mit8 02sc notes

Module 1: Fields1

Table of Contents 1.1 Action at a Distance versus Field Theory ................................................................ 1-2

1.2 Scalar Fields ......................................................................................................... 1-3

Example 1.1: Half-Frozen /Half-Baked Planet ........................................................ 1-51.2.1 Representations of a Scalar Field................................................................ 1-5

1.3 Vector Fields........................................................................................................ 1-7

1.4 Fluid Flow............................................................................................................ 1-8

1.4.1 Sources and Sinks (link) ............................................................................. 1-81.4.2 Circulations (link) ..................................................................................... 1-111.4.3 Relationship between Fluid Fields and Electromagnetic Fields ............... 1-13

1.5 Gravitational Field ............................................................................................. 1-13

1.6 Electric Fields .................................................................................................... 1-14

1.7 Magnetic Field ................................................................................................... 1-15

1.8 Representations of a Vector Field...................................................................... 1-16

1.8.1 Vector Field Representation ..................................................................... 1-171.8.2 Field Line Representation ......................................................................... 1-181.8.3 Grass Seeds and Iron Filings Representations .......................................... 1-181.8.4 Motion of Electric and Magnetic Field Lines ........................................... 1-19

1.9 Summary ............................................................................................................ 1-20

1.10 Solved Problems ................................................................................................ 1-20

1.10.1 Vector Fields............................................................................................. 1-201.10.2 Scalar Fields .............................................................................................. 1-22

These notes are excerpted Introduction to Electricity and Magnetism by Sen-Ben Liao, Peter Dourmashkin, and John Belcher, Copyright 2004, ISBN 0-536-81207-1.

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1

Fields

1.1 Action at a Distance versus Field Theory

In order therefore to appreciate the requirements of the science [of electromagnetism], the student must make himself familiar with a considerable body of most intricate mathematics, the mere retention of which in the memory materially interferes with further progress

James Clerk Maxwell [1855]

Classical electromagnetic field theory emerged in more or less complete form in 1873 in James Clerk Maxwells A Treatise on Electricity and Magnetism. Maxwell based his theory in large part on the intuitive insights of Michael Faraday. The wide acceptance of Maxwells theory has caused a fundamental shift in our understanding of physical reality. In this theory, electromagnetic fields are the mediators of the interaction between material objects. This view differs radically from the older action at a distance view that preceded field theory.

What is action at a distance? It is a worldview in which the interaction of two material objects requires no mechanism other than the objects themselves and the empty space between them. That is, two objects exert a force on each other simply because they are present. Any mutual force between them (for example, gravitational attraction or electric repulsion) is instantaneously transmitted from one object to the other through empty space. There is no need to take into account any method or agent of transmission of that force, or any finite speed for the propagation of that agent of transmission. This is known as action at a distance because objects exert forces on one another (action) with nothing but empty space (distance) between them. No other agent or mechanism is needed.

Many natural philosophers objected to the action at a distance model because in our everyday experience, forces are exerted by one object on another only when the objects are in direct contact. In the field theory view, this is always true in some sense. That is, objects that are not in direct contact (objects separated by apparently empty space) must exert a force on one another through the presence of an intervening medium or mechanism existing in the space between the objects.

The force between the two objects is transmitted by direct contact from the first object to an intervening mechanism immediately surrounding that object, and then from one element of space to a neighboring element, in a continuous manner, until the force is transmitted to the region of space contiguous to the second object, and thus ultimately to the second object itself.

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Although the two objects are not in direct contact with one another, they are in direct contact with a medium or mechanism that exists between them. The force between the objects is transmitted (at a finite speed) by stresses induced in the intervening space by the presence of the objects. The field theory view thus avoids the concept of action at a distance and replaces it by the concept of action by continuous contact. The contact is provided by a stress, or field, induced in the space between the objects by their presence.

This is the essence of field theory, and is the foundation of all modern approaches to understanding the world around us. Classical electromagnetism was the first field theory. It involves many concepts that are mathematically complex. As a result, even now it is difficult to appreciate. In this first chapter of your introduction to field theory, we discuss what a field is, and how we represent fields. We begin with scalar fields.

1.2 Scalar Fields

A scalar field is a function that gives us a single value of some variable for every point in space. As an example, the image in Figure 1.2.1 shows the nighttime temperatures measured by the Thermal Emission Spectrometer instrument on the Mars Global Surveyor (MGS). The data were acquired during the first 500 orbits of the MGS mapping mission. The coldest temperatures, shown in purple, are 120 C while the warmest, shown in white, are 65 C.

The view is centered on Isidis Planitia (15N, 270W), which is covered with warm material, indicating a sandy and rocky surface. The small, cold (blue) circular region to the right is the area of the Elysium volcanoes, which are covered in dust that cools off rapidly at night. At this season the north polar region is in full sunlight and is relatively warm at night. It is winter in the southern hemisphere and the temperatures are extremely low.

Figure 1.2.1 Nighttime temperature map for Mars

The various colors on the map represent the surface temperature. This map, however, is limited to representing only the temperature on a two-dimensional surface and thus, it does not show how temperature varies as a function of altitude. In principal, a scalar

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field provides values not only on a two-dimensional surface in space but for every point in space.

