relative, i.e., all inertial frames of reference areequally valid.

4. Causality holds, in the sense described on page 591.

5. Time depends on the state of motion of the observer.


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g / A Galilean version of trelationship between two framof reference. As in all sugraphs in this chapter, the origincoordinates, represented by tgray rectangle, have a time ax

that goes to the right, andposition axis that goes straig

up.

Most of these are not very subversive. Properties 1 and 2 dateback to the time when Galileo and Newton started applying thesame universal laws of motion to the solar system and to the earth;this contradicted Aristotle, who believed that, for example, a rockwould naturally want to move in a certain special direction (down)in order to reach a certain special location (the earth’s surface).

Property 3 is the reason that Einstein called his theory “relativity,”but Galileo and Newton believed exactly the same thing to be true,as dramatized by Galileo’s run-in with the Church over the questionof whether the earth could really be in motion around the sun.Property 4 would probably surprise most people only because itasserts in such a weak and specialized way something that they feeldeeply must be true. The only really strange item on the list is 5,but the Hafele-Keating experiment forces it upon us.

If it were not for property 5, we could imagine that figure g wouldgive the correct transformation between frames of reference in mo-tion relative to one another. Let’s say that observer 1, whose grid

coincides with the gray rectangle, is a hitch-hiker standing by theside of a road. Event A is a raindrop hitting his head, and event Bis another raindrop hitting his head. He says that A and B occur atthe same location in space. Observer 2 is a motorist who drives bywithout stopping; to him, the passenger compartment of his car isat rest, while the asphalt slides by underneath. He says that A andB occur at different points in space, because during the time be-tween the first raindrop and the second, the hitch-hiker has movedbackward. On the other hand, observer 2 says that events A and Coccur in the same place, while the hitch-hiker disagrees. The slopeof the grid-lines is simply the velocity of the relative motion of each

observer relative to the other.Figure g has familiar, comforting, and eminently sensible behavior,but it also happens to be wrong, because it violates property 5.The distortion of the coordinate grid has only moved the verticallines up and down, so both observers agree that events like B andC are simultaneous. If this was really the way things worked, thenall observers could synchronize all their clocks with one another foronce and for all, and the clocks would never get out of sync. Thiscontradicts the results of the Hafele-Keating experiment, in whichall three clocks were initially synchronized in Washington, but laterwent out of sync because of their different states of motion.

It might seem as though we still had a huge amount of wiggle roomavailable for the correct form of the distortion. It turns out, however,that properties 1-5 are sufficient to prove that there is only oneanswer, which is the one found by Einstein in 1905. To see why thisis, let’s work by a process of elimination.


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h / A transformation that leadsto disagreements about whethertwo events occur at the sametime and place. This is not justa matter of opinion. Either thearrow hit the bull’s-eye or it didn’t.

i / A nonlinear transformation.

Figure h shows a transformation that might seem at first glance tobe as good a candidate as any other, but it violates property 3, thatmotion is relative, for the following reason. In observer 2’s frameof reference, some of the grid lines cross one another. This meansthat observers 1 and 2 disagree on whether or not certain events arethe same. For instance, suppose that event A marks the arrival of

an arrow at the bull’s-eye of a target, and event B is the locationand time when the bull’s-eye is punctured. Events A and B occurat the same location and at the same time. If one observer saysthat A and B coincide, but another says that they don’t, we havea direct contradiction. Since the two frames of reference in figureh give contradictory results, one of them is right and one is wrong.This violates property 3, because all inertial frames of reference aresupposed to be equally valid. To avoid problems like this, we clearlyneed to make sure that none of the grid lines ever cross one another.

The next type of transformation we want to kill off is shown in figurei, in which the grid lines curve, but never cross one another. The

trouble with this one is that it violates property 1, the uniformityof time and space. The transformation is unusually “twisty” at A,whereas at B it’s much more smooth. This can’t be correct, becausethe transformation is only supposed to depend on the relative stateof motion of the two frames of reference, and that given informationdoesn’t single out a special role for any particular point in spacetime.If, for example, we had one frame of reference rotating relative tothe other, then there would be something special about the axis of rotation. But we’re only talking about inertial frames of referencehere, as specified in property 3, so we can’t have rotation; eachframe of reference has to be moving in a straight line at constant

speed. For frames related in this way, there is nothing that couldsingle out an event like A for special treatment compared to B, sotransformation i violates property 1.

The examples in figures h and i show that the transformation we’relooking for must be linear, meaning that it must transform linesinto lines, and furthermore that it has to take parallel lines to par-allel lines. Einstein wrote in his 1905 paper that “. . . on account of the property of homogeneity [property 1] which we ascribe to timeand space, the [transformation] must be linear.”1 Applying this toour diagrams, the original gray rectangle, which is a special typeof parallelogram containing right angles, must be transformed into

another parallelogram. There are three types of transformations,figure j, that have this property. Case I is the Galilean transforma-tion of figure g on page 627, which we’ve already ruled out.

1A. Einstein, “On the Electrodynamics of Moving Bodies,” Annalen der Physik 17 (1905), p. 891, tr. Saha and Bose, 1920


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k / In the units that are moconvenient for relativity, the tran

formation has symmetry abou45-degree diagonal line.

j / Three types of transformations that preserve parallelism. Theirdistinguishing feature is what they do to simultaneity, as shown by what

happens to the left edge of the original rectangle. In I, the left edgeremains vertical, so simultaneous events remain simultaneous. In II, theleft edge turns counterclockwise. In III, it turns clockwise.

Case II can also be discarded. Here every point on the grid rotates

counterclockwise. What physical parameter would determine theamount of rotation? The only thing that could be relevant wouldbe v, the relative velocity of the motion of the two frames of referencewith respect to one another. But if the angle of rotation was pro-portional to v, then for large enough velocities the grid would haveleft and right reversed, and this would violate property 4, causality:one observer would say that event A caused a later event B, butanother observer would say that B came first and caused A.