Figure 1.2.2 illustrates the variation of temperature as a function of height above the surface of the Earth, which is a third dimension which complements the two dimensions shown in Figure 1.2.1.

Figure 1.2.2 Atmospheric temperature variation as a function of altitude above the Earths surface

How do we represent three-dimensional scalar fields? In principle, one could create a three-dimensional atmospheric volume element and color it to represent the temperature variation.

Figure 1.2.3 Spherical coordinates

Another way is to simply represent the temperature variation by a mathematical function. For the Earth we shall use spherical coordinates (r,,) shown in Figure 1.2.3 with the origin chosen to coincide with the center of the Earth. The temperature at any point is characterized by a function T (r, , ) . In other words, the value of this function at the point with coordinates (r,,) is a temperature with given units. The temperature function ( , , )T r is an example of a scalar field. The term scalar implies that temperature at any point is a number rather than a vector (a vector has both magnitude and direction).

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Example 1.1: Half-Frozen /Half-Baked Planet

As an example of a scalar field, consider a planet with an atmosphere that rotates with the same angular frequency about its axis as the planet orbits about a nearby star, i.e., one hemisphere always faces the star. Let R denote the radius of the planet. Use spherical coordinates (r,,) with the origin at the center of the planet, and choose = 2 for the center of the hemisphere facing the star. A simplistic model for the temperature variation at any point is given by

T r e ) (1.2.1)( , , ) = T0 + T1 sin 2 + T2 (1 + sin )

(r R

where T0 , T1 , T2 , and are constants. The dependence on the variable r in the term e (r R ) indicates that the temperature decreases exponentially as we move radially away from the surface of the planet. The dependence on the variable in the term sin2 implies that the temperature decreases as we move toward the poles. Finally, the dependence in the term (1+ sin ) indicates that the temperature decreases as we move away from the center of the hemisphere facing the star.

A scalar field can also be used to describe other physical quantities such as the atmospheric pressure. However, a single number (magnitude) at every point in space is not sufficient to characterize quantities such as the wind velocity since a direction at every point in space is needed as well.

1.2.1 Representations of a Scalar Field

A field, as stated earlier, is a function that has a different value at every point in space. A scalar field is a field for which there is a single number associated with every point in space. We have seen that the temperature of the Earths atmosphere at the surface is an example of a scalar field. Another example is

( ,x y z , ) = 1 1/ 3 (1.2.2) x2 + ( y d+ )2 + z2 x2 + ( y d )2 + z2

This expression defines the value of the scalar function at every point ( ,x y z, ) in space. How do visually represent a scalar field defined by an equation such as Eq. (1.2.2)? Below we discuss three possible representations.

1. Contour Maps

One way is to fix one of our independent variables (z, for example) and then show a contour map for the two remaining dimensions, in which the curves represent lines of constant values of the function . A series of these maps for various (fixed) values of z

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then will give a feel for the properties of the scalar function. We show such a contour map in the xy-plane at z = 0 for Eq. (1.2.2), namely,

( ,x y,0) = 1 1/ 3 (1.2.3) x2 + ( y d )2 x2 + ( y )2+ d

Various contour levels are shown in Figure 1.2.4, for d = 1, labeled by the value of the function at that level.

Figure 1.2.4 A contour map in the xy-plane of the scalar field given by Eq. (1.2.3).

2. Color-Coding

Another way we can represent the values of the scalar field is by color-coding in two dimensions for a fixed value of the third. This was the scheme used for illustrating the temperature fields in Figures 1.2.1 and 1.2.2. In Figure 1.2.5 a similar map is shown for the scalar field (x, y,0) . Different values of (x, y,0) are characterized by different colors in the map.

Figure 1.2.5 A color-coded map in the xy-plane of the scalar field given by Eq. (1.2.3).

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3. Relief Maps

A third way to represent a scalar field is to fix one of the dimensions, and then plot the value of the function as a height versus the remaining spatial coordinates, say x and y, that is, as a relief map. Figure 1.2.6 shows such a map for the same function ( , x y,0) .

Figure 1.2.6 A relief map of the scalar field given by Eq. (1.2.3).

1.3 Vector Fields

A vector is a quantity which has both a magnitude and a direction in space. Vectors are used to describe physical quantities such as velocity, momentum, acceleration and force, associated with an object. However, when we try to describe a system which consists of a large number of objects (e.g., moving water, snow, rain,) we need to assign a vector to each individual object.

As an example, lets consider falling snowflakes, as shown in Figure 1.3.1. As snow falls, each snowflake moves in a specific direction. The motion of the snowflakes can be analyzed by taking a series of photographs. At any instant in time, we can assign, to each snowflake, a velocity vector which characterizes its movement.

Figure 1.3.1 Falling snow.

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The falling snow is an example of a collection of discrete bodies. On the other hand, if we try to analyze the motion of continuous bodies such as fluids, a velocity vector then needs to be assigned to every point in the fluid at any instant in time. Each vector describes the direction and magnitude of the velocity at a particular point and time. The collection of all the velocity vectors is called the velocity vector field. An important distinction between a vector field and a scalar field is that the former contains information about both the direction and the magnitude at every point in space, while only a single variable is specified for the latter. An example of a system of continuous bodies is air flow. Figure 1.3.2 depicts a scenario of the variation of the jet stream, which is the wind velocity as a function of position. Note that the value of the height is fixed at 34,000 ft.

Figure 1.3.2 Jet stream with arrows indicating flow velocity. The streamlines are formed by joining arrows from head to tail.

1.4 Fluid Flow

1.4.1 Sources and Sinks (link) r

In general, a vector field F( , , )x y z can be written as

r(x, y z = ( , , )i + ( , , ) j + F x y z ( , , ) F , ) F x y z F x y z k (1.4.1)x y z

where the components are scalar fields. Below we use fluids to examine the properties associated with a vector field since fluid flows are the easiest vector fields to visualize.

In Figure 1.4.1 we show physical examples of a fluid flow field, where we represent the fluid by a finite number of particles to show the structure of the flow. In Figure1.4.1(a),

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particles (fluid elements) appear at the center of a cone (a source) and then flow downward under the effect of gravity. That is, we create particles at the origin, and they subsequently flow away from their creation point. We also call this a diverging flow, since the particles appear to diverge from the creation point. Figure 1.4.1(b) is the converse of this, a converging flow, or a sink of particles.

Figure 1.4.1 (a) An example of a source of particles and the flow associated with a source, (b) An example of a sink of particles and the flow associated with a sink.

Another representation of a diverging flow is in depicted in Figure 1.4.2.

Figure 1.4.2 Representing the flow field associated with a source using textures.

Here the direction of the flow is represented by a texture pattern in which the direction of correlation in the texture is along the field direction.

Figure 1.4.3(a) shows a source next to a sink of lesser magnitude, and Figure 1.4.3(b) shows two sources of unequal strength.

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Figure 1.4.3 The flow fields associated with (a) a source (lower) and a sink (upper) where the sink is smaller than the source, and (b) two sources of unequal strength.

Finally, in Figure 1.4.4, we illustrate a constant downward flow interacting with a diverging flow (source). The diverging flow is able to make some headway upwards against the downward constant flow, but eventually turns and flows downward, overwhelmed by the strength of the downward flow.

Figure 1.4.4 A constant downward flow interacting with a diverging flow (source).

In the language of vector calculus, we represent the flow field of a fluid by

v r = vx i + vy j + vzk (1.4.2)

A point ( ,x y z, ) is a source if the divergence of v r( ,x y z, ) is greater than zero. That is,

r vx vy vz ( , , ) + + > 0 (1.4.3)v x y z = x y z

where

= x

i + y

k + z

k (1.4.4)

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x y z r(x y zis the del operator. On the other hand, ( , , ) is a sink if the divergence of v , , ) is less than zero. When v r( , x y z ) = 0 x y z) is neither a source nor a , , then the point ( , , sink. A fluid whose flow field has zero divergence is said to be incompressible.

1.4.2 Circulations (link)

A flow field which is neither a source nor a sink may exhibit another class of behavior - circulation. In Figure 1.4.5(a) we show a physical example of a circulating flow field where particles are not created or destroyed (except at the beginning of the animation), but merely move in circles. The purely circulating flow can also be represented by textures, as shown in Figure 1.4.5(b).

Figure 1.4.5 (a) An example of a circulating fluid. (b) Representing a circulating flow using textures.

A flow field can have more than one system of circulation centered about different points in space. In Figure 1.4.6(a) we show a flow field with two circulations. The flows are in opposite senses, and one of the circulations is stronger than the other. In Figure 1.4.6(b) we have the same situation, except that now the two circulations are in the same sense.

Figure 1.4.6 A flow with two circulation centers with (a) opposite directions of circulation. (b) the same direction of circulation

In Figure 1.4.7, we show a constant downward flow interacting with a counter-clockwise circulating flow. The circulating flow is able to make some headway against the

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downward constant flow, but eventually is overwhelmed by the strength of the downward flow.

Figure 1.4.7 A constant downward flow interacting with a counter-clockwise circulating flow.

In the language of vector calculus, the flows shown in Figures 1.4.5 through 1.4.7 are said to have a non-zero curl, but zero divergence. In contrast, the flows shown in Figures 1.4.2 through 1.4.4 have a zero curl (they do not move in circles) and a non-zero divergence (particles are created or destroyed).

Finally, in Figure 1.4.8, we show a fluid flow field that has both a circulation and a divergence (both the divergence and the curl of the vector field are non-zero). Any vector field can be written as the sum of a curl-free part (no circulation) and a divergence-free part (no source or sink). We will find in our study of electrostatics and magnetostatics that the electrostatic fields are curl free (e.g. they look like Figures 1.4.2 through 1.4.4) and the magnetic fields are divergence free (e.g. they look like Figures 1.4.5 and 1.4.6). Only when dealing with time-varying situations will we encounter electric fields that have both a divergence and a curl. Figure 1.4.8 depicts a field whose curl and divergence are non-vanishing. As far as we know even in time-varying situations magnetic fields always remain divergence-free. Therefore, magnetic fields will always look like the patterns shown in Figures 1.4.5 through 1.4.7.