The only remaining possibility is case III, which I’ve redrawn infigure k with a couple of changes. This is the one that Einsteinpredicted in 1905. The transformation is known as the Lorentztransformation, after Hendrik Lorentz (1853-1928), who partiallyanticipated Einstein’s work, without arriving at the correct inter-pretation. The distortion is a kind of smooshing and stretching, assuggested by the hands. Also, we’ve already seen in figures d-f onpage 626 that we’re free to stretch or compress everything as muchas we like in the horizontal and vertical directions, because this sim-ply corresponds to changing the units of measurement for time anddistance. In figure k I’ve chosen units that give the whole drawing

a convenient symmetry about a 45-degree diagonal line. Ordinarilyit wouldn’t make sense to talk about a 45-degree angle on a graphwhose axes had different units. But in relativity, the symmetric ap-pearance of the transformation tells us that space and time oughtto be treated on the same footing, and measured in the same units.


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l / Interpretation of the Lorentztransformation. The slope in-dicated in the figure gives therelative velocity of the two framesof reference. Events A and B that

were simultaneous in frame 1are not simultaneous in frame 2,

where event A occurs to the rightof the t = 0 line represented bythe left edge of the grid, but eventB occurs to its left.

As in our discussion of the Galilean transformation, slopes are in-terpreted as velocities, and the slope of the near-horizontal lines infigure l is interpreted as the relative velocity of the two observers.The difference between the Galilean version and the relativistic oneis that now there is smooshing happening from the other side aswell. Lines that were vertical in the original grid, representing si-

multaneous events, now slant over to the right. This tells us that, asrequired by property 5, different observers do not agree on whetherevents that occur in different places are simultaneous. The Hafele-Keating experiment tells us that this non-simultaneity effect is fairlysmall, even when the velocity is as big as that of a passenger jet,and this is what we would have anticipated by the correspondenceprinciple. The way that this is expressed in the graph is that if wepick the time unit to be the second, then the distance unit turns outto be hundreds of thousands of miles. In these units, the velocityof a passenger jet is an extremely small number, so the slope v in afigure like l is extremely small, and the amount of distortion is tiny— it would be much too small to see on this scale.

The only thing left to determine about the Lorentz transformationis the size of the transformed parallelogram relative to the size of the original one. Although the drawing of the hands in figure kmay suggest that the grid deforms like a framework made of rigidcoat-hanger wire, that is not the case. If you look carefully at thefigure, you’ll see that the edges of the smooshed parallelogram areactually a little longer than the edges of the original rectangle. Infact what stays the same is not lengths but areas, as proved in thecaption to figure m.


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n / The clock is at rest in t

original frame of reference, andmeasures a time interval t . In tnew frame of reference, the timinterval is greater by a factor thwe notate as γ.

m / Proof that Lorentz transformations don’t change area: We first subject a square to a transformation

with velocity v , and this increases its area by a factor R (v ), which we want to prove equals 1. We chop theresulting parallelogram up into little squares and finally apply a −v transformation; this changes each littlesquare’s area by a factor R (−v ), so the whole figure’s area is also scaled by R (−v ). The final result is to restorethe square to its original shape and area, so R (v )R (−v ) = 1. But R (v ) = R (−v ) by property 2 of spacetime on

page 626, which states that all directions in space have the same properties, so R (v ) = 1.

The γ factor

Figure l showed us that observers in different frames disagree onwhether different events are simultaneous. This is an indication thattime is not absolute, so we shouldn’t be surprised that time’s rateof flow is also different for different observers. We use the symbol γ (Greek letter gamma) defined in the figure n to measure this unequalrate of flow. With a little algebra and geometry (homework problem2, page 642), one can use the equal-area property to show that thisratio is given by

γ =1√

1 − v2.

If you’ve had good training in physics, the first thing you probablythink when you look at this equation is that it must be nonsense,because its units don’t make sense. How can we take somethingwith units of velocity squared, and subtract it from a unitless 1?But remember that this is expressed in our special relativistic units,in which the same units are used for distance and time. In thissystem, velocities are always unitless. This sort of thing happensfrequently in physics. For instance, before James Joule discoveredconservation of energy, nobody knew that heat and mechanical en-

ergy were different forms of the same thing, so instead of measuringthem both in units of joules as we would do now, they measuredheat in one unit (such as calories) and mechanical energy in another(such as foot-pounds). In ordinary metric units, we just need anextra conversion factor c, and the equation becomes

γ =1

1 − vc

2

.


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The conversion factor c has units of velocity. As argued on p. 592,cause and effect can never be propagated instantaneously; c turnsout to be the specific numerical speed limit on cause and effect. Inparticular, we’ll see in section 24.3 that light travels at c, which hasa numerical value of 3.0 × 108 m/s.

Because γ is always greater than 1, we have the following interpre-tation:

Time dilation

A clock runs fastest in the frame of reference of an observerwho is at rest relative to the clock. An observer in motionrelative to the clock at speed v perceives the clock as runningmore slowly by a factor of γ .

Since the Lorentz transformation treats time and distance entirelysymmetrically, we could just as well have defined γ using the uprightx axis in figure n, and we therefore have a similar interpretation interms of space:

Length contraction

A meter-stick appears longest to an observer who is at restrelative to it. An observer moving relative to the meter-stickat v observes the stick to be shortened by a factor of γ .

self-check A

What is γ when v = 0? What does this mean? Answer, p. 969

The correspondence principle example 1

The correspondence principle requires that γ be close to 1 for the

velocities much less than c encountered in everyday life. Let’s

explicitly find the amount by which γ differs from 1, when v is

small. Let γ = 1 + . The definition of γ gives 1 = γ2(1 − v 2/c 2),so 1 = (1 + 2 + 2)(1 − v 2/c 2) ≈ 1 + 2 − v 2/c 2, where the

approximation comes from discarding very small terms such as

2 and v 2/c 2. We find = v 2/2c 2. As expected, this will be

small when v is small compared to c .


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p / Example 2: Muons a

erated to nearly c undradioactive decay much mslowly than they would accing to an observer at rest respect to the muons. The

two data-points (unfilled cirwere subject to large systemerrors.

o / Apparatus used for the tof relativistic time dilation d

scribed in example 2. The promnent black and white blocks alarge magnets surrounding a ccular pipe with a vacuum insid(c) 1974 by CERN.

Large time dilation example 2

The time dilation effect in the Hafele-Keating experiment was very

small. If we want to see a large time dilation effect, we can’t doit with something the size of the atomic clocks they used; the ki-

netic energy would be greater than the total megatonnage of all

the world’s nuclear arsenals. We can, however, accelerate sub-

atomic particles to speeds at which γ is large. For experimental

particle physicists, relativity is something you do all day before

heading home and stopping off at the store for milk. An early, low-

precision experiment of this kind was performed by Rossi and Hall

in 1941, using naturally occurring cosmic rays. Figure o shows a

1974 experiment2 of a similar type which verified the time dilation

predicted by relativity to a precision of about one part per thou-

sand.Particles called muons (named after the Greek letter µ , “myoo”)

were produced by an accelerator at CERN, near Geneva. A muon

is essentially a heavier version of the electron. Muons undergo

radioactive decay, lasting an average of only 2.197 µ s before they

evaporate into an electron and two neutrinos. The 1974 experi-

ment was actually built in order to measure the magnetic proper-

ties of muons, but it produced a high-precision test of time dilation

as a byproduct. Because muons have the same electric charge

as electrons, they can be trapped using magnetic fields. Muons

were injected into the ring shown in figure o, circling around it un-

til they underwent radioactive decay. At the speed at which thesemuons were traveling, they had γ = 29.33, so on the average they

lasted 29.33 times longer than the normal lifetime. In other words,

they were like tiny alarm clocks that self-destructed at a randomly

selected time. Figure p shows the number of radioactive decays

counted, as a function of the time elapsed after a given stream of

muons was injected into the storage ring. The two dashed lines

2Bailey at al., Nucl. Phys. B150(1979) 1


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Example 4: In the garage’sframe of reference, 1, the bus ismoving, and can fit in the garage.In the bus’s frame of reference,the garage is moving, and can’thold the bus.

show the rates of decay predicted with and without relativity. The

relativistic line is the one that agrees with experiment.

An example of length contraction example 3

Figure q shows an artist’s rendering of the length contraction for

the collision of two gold nuclei at relativistic speeds in the RHIC

accelerator in Long Island, New York, which went on line in 2000.

The gold nuclei would appear nearly spherical (or just slightlylengthened like an American football) in frames moving along with

them, but in the laboratory’s frame, they both appear drastically

foreshortened as they approach the point of collision. The later

pictures show the nuclei merging to form a hot soup, in which

experimenters hope to observe a new form of matter.

Example 3: Colliding nuclei showrelativistic length contraction.

The garage paradox example 4

One of the most famous of all the so-called relativity paradoxes

has to do with our incorrect feeling that simultaneity is well de-

fined. The idea is that one could take a schoolbus and drive it at

relativistic speeds into a garage of ordinary size, in which it nor-mally would not fit. Because of the length contraction, the bus

would supposedly fit in the garage. The paradox arises when we

shut the door and then quickly slam on the brakes of the bus.

An observer in the garage’s frame of reference will claim that the

bus fit in the garage because of its contracted length. The driver,

however, will perceive the garage as being contracted and thus

even less able to contain the bus. The paradox is resolved when

we recognize that the concept of fitting the bus in the garage “all

at once” contains a hidden assumption, the assumption that it

makes sense to ask whether the front and back of the bus can

simultaneously be in the garage. Observers in different framesof reference moving at high relative speeds do not necessarily

agree on whether things happen simultaneously. The person in

the garage’s frame can shut the door at an instant he perceives to

be simultaneous with the front bumper’s arrival at the back wall of

the garage, but the driver would not agree about the simultaneity

of these two events, and would perceive the door as having shut

long after she plowed through the back wall.


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Discussion question B


A A person in a spaceship moving at 99.99999999% of the speedof light relative to Earth shines a flashlight forward through dusty air, sothe beam is visible. What does she see? What would it look like to anobserver on Earth?

B A question that students often struggle with is whether time andspace can really be distorted, or whether it just seems that way. Compare

with optical illusions or magic tricks. How could you verify, for instance,that the lines in the figure are actually parallel? Are relativistic effects thesame, or not?

C On a spaceship moving at relativistic speeds, would a lecture seemeven longer and more boring than normal?

D Mechanical clocks can be affected by motion. For example, it was

a significant technological achievement to build a clock that could sailaboard a ship and still keep accurate time, allowing longitude to be deter-mined. How is this similar to or different from relativistic time dilation?

E Figure q from page 634, depicting the collision of two nuclei at theRHIC accelerator, is reproduced below. What would the shapes of the twonuclei look like to a microscopic observer riding on the left-hand nucleus?To an observer riding on the right-hand one? Can they agree on what is

happening? If not, why not — after all, shouldn’t they see the same thing

if they both compare the two nuclei side-by-side at the same instant intime?

Discussion question E: colliding nuclei show relativistic length contraction.

F If you stick a piece of foam rubber out the window of your car whiledriving down the freeway, the wind may compress it a little. Does it makesense to interpret the relativistic length contraction as a type of strainthat pushes an object’s atoms together like this? How does this relate to

discussion question E?