Figure 1.4.8 A flow field that has both a source (divergence) and a circulation (curl).

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1.4.3 Relationship between Fluid Fields and Electromagnetic Fields

Vector fields that represent fluid flow have an immediate physical interpretation: the vector at every point in space represents a direction of motion of a fluid element, and we can construct animations of those fields, as above, which show that motion. A more general vector field, for example the electric and magnetic fields discussed below, do not have that immediate physical interpretation of a flow field. There is no flow of a fluid along an electric field or magnetic field.

However, even though the vectors in electromagnetism do not represent fluid flow, we carry over many of the terms we use to describe fluid flow to describe electromagnetic fields as well. For example we will speak of the flux (flow) of the electric field through a surface. If we were talking about fluid flow, flux would have a well-defined physical meaning, in that the flux would be the amount of fluid flowing across a given surface per unit time. There is no such meaning when we talk about the flux of the electric field through a surface, but we still use the same term for it, as if we were talking about fluid flow. Similarly we will find that magnetic vector field exhibit patterns like those shown above for circulating flows, and we will sometimes talk about the circulation of magnetic fields. But there is no fluid circulating along the magnetic field direction.

We use much of the terminology of fluid flow to describe electromagnetic fields because it helps us understand the structure of electromagnetic fields intuitively. However, we must always be aware that the analogy is limited.

1.5 Gravitational Field

The gravitational field of the Earth is another example of a vector field which can be used to describe the interaction between a massive object and the Earth. According to Newtons universal law of gravitation, the gravitational force between two masses m and M is given by

F r

g = GM

2

m r (1.5.1)r

where r is the distance between the two masses and r is the unit vector located at the position of m that points from M towards m . The constant of proportionality is the gravitational constant G = 6.671011 N m 2 / kg 2 . Notice that the force is always attractive, with its magnitude being proportional to the inverse square of the distance between the masses.

As an example, if M is the mass of the Earth, the gravitational field g r at a point P in space, defined as the gravitational force per unit mass, can be written as

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r

g r = lim Fg = G M 2 r (1.5.2)m0 m r

From the above expression, we see that the field is radial and points toward the center of the Earth, as shown in Figure 1.5.1.

Figure 1.5.1 Gravitational field of the Earth.

Near the Earths surface, the gravitational field g r is approximately constant: g r = gr , where

g G M 2 9.8 m/s 2= (1.5.3)

RE

and RE is the radius of Earth. The gravitational field near the Earths surface is depicted in Figure 1.5.2.

Figure 1.5.2 Uniform gravitational field near the surface of the Earth.

Notice that a mass in a constant gravitational field does not necessarily move in the direction of the field. This is true only when its initial velocity is in the same direction as the field. On the other hand, if the initial velocity has a component perpendicular to the gravitational field, the trajectory will be parabolic.

1.6 Electric Fields

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The interaction between electric charges at rest is called the electrostatic force. However, unlike mass in gravitational force, there are two types of electric charge: positive and negative. Electrostatic force between charges falls off as the inverse square of their distance of separation, and can be either attractive or repulsive. Electric charges exert forces on each other in a manner that is analogous to gravitation. Consider an object which has charge Q . A test charge that is placed at a point P a distance r from Q will experience a Coulomb force:

F r

e = ke Qq

2 r (1.6.1)r

where r is the unit vector that points from Q to q . The constant of proportionality k 9 2 2= 9.010 N m / C is called the Coulomb constant. The electric field at P is defined e as

r

E r

= lim Fe = ke Q r (1.6.2)

q0 q r 2

The SI unit of electric field is newtons/coulomb (N/C) . If Q is positive, its electric field points radially away from the charge; on the other hand, the field points radially inward ifQ is negative (Figure 1.6.1). In terms of the field concept, we may say that the charge

r r r Q creates an electric field E which exerts a force Fe = qE on q.

Figure 1.6.1 Electric field for positive and negative charges

1.7 Magnetic Field

Magnetic field is another example of a vector field. The most familiar source of magnetic fields is a bar magnet. One end of the bar magnet is called the North pole and the other, the South pole. Like poles repel while opposite poles attract (Figure 1.7.1).

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Figure 1.7.1 Magnets attracting and repelling

If we place some compasses near a bar magnet, the needles will align themselves along the direction of the magnetic field, as shown in Figure 1.7.2.

Figure 1.7.2 Magnetic field of a bar magnet

The observation can be explained as follows: A magnetic compass consists of a tiny bar magnet that can rotate freely about a pivot point passing through the center of the magnet. When a compass is placed near a bar magnet which produces an external magnetic field, it experiences a torque which tends to align the north pole of the compass with the external magnetic field.

The Earths magnetic field behaves as if there were a bar magnet in it (Figure 1.7.3). Note that the south pole of the magnet is located in the northern hemisphere.

Figure 1.7.3 Magnetic field of the Earth

1.8 Representations of a Vector Field

How do we represent vector fields? Since there is much more information (magnitude and direction) in a vector field, our visualizations are correspondingly more complex when compared to the representations of scalar fields.