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r / 1. When the circuit is in-

complete, no current flowsthrough the wire, and the magnetis unaffected. It points in thedirection of the Earth’s magneticfield. 2. The circuit is completed,and current flows through thewire. The wire has a strongeffect on the magnet, which turnsalmost perpendicular to it. If theearth’s field could be removed

entirely, the compass would pointexactly perpendicular to the wire;this is the direction of the wire’s

field.

s / A schematic representa-

tion of an unmagnetized material,1, and a magnetized one, 2.

23.2 Magnetic interactions

Think not that I am come to destroy the law, or the prophets: I amnot come to destroy, but to fulfill. Matthew 5:17

At this stage, you understand roughly as much about the classi-fication of interactions as physicists understood around the year

1800. There appear to be three fundamentally different types of interactions: gravitational, electrical, and magnetic. Many typesof interactions that appear superficially to be distinct — stickiness,chemical interactions, the energy an archer stores in a bow — arereally the same: they’re manifestations of electrical interactions be-tween atoms. Is there any way to shorten the list any further? Theprospects seem dim at first. For instance, we find that if we rub apiece of fur on a rubber rod, the fur does not attract or repel a mag-net. The fur has an electric field, and the magnet has a magneticfield. The two are completely separate, and don’t seem to affectone another. Likewise we can test whether magnetizing a piece of iron changes its weight. The weight doesn’t seem to change by anymeasurable amount, so magnetism and gravity seem to be unrelated.

That was where things stood until 1820, when the Danish physicistHans Christian Oersted was delivering a lecture at the Universityof Copenhagen, and he wanted to give his students a demonstrationthat would illustrate the cutting edge of research. He generated acurrent in a wire by making a short circuit across a battery, andheld the wire near a magnetic compass. The ideas was to give anexample of how one could search for a previously undiscovered linkbetween electricity (the electric current in the wire) and magnetism.One never knows how much to believe from these dramatic legends,

but the story is

3

that the experiment he’d expected to turn outnegative instead turned out positive: when he held the wire nearthe compass, the current in the wire caused the compass to twist!

People had tried similar experiments before, but only with staticelectricity, not with a moving electric current. For instance, they hadhung batteries so that they were free to rotate in the earth’s mag-netic field, and found no effect; since the battery was not connectedto a complete circuit, there was no current flowing. With Oersted’sown setup, r, the effect was only produced when the “circuit wasclosed, but not when open, as certain very celebrated physicists invain attempted several years ago.”4

Oersted was eventually led to the conclusion that magnetism wasan interaction between moving charges and other moving charges,i.e., between one current and another. A permanent magnet, he in-

3Oersted’s paper describing the phenomenon says that “The first experimentson the subject . . . were set on foot in the classes for electricity, galvanism, andmagnetism, which were held by me in the winter just past,” but that doesn’t tellus whether the result was really a surprise that occurred in front of his students.

4All quotes are from the 1876 translation are by J.E. Kempe.


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t / Magnetism is an interation between moving chargand moving charges. The movicharges in the wire attract tmoving charges in the beam charged particles in the vacuutube.

ferred, contained currents on a microscopic scale that simply weren’tpractical to measure with an ammeter. Today this seems naturalto us, since we’re accustomed to picturing an atom as a tiny solarsystem, with the electrons whizzing around the nucleus in circles.As shown in figure s, a magnetized piece of iron is different from anunmagnetized piece because the atoms in the unmagnetized piece

are jumbled in random orientations, whereas the atoms in the mag-netized piece are at least partially organized to face in a certaindirection.

Figure t shows an example that is conceptually simple, but not verypractical. If you try this with a typical vacuum tube, like a TVor computer monitor, the current in the wire probably won’t beenough to produce a visible effect. A more practical method is tohold a magnet near the screen. We still have an interaction betweenmoving charges and moving charges, but the swirling electrons inthe atoms in the magnet are now playing the role played by themoving charges in the wire in figure t. Warning: if you do this,

make sure your monitor has a demagnetizing button! If not, thenyour monitor may be permanently ruined.

Relativity requires magnetism

So magnetism is an interaction between moving charges and mov-ing charges. But how can that be? Relativity tells us that motionis a matter of opinion. Consider figure u. In this figure and infigure v, the dark and light coloring of the particles represents thefact that one particle has positive charge and the other negative.Observer u/2 sees the two particles as flying through space side byside, so they would interact both electrically (simply because they’recharged) and magnetically (because they’re charges in motion). Butan observer moving along with them, u/1, would say they were bothat rest, and would expect only an electrical interaction. This seemslike a paradox. Magnetism, however, comes not to destroy relativitybut to fulfill it. Magnetic interactions must exist according to thetheory of relativity. To understand how this can be, consider howtime and space behave in relativity. Observers in different framesof reference disagree about the lengths of measuring sticks and thespeeds of clocks, but the laws of physics are valid and self-consistentin either frame of reference. Similarly, observers in different framesof reference disagree about what electric and magnetic fields thereare, but they agree about concrete physical events. An observer in

frame of reference u/1 says there are electric fields around the par-ticles, and predicts that as time goes on, the particles will begin toaccelerate towards one another, eventually colliding. She explainsthe collision as being due to the electrical attraction between theparticles. A different observer, u/2, says the particles are moving.This observer also predicts that the particles will collide, but ex-plains their motion in terms of both an electric field and a magneticfield. As we’ll see shortly, the magnetic field is required in order

Section 23.2 Magnetic interactions 6

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u / One observer sees an electricfield, while the other sees both anelectric field and a magnetic one.

v / A model of a charged particleand a current-carrying wire,seen in two different frames of

reference. The relativistic lengthcontraction is highly exaggerated.The force on the lone particle ispurely magnetic in 1, and purelyelectric in 2.

to maintain consistency between the predictions made in the twoframes of reference.

To see how this really works out, we need to find a nice simpleexample. An example like figure u is not easy to handle, because inthe second frame of reference, the moving charges create fields thatchange over time at any given location, like when the V-shaped wake

of a speedboat washes over a buoy. Examples like figure t are easier,because there is a steady flow of charges, and all the fields stay thesame over time. Figure v/1 shows a simplified and idealized modelof figure t. The charge by itself is like one of the charged particles inthe vacuum tube beam of figure t, and instead of the wire, we havetwo long lines of charges moving in opposite directions. Note that,as discussed in discussion question C on page 547, the currents of the two lines of charges do not cancel out. The dark and light ballsrepresent particles with opposite charges. Because of this, the totalcurrent in the “wire” is double what it would be if we took awayone line.

As a model of figure t, figure v/1 is partly realistic and partly un-realistic. In a real piece of copper wire, there are indeed chargedparticles of both types, but it turns out that the particles of onetype (the protons) are locked in place, while only some of the othertype (the electrons) are free to move. The model also shows theparticles moving in a simple and orderly way, like cars on a two-laneroad, whereas in reality most of the particles are organized into cop-per atoms, and there is also a great deal of random thermal motion.The model’s unrealistic features aren’t a problem, because the pointof this exercise is only to find one particular situation that showsmagnetic effects must exist based on relativity.

What electrical force does the lone particle in figure v/1 feel? Sincethe density of “traffic” on the two sides of the “road” is equal, thereis zero overall electrical force on the lone particle. Each “car” thatattracts the lone particle is paired with a partner on the other sideof the road that repels it. If we didn’t know about magnetism, we’dthink this was the whole story: the lone particle feels no force at allfrom the wire.

Figure v/2 shows what we’d see if we were observing all this from aframe of reference moving along with the lone charge. Here’s wherethe relativity comes in. Relativity tells us that moving objects ap-pear contracted to an observer who is not moving along with them.Both lines of charge are in motion in both frames of reference, butin frame 1 they were moving at equal speeds, so their contractionswere equal. In frame 2, however, their speeds are unequal. Thedark charges are moving more slowly than in frame 1, so in frame 2they are less contracted. The light-colored charges are moving morequickly, so their contraction is greater now. The “cars” on the twosides of the “road” are no longer paired off, so the electrical forces


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w / Magnetic interactions volving only two particles attime. In these figures, unlfigure v/1, there are electriforces as well as magnetic one

The electrical forces are nshown here. Don’t memorthese rules!

on the lone particle no longer cancel out as they did in v/1. Thelone particle is attracted to the wire, because the particles attract-ing it are more dense than the ones repelling it. Furthermore, theattraction felt by the lone charge must be purely electrical, since thelone charge is at rest in this frame of reference, and magnetic effectsoccur only between moving charges and other moving charges.

Now observers in frames 1 and 2 disagree about many things, butthey do agree on concrete events. Observer 2 is going to see the loneparticle drift toward the wire due to the wire’s electrical attraction,gradually speeding up, and eventually hit the wire. If 2 sees thiscollision, then 1 must as well. But 1 knows that the total electricalforce on the lone particle is exactly zero. There must be some newtype of force. She invents a name for this new type of force: mag-netism. This was a particularly simple example, because the forcewas purely magnetic in one frame of reference, and purely electricalin another. In general, an observer in a certain frame of referencewill measure a mixture of electric and magnetic fields, while an ob-

server in another frame, in motion with respect to the first, saysthat the same volume of space contains a different mixture.

We therefore arrive at the conclusion that electric and magnetic phe-nomena aren’t separate. They’re different sides of the same coin.We refer to electric and magnetic interactions collectively as elec-tromagnetic interactions. Our list of the fundamental interactionsof nature now has two items on it instead of three: gravity andelectromagnetism.

Oersted found that magnetism was an interaction between movingcharges and other moving charges. We can see this in the situationdescribed in figure v/2, in which the result of the argument dependedon the fact that both the lone charge and the charges in the wire weremoving. To see this in a different way, we can apply the result of example 1 on p. 632, that for small velocities the γ factor differs from1 by about v2/2c2. Let the lone charge in figure v/2 have velocity u,the ones in the wire v. As we’ll see on p. 652, velocities in relativemotion don’t exactly add and subtract relativistically, but as longas we assume that u and v are small, the correspondence principleguarantees that they will approximately add and subtract. Then thevelocities in the lone charge’s rest frame, v/2, are approximately 0,v − u, and −v − u. The nonzero charge density of the wire in framev/2 is then proportional to the difference in the length contractions

γ −v−u−γ v−u ≈ 2uv. This depends on the product of the velocities uand v, which is as expected if magnetism is an interaction of movingcharges with moving charges.

The basic rules for magnetic attractions and repulsions, shown infigure w, aren’t quite as simple as the ones for gravity and electric-ity. Rules w/1 and w/2 follow directly from our previous analysis of figure v. Rules 3 and 4 are obtained by flipping the type of charge

Section 23.2 Magnetic interactions 6

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Example 5

that the bottom particle has. For instance, rule 3 is like rule 1,except that the bottom charge is now the opposite type. This turnsthe attraction into a repulsion. (We know that flipping the chargereverses the interaction, because that’s the way it works for elec-tric forces, and magnetic forces are just electric forces viewed in adifferent frame of reference.)

A magnetic weathervane placed near a current. example 5 Figure 23.2.1 shows a magnetic weathervane, consisting of two

charges that spin in circles around the axis of the arrow. (The

magnetic field doesn’t cause them to spin; a motor is needed to

get them to spin in the first place.) Just like the magnetic compass

in figure r, the weathervane’s arrow tends to align itself in the

direction perpendicular to the wire. This is its preferred orientation

because the charge close to the wire is attracted to the wire, while

the charge far from the wire is repelled by it.