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Let us introduce an analytic form for a vector field and discuss the various ways that we represent it. Let

r x i + ( y + d ) j + z k 1 x i + ( y d ) j+ z k E( ,x y z , ) = 2 2 2 3/ 2 2 2 2 3/ 2 (1.8.1)[x + ( y + d ) + z ] 3 [x + ( y d ) + z ]

This field is proportional to the electric field of two point charges of opposite signs, with the magnitude of the positive charge three times that of the negative charge. The positive charge is located at (0, d ,0) and the negative charge is located at (0, d ,0) . We discuss how this field is calculated in Section 2.7.

1.8.1 Vector Field Representation

Figure 1.8.1 is an example of a vector field representation of Eq. (1.8.1), in the plane where z = 0. We show the charges that would produce this field if it were an electric field, one positive (the orange charge) and one negative (the blue charge). We will always use this color scheme to represent positive and negative charges.

Figure 1.8.1 A vector field representation of the field of two point charges (link), one negative and one positive, with the magnitude of the positive charge three times that of the negative charge. In the applet linked to this figure, one can vary the magnitude of the charges and the spacing of the vector field grid, and move the charges about.

In the vector field representation, we put arrows representing the field direction on a rectangular grid. The direction of the arrow at a given location represents the direction of the vector field at that point. In many cases, we also make the length of the vector proportional to the magnitude of the vector field at that point. But we also may show only the direction with the vectors (that is make all vectors the same length), and color-code the arrows according to the magnitude of the vector. Or we may not give any information about the magnitude of the field at all, but just use the arrows on the grid to indicate the direction of the field at that point.

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Figure 1.8.1 is an example of the latter situation. That is, we use the arrows on the vector field grid to simply indicate the direction of the field, with no indication of the magnitude of the field, either by the length of the arrows or their color. Note that the arrows point away from the positive charge (the positive charge is a source for electric field) and towards the negative charge (the negative charge is a sink for electric field).

1.8.2 Field Line Representation

There are other ways to represent a vector field. One of the most common is to draw field lines. Faraday called the field lines for electric field lines of force. To draw a field line, start out at any point in space and move a very short distance in the direction of the local vector field, drawing a line as you do so. After that short distance, stop, find the new direction of the local vector field at the point where you stopped, and begin moving again in that new direction. Continue this process indefinitely. Thereby you construct a line in space that is everywhere tangent to the local vector field. If you do this for different starting points, you can draw a set of field lines that give a good representation of the properties of the vector field. Figure 1.8.2 below is an example of a field line representation for the same two charges we used in Figure 1.8.1.

Figure 1.8.2 A vector field representation of the field of two point charges (link), and a field line representation of the same field. The field lines are everywhere tangent to the local field direction

In summary, the field lines are a representation of the collection of vectors that constitute the field, and they are drawn according to the following rules:

(1) The direction of the field line at any point in space is tangent to the field at that point.

(2) The field lines never cross each other, otherwise there would be two different field directions at the point of intersection.

1.8.3 Grass Seeds and Iron Filings Representations

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The final representation of vector fields is the grass seeds representation or the iron filings representation. For an electric field, this name derives from the fact that if you scatter grass seeds in a strong electric field, they will orient themselves with the long axis of the seed parallel to the local field direction. They thus provide a dense sampling of the shape of the field. Figure 1.8.4 is a grass seeds representation of the electric field for the same two charges in Figures 1.8.1 and 1.8.2.

Figure 1.8.4: A grass seeds representation of the electric field that we considered in Figures 1.8.1 and 1.8.2. In the applet linked to this figure (link), one can generate grass seeds representations for different amounts of charge and different positions.

The local field direction is in the direction in which the texture pattern in this figure is correlated. This grass seeds representation gives by far the most information about the spatial structure of the field.

We will also use this technique to represent magnetic fields, but when used to represent magnetic fields we call it the iron filings representation. This name derives from the fact that if you scatter iron filings in a strong magnetic field, they will orient themselves with their long axis parallel to the local field direction. They thus provide a dense sampling of the shape of the magnetic field.

A frequent question from the student new to electromagnetism is What is between the field lines? Figures 1.8.2 and 1.8.4 make the answer to that question clear. What is between the field lines are more field lines that we have chosen not to draw. The field itself is a continuous feature of the space between the charges.

1.8.4 Motion of Electric and Magnetic Field Lines

In this course we will show the spatial structure of electromagnetic fields using all of the methods discussed above. In addition, for the field line and the grass seeds and iron filings representation, we will frequently show the time evolution of the fields. We do this by having the field lines and the grass seed patterns or iron filings patterns move in the direction of the energy flow in the electromagnetic field at a given point in space.

r r rThe flow is in the direction of , the cross product of the electric field E and the E B

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r r rmagnetic field B , and is perpendicular to both E and B . This is very different from our representation of fluid flow fields above, where the direction of the flow is in the same direction as the velocity field itself. We will discuss the concept on energy flow in electromagnetic fields toward the end of the course.