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Summary

Notation

γ . . . . . . . . . . an abbreviation for 1/

1 − v2/c2

Summary

Experiments show that space and time do not have the properties

claimed by Galileo and Newton. Time and space as seen by oneobserver are distorted compared to another observer’s perceptions if they are moving relative to each other. This distortion is quantifiedby the factor

γ =1

1 − v2

c2

,

where v is the relative velocity of the two observers, and c is auniversal velocity that is the same in all frames of reference. Lighttravels at c. A clock appears to run fastest to an observer who isnot in motion relative to it, and appears to run too slowly by a

factor of γ to an observer who has a velocity v relative to the clock.Similarly, a meter-stick appears longest to an observer who sees itat rest, and appears shorter to other observers. Time and space arerelative, not absolute.

As a consequence of relativity, we must have not just electrical in-teractions of charges with charges, but also an additional magneticinteraction of moving charges with other moving charges.

Exploring further

Relativity Simply Explained, Martin Gardner. A beautifullyclear, nonmathematical introduction to the subject, with entertain-

ing illustrations. A postscript, written in 1996, follows up on recentdevelopments in some of the more speculative ideas from the 1967edition.

Was Einstein Right? — Putting General Relativity to the

Test, Clifford M. Will. This book makes it clear that generalrelativity is neither a fantasy nor holy scripture, but a scientifictheory like any other.

Summary 6

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Problem 2.

Problems

Key√ A computerized answer check is available online. A problem that requires calculus.

A difficult problem.

1What happens in the equation for γ when you put in a negativenumber for v? Explain what this means physically, and why it makes

sense.

2 In this homework problem, you’ll fill in the steps of the algebrarequired in order to find the equation for γ on page 631. To keep thealgebra simple, let the time t in figure n equal 1, as suggested in thefigure accompanying this homework problem. The original squarethen has an area of 1, and the transformed parallelogram must alsohave an area of 1. (a) Prove that point P is at x = vγ , so that its(t,x) coordinates are (γ , vγ ). (b) Find the (t, x) coordinates of point

Q. (c) Find the length of the short diagonal connecting P and Q.(d) Average the coordinates of P and Q to find the coordinates of the midpoint C of the parallelogram, and then find distance OC. (e)Find the area of the parallelogram by computing twice the area of triangle PQO. [Hint: You can take PQ to be the base of the triangle.](f) Set this area equal to 1 and solve for γ to prove γ = 1/

√1 − v2.

√

3 Astronauts in three different spaceships are communicatingwith each other. Those aboard ships A and B agree on the rate atwhich time is passing, but they disagree with the ones on ship C.

(a) Alice is aboard ship A. How does she describe the motion of herown ship, in its frame of reference?(b) Describe the motion of the other two ships according to Alice.(c) Give the description according to Betty, whose frame of referenceis ship B.(d) Do the same for Cathy, aboard ship C.

4 The Voyager 1 space probe, launched in 1977, is moving fasterrelative to the earth than any other human-made object, at 17,000meters per second.(a) Calculate the probe’s γ .

(b) Over the course of one year on earth, slightly less than one yearpasses on the probe. How much less? (There are 31 million secondsin a year.)

√

5 The earth is orbiting the sun, and therefore is contractedrelativistically in the direction of its motion. Compute the amountby which its diameter shrinks in this direction.


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Exercise 23: Polarization

Apparatus:

calcite (Iceland spar) crystal

polaroid film

1. Lay the crystal on a piece of paper that has print on it. You will observe a double image.See what happens if you rotate the crystal.

Evidently the crystal does something to the light that passes through it on the way from thepage to your eye. One beam of light enters the crystal from underneath, but two emerge fromthe top; by conservation of energy the energy of the original beam must be shared betweenthem. Consider the following three possible interpretations of what you have observed:

(a) The two new beams differ from each other, and from the original beam, only in energy.Their other properties are the same.

(b) The crystal adds to the light some mysterious new property (not energy), which comes intwo flavors, X and Y. Ordinary light doesn’t have any of either. One beam that emerges fromthe crystal has some X added to it, and the other beam has Y.

(c) There is some mysterious new property that is possessed by all light. It comes in two flavors,X and Y, and most ordinary light sources make an equal mixture of type X and type Y light.The original beam is an even mixture of both types, and this mixture is then split up by thecrystal into the two purified forms.

In parts 2 and 3 you’ll make observations that will allow you to figure out which of these iscorrect.

2. Now place a polaroid film over the crystal and see what you observe. What happens whenyou rotate the film in the horizontal plane? Does this observation allow you to rule out any of the three interpretations?

3. Now put the polaroid film under the crystal and try the same thing. Putting together allyour observations, which interpretation do you think is correct?

4. Look at an overhead light fixture through the polaroid, and try rotating it. What do youobserve? What does this tell you about the light emitted by the lightbulb?

5. Now position yourself with your head under a light fixture and directly over a shiny surface,such as a glossy tabletop. You’ll see the lamp’s reflection, and the light coming from the lamp

Exercise 23: Polarization 6

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to your eye will have undergone a reflection through roughly a 180-degree angle (i.e. it verynearly reversed its direction). Observe this reflection through the polaroid, and try rotating it.Finally, position yourself so that you are seeing glancing reflections, and try the same thing.Summarize what happens to light with properties X and Y when it is reflected. (This is theprinciple behind polarizing sunglasses.)


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a / Breaking a bar magnet in hdoesn’t create two monopoles

creates two smaller dipoles.

b / An explanation at the atomlevel.