We adopt this representation for time-changing electromagnetic fields because these fields can both support the flow of energy and can store energy as well. We will discuss quantitatively how to compute this energy flow later, when we discuss the Poynting vector in Chapter 13. For now we simply note that when we animate the motion of the field line or grass seeds or iron filings representations, the direction of the pattern motion indicates the direction in which energy in the electromagnetic field is flowing.

1.9 Summary

In this chapter, we have discussed the concept of fields. A scalar field ( , ,T x y z) is a function on all the coordinates of space. Examples of a scalar field include temperature r and pressure. On the other hand, a vector field F( , , )x y z is a vector each of whose

r components is a scalar field. A vector field F(x, y z, ) has both magnitude and direction at every point ( , x y z, ) in space. Gravitational, electric and magnetic fields are all examples of vector fields.

1.10 Solved Problems

1.10.1 Vector Fields

Make a plot of the following vector fields:

(a) v r = i j3 5

This is an example of a constant vector field in two dimensions. The plot is depicted in Figure 1.10.1:

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Figure 1.10.1

Figure 1.10.2

rr=v r(b)

rv = r 2r

(c)

rr = x i + y j r, v can be written asIn two dimensions, using the Cartesian coordinates where

r r rr x i j+yv = = = 2 3 2 2 3/ 2 ( )+yr r x

r

rThe plot is shown in Figure 1.10.3(a). Both the gravitational field of the Earth g electric field E due to a point charge have the same characteristic behavior as v

and the r

. In three rdimensions where r = x i + y j + zk , the plot looks like that shown in Figure 1.10.3(b).

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(a) (b)

Figure 1.10.3

(d) v r = 3x5 y i + 2y

2 5

x2 j r r

Figure 1.10.4

The plot is characteristic of the electric field due to a point electric dipole located at the origin.

1.10.2 Scalar Fields

Make a plot of the following scalar functions in two dimensions:

1(a) f r( ) = r

In two dimensions, we may write r = 2 2x y+ .

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Figure 1.10.5 can be used to represent the electric potential due to a point charge located at the origin. Notice that the mesh size has been adjusted so that the singularity at r = 0 is not shown.

1 1(b) ( , ) =f x y x2 + ( y 1) 2 x2 + ( y +1) 2

-4 -2

0 2

4 -4

-2

0

2

4

0.2

0.4

0.6

-4-2

02

4 Figure 1.10.5

-4 -2

0 2

4 -4

-2

0

2

4

-0.4 -0.2

0 0.2

0.4

-4-2

02

4 Figure 1.10.6

This plot represents the potential due to a dipole with the positive charge located at y =1and the negative charge at y = 1. Again, singularities at ( , )x y = (0, 1) are not shown.

1-23

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A. Vector Analysis 0

Module 2: Math Review

A. Vector Analysis A.1 Vectors A.1.1 Introduction Certain physical quantities such as mass or the absolute temperature at some point only have magnitude. These quantities can be represented by numbers alone, with the appropriate units, and they are called scalars. There are, however, other physical quantities which have both magnitude and direction; the magnitude can stretch or shrink, and the direction can reverse. These quantities can be added in such a way that takes into account both direction and magnitude. Force is an example of a quantity that acts in a certain direction with some magnitude that we measure in newtons. When two forces act on an object, the sum of the forces depends on both the direction and magnitude of the two forces. Position, displacement, velocity, acceleration, force, momentum and torque are all physical quantities that can be represented mathematically by vectors. We shall begin by defining precisely what we mean by a vector. A.1.2 Properties of a Vector A vector is a quantity that has both direction and magnitude. Let a vector be denoted by the symbol A

. The magnitude of A

is | AA |

. We can represent vectors as geometric objects using arrows. The length of the arrow corresponds to the magnitude of the vector. The arrow points in the direction of the vector (Figure A.1.1).

Figure A.1.1 Vectors as arrows. There are two defining operations for vectors: (1) Vector Addition: Vectors can be added.


Let A

and B

be two vectors. We define a new vector, = +C A B

, the vector addition of A

and B

, by a geometric construction. Draw the arrow that represents A

. Place the tail of the arrow that represents B

at the tip of the arrow for A

as shown in Figure A.1.2(a). The arrow that starts at the tail of A

and goes to the tip of B

is defined to be the vector addition = +C A B

. There is an equivalent construction for the law of vector addition. The vectors A

and B

can be drawn with their tails at the same point. The two vectors form the sides of a parallelogram. The diagonal of the parallelogram corresponds to the vector = +C A B

, as shown in Figure A.1.2(b).

Figure A.1.2 Geometric sum of vectors.

Vector addition satisfies the following four properties: (i) Commutivity: The order of adding vectors does not matter. + = +A B B A

(A.1.1) Our geometric definition for vector addition satisfies the commutivity property (i) since in the parallelogram representation for the addition of vectors, it doesnt matter which side you start with as seen in Figure A.1.3.

Figure A.1.3 Commutative property of vector addition (ii) Associativity: When adding three vectors, it doesnt matter which two you start with


( ) ( )+ + = + +A B C A B C

(A.1.2) In Figure A.1.4(a), we add ( )+ +A B C

, while in Figure A.1.4(b) we add ( )+ +A B C

. We arrive at the same vector sum in either case.