Chapter 24

Electromagnetism24.1 The magnetic field

No magnetic monopoles

If you could play with a handful of electric dipoles and a handful of bar magnets, they would appear very similar. For instance, a pairof bar magnets wants to align themselves head-to-tail, and a pair of electric dipoles does the same thing. (It is unfortunately not thateasy to make a permanent electric dipole that can be handled like

this, since the charge tends to leak.)You would eventually notice an important difference between thetwo types of ob jects, however. The electric dipoles can be brokenapart to form isolated positive charges and negative charges. Thetwo-ended device can be broken into parts that are not two-ended.But if you break a bar magnet in half, a, you will find that you havesimply made two smaller two-ended objects.

The reason for this behavior is not hard to divine from our micro-scopic picture of permanent iron magnets. An electric dipole hasextra positive “stuff” concentrated in one end and extra negative inthe other. The bar magnet, on the other hand, gets its magnetic

properties not from an imbalance of magnetic “stuff” at the twoends but from the orientation of the rotation of its electrons. Oneend is the one from which we could look down the axis and see theelectrons rotating clockwise, and the other is the one from whichthey would appear to go counterclockwise. There is no differencebetween the “stuff” in one end of the magnet and the other, b.

Nobody has ever succeeded in isolating a single magnetic pole. Intechnical language, we say that magnetic monopoles do not seem toexist. Electric monopoles do exist — that’s what charges are.

Electric and magnetic forces seem similar in many ways. Both

act at a distance, both can be either attractive or repulsive, andboth are intimately related to the property of matter called charge.(Recall that magnetism is an interaction between moving charges.)Physicists’s aesthetic senses have been offended for a long time be-cause this seeming symmetry is broken by the existence of elec-tric monopoles and the absence of magnetic ones. Perhaps someexotic form of matter exists, composed of particles that are mag-netic monopoles. If such particles could be found in cosmic rays

6

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The unit of magnetic field,the tesla, is named after Serbian-American inventor Nikola Tesla.

A standard dipole made from

a square loop of wire shortingacross a battery. It acts very

much like a bar magnet, but itsstrength is more easily quantified.

A dipole tends to align itselfto the surrounding magnetic field.

or moon rocks, it would be evidence that the apparent asymmetrywas only an asymmetry in the composition of the universe, not inthe laws of physics. For these admittedly subjective reasons, therehave been several searches for magnetic monopoles. Experimentshave been performed, with negative results, to look for magneticmonopoles embedded in ordinary matter. Soviet physicists in the

1960s made exciting claims that they had created and detected mag-netic monopoles in particle accelerators, but there was no successin attempts to reproduce the results there or at other accelerators.The most recent search for magnetic monopoles, done by reanalyz-ing data from the search for the top quark at Fermilab, turned upno candidates, which shows that either monopoles don’t exist innature or they are extremely massive and thus hard to create inaccelerators.

Definition of the magnetic field

Since magnetic monopoles don’t seem to exist, it would not make

much sense to define a magnetic field in terms of the force on atest monopole. Instead, we follow the philosophy of the alternativedefinition of the electric field, and define the field in terms of thetorque on a magnetic test dipole. This is exactly what a magneticcompass does: the needle is a little iron magnet which acts like amagnetic dipole and shows us the direction of the earth’s magneticfield.

To define the strength of a magnetic field, however, we need someway of defining the strength of a test dipole, i.e., we need a definitionof the magnetic dipole moment. We could use an iron permanentmagnet constructed according to certain specifications, but such an

object is really an extremely complex system consisting of manyiron atoms, only some of which are aligned. A more fundamentalstandard dipole is a square current loop. This could be little resistivecircuit consisting of a square of wire shorting across a battery.

We will find that such a loop, when placed in a magnetic field,experiences a torque that tends to align plane so that its face pointsin a certain direction. (Since the loop is symmetric, it doesn’t careif we rotate it like a wheel without changing the plane in which itlies.) It is this preferred facing direction that we will end up definingas the direction of the magnetic field.

Experiments show if the loop is out of alignment with the field,

the torque on it is proportional to the amount of current, and alsoto the interior area of the loop. The proportionality to currentmakes sense, since magnetic forces are interactions between movingcharges, and current is a measure of the motion of charge. Theproportionality to the loop’s area is also not hard to understand,because increasing the length of the sides of the square increasesboth the amount of charge contained in this circular “river” and

646 Chapter 24 Electromagnetism

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c / The magnetic field patteof a bar magnet. This picture wmade by putting iron filings a piece of paper, and bringia bar magnet up underneathNote how the field pattern passacross the body of the magn

forming closed loops, as in figud/2. There are no sources sinks.

d / Electric fields, 1, have sourcand sinks, but magnetic fields,don’t.

the amount of leverage supplied for making torque. Two separatephysical reasons for a proportionality to length result in an overallproportionality to length squared, which is the same as the area of the loop. For these reasons, we define the magnetic dipole momentof a square current loop as

Dm = IA , [definition of the magneticdipole moment of a square current loop]

We now define the magnetic field in a manner entirely analogous tothe second definition of the electric field:

definition of the magnetic field

The magnetic field vector, B, at any location in space is de-fined by observing the torque exerted on a magnetic test dipoleDmt consisting of a square current loop. The field’s magnitudeis |B| = τ/Dmt sin θ, where θ is the angle by which the loop ismisaligned. The direction of the field is perpendicular to the

loop; of the two perpendiculars, we choose the one such thatif we look along it, the loop’s current is counterclockwise.

We find from this definition that the magnetic field has units of N·m/A·m2 = N/A·m. This unwieldy combination of units is abbre-viated as the tesla, 1 T = 1 N/A·m. Refrain from memorizing thepart about the counterclockwise direction at the end; in section 24.5we’ll see how to understand this in terms of more basic principles.

The nonexistence of magnetic monopoles means that unlike an elec-tric field, d/1, a magnetic one, d/2, can never have sources or sinks.The magnetic field vectors lead in paths that loop back on them-

selves, without ever converging or diverging at a point.