Figure A.1.4 Associative law. (iii) Identity Element for Vector Addition: There is a unique vector, 0

, that acts as an identity element for vector addition. This means that for all vectors A

, + = + =A 0 0 A A

(A.1.3) (iv) Inverse element for Vector Addition: For every vector A

, there is a unique inverse vector ( )1 A A

(A.1.4) such that

( )+ =A A 0 This means that the vector A

has the same magnitude as A

, | | | | A= =A A

, but they point in opposite directions (Figure A.1.5).


Figure A.1.5 additive inverse. (2) Scalar Multiplication of Vectors: Vectors can be multiplied by real numbers. Let A

be a vector. Let

c be a real positive number. Then the multiplication of A

by

c is a new vector which we denote by the symbol cA

. The magnitude of cA

is

c times the magnitude of A

(Figure A.1.6a), cA Ac= (A.1.5) Since

c > 0, the direction of cA

is the same as the direction of A

. However, the direction of c A

is opposite of A

(Figure A.1.6b).

Figure A.1.6 Multiplication of vector A

by (a) 0c > , and (b) 0c < . Scalar multiplication of vectors satisfies the following properties: (i) Associative Law for Scalar Multiplication: The order of multiplying numbers is doesnt matter. Let b and c be real numbers. Then ( ) ( ) ( ) ( )b c bc cb c b= = =A A A A

(A.1.6) (ii) Distributive Law for Vector Addition: Vector addition satisfies a distributive law for multiplication by a number. Let c be a real number. Then ( )c c c+ = +A B A B

(A.1.7) Figure A.1.7 illustrates this property.


Figure A.1.7 Distributive Law for vector addition. (iii) Distributive Law for Scalar Addition: The multiplication operation also satisfies a distributive law for the addition of numbers. Let b and c be real numbers. Then ( )b c b c+ = +A A A

(A.1.8) Our geometric definition of vector addition satisfies this condition as seen in Figure A.1.8.

Figure A.1.8 Distributive law for scalar multiplication (iv) Identity Element for Scalar Multiplication: The number 1 acts as an identity element for multiplication, 1 =A A

(A.1.9) A.1.3 Application of Vectors When we apply vectors to physical quantities its nice to keep in the back of our minds all these formal properties. However from the physicists point of view, we are interested in representing physical quantities such as displacement, velocity, acceleration, force, impulse, momentum, torque, and angular momentum as vectors. We cant add force to velocity or subtract momentum from torque. We must always understand the physical context for the vector quantity. Thus, instead of approaching vectors as formal


mathematical objects we shall instead consider the following essential properties that enable us to represent physical quantities as vectors. (1) Vectors can exist at any point P in space. (2) Vectors have direction and magnitude. (3) Vector Equality: Any two vectors that have the same direction and magnitude are equal no matter where in space they are located. (4) Vector Decomposition: Choose a coordinate system with an origin and axes. We can decompose a vector into component vectors along each coordinate axis. In Figure A.1.9 we choose Cartesian coordinates for the -x y plane (we ignore the z -direction for simplicity but we can extend our results when we need to). A vector A

at P can be decomposed into the vector sum, x y= +A A A

(A.1.10) where xA

is the x -component vector pointing in the positive or negative x -direction,

and yA

is the y -component vector pointing in the positive or negative y -direction (Figure A.1.9).

Figure A.1.9 Vector decomposition (5) Unit vectors: The idea of multiplication by real numbers allows us to define a set of unit vectors at each point in space. We associate to each point P in space, a set of three unit vectors ( ) , ,i j k . A unit vector means that the magnitude is one: 1| |=i , 1| |=j , and

1| |=k . We assign the direction of i to point in the direction of the increasing x -coordinate at the point P . We call i the unit vector at P pointing in the + x -direction. Unit vectors j and k can be defined in a similar manner (Figure A.1.10).


Figure A.1.10 Choice of unit vectors in Cartesian coordinates. (6) Vector Components: Once we have defined unit vectors, we can then define the x -component and y -component of a vector. Recall our vector decomposition,

x y= +A A A

. We can write the x-component vector, xA

, as x xA=A i

(A.1.11) In this expression the term

Ax , (without the arrow above) is called the x-component of the vector A

. The x -component

Ax can be positive, zero, or negative. It is not the magnitude of xA

which is given by 2 1/ 2( )xA . Note the difference between the x -component,

Ax , and

the x -component vector, xA

. In a similar fashion we define the y -component,

Ay , and the z -component,

Az , of the

vector A

y y z z A , A= =A j A k

(A.1.12) A vector A

can be represented by its three components ( )x y zA ,A ,A=A

. We can also write the vector as x y z A A A= + +A i j k

(A.1.13) (7) Magnitude: In Figure A.1.10, we also show the vector components ( )x y zA ,A ,A=A

.