Section 24.1 The magnetic field 6

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24.2 Calculating magnetic fields and forces

Magnetostatics

Our study of the electric field built on our previous understandingof electric forces, which was ultimately based on Coulomb’s law forthe electric force between two point charges. Since magnetism is

ultimately an interaction between currents, i.e., between movingcharges, it is reasonable to wish for a magnetic analog of Coulomb’slaw, an equation that would tell us the magnetic force between anytwo moving point charges.

Such a law, unfortunately, does not exist. Coulomb’s law describesthe special case of electrostatics: if a set of charges is sitting aroundand not moving, it tells us the interactions among them. Coulomb’slaw fails if the charges are in motion, since it does not incorporateany allowance for the time delay in the outward propagation of achange in the locations of the charges.

A pair of moving point charges will certainly exert magnetic forces

on one another, but their magnetic fields are like the v-shaped bowwaves left by boats. Each point charge experiences a magnetic fieldthat originated from the other charge when it was at some previousposition. There is no way to construct a force law that tells us theforce between them based only on their current positions in space.

There is, however, a science of magnetostatics that covers a greatmany important cases. Magnetostatics describes magnetic forcesamong currents in the special case where the currents are steadyand continuous, leading to magnetic fields throughout space thatdo not change over time.

The magnetic field of a long, straight wire is one example that we cansay something about without resorting to fancy mathematics. Wesaw in examples 4 on p. 599 and 14 on p. 611 that the electric fieldof a uniform line of charge is E = 2kq/Lr, where r is the distancefrom the line and q/L is the charge per unit length. In a frameof reference moving at velocity v parallel to the line, this electricfield will be observed as a combination of electric and magneticfields. It therefore follows that the magnetic field of a long, straight,current-carrying wire must be proportional to 1/r. We also expectthat it will be proportional to the Coulomb constant, which sets thestrength of electric and magnetic interactions, and to the current I in

the wire. The complete expression turns out to be B = (k/c

2

)(2I/r).This is identical to the expression for E except for replacement of q/L with I and an additional factor of 1/c2. The latter occursbecause magnetism is a purely relativistic effect, and the relativisticlength contraction depends on v2/c2.

648 Chapter 24 Electromagnetism

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e / Some magnetic fields.

Figure e shows the equations for some of the more commonly en-countered configurations, with illustrations of their field patterns.They all have a factor of k/c2 in front, which shows that magnetismis just electricity (k) seen through the lens of relativity (1/c2). Aconvenient feature of SI units is that k/c2 has a numerical value of exactly 10−7, with units of N/A2.

Field created by a long, straight wire carrying current I:

B =k

c 2· 2I

r

Here r is the distance from the center of the wire. The field vectorstrace circles in planes perpendicular to the wire, going clockwise whenviewed from along the direction of the current.

Field created by a single circular loop of current: The field vectors form a dipole-like pattern, coming through the loopand back around on the outside. Each oval path traced out by the fieldvectors appears clockwise if viewed from along the direction the currentis going when it punches through it. There is no simple equation for afield at an arbitrary point in space, but for a point lying along the central

axis perpendicular to the loop, the field is

B =k

c 2· 2π Ib 2

b 2 + z 2

−3/2

,

where b is the radius of the loop and z is the distance of the point fromthe plane of the loop.

Field created by a solenoid (cylindrical coil): The field pattern is similar to that of a single loop, but for a long solenoidthe paths of the field vectors become very straight on the inside of thecoil and on the outside immediately next to the coil. For a sufficientlylong solenoid, the interior field also becomes very nearly uniform, witha magnitude of

B =

k

c 2 · 4π IN / ,

where N is the number of turns of wire and is the length of the solenoid.The field near the mouths or outside the coil is not constant, and ismore difficult to calculate. For a long solenoid, the exterior field is muchsmaller than the interior field.

Don’t memorize the equations!

Section 24.2 Calculating magnetic fields and forces 6

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Force on a charge moving through a magnetic field

We now know how to calculate magnetic fields in some typical sit-uations, but one might also like to be able to calculate magneticforces, such as the force of a solenoid on a moving charged particle,or the force between two parallel current-carrying wires.

We will restrict ourselves to the case of the force on a charged par-ticle moving through a magnetic field, which allows us to calculatethe force between two objects when one is a moving charged parti-cle and the other is one whose magnetic field we know how to find.An example is the use of solenoids inside a TV tube to guide theelectron beam as it paints a picture.

Experiments show that the magnetic force on a moving chargedparticle has a magnitude given by

|F| = q|v||B| sin θ ,

where v is the velocity vector of the particle, and θ is the angle be-

tween the v and B vectors. Unlike electric and gravitational forces,magnetic forces do not lie along the same line as the field vector.The force is always perpendicular to both v and B. Given two vec-tors, there is only one line perpendicular to both of them, so theforce vector points in one of the two possible directions along thisline. For a positively charged particle, the direction of the forcevector can be found as follows. First, position the v and B vectorswith their tails together. The direction of F is such that if you sightalong it, the B vector is clockwise from the v vector; for a nega-tively charged particle the direction of the force is reversed. Notethat since the force is perpendicular to the particle’s motion, themagnetic field never does work on it.

Magnetic levitation example 1

In figure 24.2.2, a small, disk-shaped permanent magnet is stuck

on the side of a battery, and a wire is clasped loosely around the

battery, shorting it. A large current flows through the wire. The

electrons moving through the wire feel a force from the magnetic

field made by the permanent magnet, and this force levitates the

wire.

From the photo, it’s possible to find the direction of the magnetic

field made by the permanent magnet. The electrons in the copper

wire are negatively charged, so they flow from the negative (flat)

terminal of the battery to the positive terminal (the one with thebump, in front). As the electrons pass by the permanent magnet,

we can imagine that they would experience a field either toward

the magnet, or away from it, depending on which way the magnet

was flipped when it was stuck onto the battery. Imagine sighting

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