Using the Pythagorean theorem, the magnitude of the A

is, 2 2 2x y zA A A A= + + (A.1.14) (8) Direction: Lets consider a vector ( 0)x yA ,A ,=A

. Since the z -component is zero, the

vector A

lies in the -x y plane. Let

denote the angle that the vector A

makes in the


counterclockwise direction with the positive x -axis (Figure A.1.12). Then the x -component and y -components are cos sinx yA A , A A = = (A.1.15)

Figure A.1.12 Components of a vector in the x-y plane. We can now write a vector in the -x y plane as cos sinA A = +A i j

(A.1.16) Once the components of a vector are known, the tangent of the angle

can be determined by

sin tancos

y

x

A AA A

= = (A.1.17)

which yields

1tan yx

AA

=

(A.1.18)

Clearly, the direction of the vector depends on the sign of xA and yA . For example, if both 0xA > and 0yA > , then 0 / 2 < < , and the vector lies in the first quadrant. If, however, 0xA > and 0yA < , then / 2 0 < < , and the vector lies in the fourth quadrant. (9) Vector Addition: Let A

and B

be two vectors in the x-y plane. Let

A and

B denote the angles that the vectors A

and B

make (in the counterclockwise direction) with the positive x-axis. Then


cos sinA A A A = +A i j

(A.1.19) cos sinB B B B = +B i j

(A.1.20) In Figure A.1.13, the vector addition = +C A B

is shown. Let

C denote the angle that the vector C

makes with the positive x-axis.

Figure A.1.13 Vector addition with components Then the components of C

are x x x y y yC A B , C A B= + = + (A.1.21) In terms of magnitudes and angles, we have

cos cos cossin sin sin

x C A B

y C A B

C C A BC C A B

= = += = +

(A.1.22)

We can write the vector C

as

( ) ( ) (cos sin )x x y y C C A B A B C = + + + = +C i j i j

(A.1.23) A.2 Dot Product (link) A.2.1 Introduction We shall now introduce a new vector operation, called the dot product or scalar product that takes any two vectors and generates a scalar quantity (a number). We shall see that the physical concept of work can be mathematically described by the dot product between the force and the displacement vectors.
http://web.mit.edu/viz/EM/visualizations/vectorfields/DotCrossProduct/DotProduct/dotProd.htm


Let A

and B

be two vectors. Since any two non-collinear vectors form a plane, we define the angle

to be the angle between the vectors A

and B

as shown in Figure A.2.1. Note that

can vary from

0 to

.

Figure A.2.1 Dot product geometry.

A.2.2 Definition The dot product A B

of the vectors A

and B

is defined to be product of the magnitude of the vectors A

and B

with the cosine of the angle

between the two vectors: cosAB =A B

(A.2.1) Where | |A = A

and | |B = B

represent the magnitude of A

and B

respectively. The dot product can be positive, zero, or negative, depending on the value of

cos . The dot product is always a scalar quantity. We can give a geometric interpretation to the dot product by writing the definition as ( cos )A B =A B

(A.2.2) In this formulation, the term cosA is the projection of the vector A

in the direction of the vector B

. This projection is shown in Figure A.2.2a. So the dot product is the product of the projection of the length of A

in the direction of B

with the length of B

. Note that we could also write the dot product as ( cos )A B =A B

(A.2.3) Now the term cosB is the projection of the vector B

in the direction of the vector A

as shown in Figure A.2.2b.From this perspective, the dot product is the product of the projection of the length of B

in the direction of A

with the length of A

.


Figure A.2.2a and A.2.2b Projection of vectors and the dot product. From our definition of the dot product we see that the dot product of two vectors that are perpendicular to each other is zero since the angle between the vectors is / 2 and cos( / 2) 0 = . A.2.3 Properties of Dot Product The first property involves the dot product between a vector cA

where c is a scalar and a vector B

, (1a) ( )c c = A B A B

(A.2.4) The second involves the dot product between the sum of two vectors A

and B

with a vector C

, (2a) ( )+ = + A B C A C B C

(A.2.5) Since the dot product is a commutative operation = A B B A

(A.2.6) the similar definitions hold (1b) ( )c c = A B A B

(A.2.7) (2b) ( ) + = + C A B C A C B

(A.2.8) A.2.4 Vector Decomposition and the Dot Product With these properties in mind we can now develop an algebraic expression for the dot product in terms of components. Lets choose a Cartesian coordinate system with the


vector B

pointing along the positive x -axis with positive x -component

Bx, i.e., xB=B i

. The vector A

can be written as x y zA A A= + +A i j k

(A.2.9)

We first calculate that the dot product of the unit vector i with itself is unity: | || | cos(0) 1 = =i i i i (A.2.10) since the unit vector has magnitude 1| |=i and cos(0) 1= . We note that the same rule applies for the unit vectors in the y and z directions: 1 = =j j k k (A.2.11)

The dot product of the unit vector i with the unit vector j is zero because the two unit vectors are perpendicular to each other:

cos( /2) 0 | || | = =i j i j (A.2.12)

Similarly, the dot product of the unit vector i with the unit vector k , and the unit vector j with the unit vector k are also zero: 0 = =i k j k (A.2.13) The dot product of the two vectors now becomes

( ) property (2a)

( ) ( ) ( ) property (1a) and (1b)

x y z x

x x y x z x

x x y x z x

x x

A A A B

A B A B A B

A B A B A BA B

= + +

= + +

= + +

=

A B i j k i

i i j i k i

i i j i k i

(A.2.14)

This third step is the crucial one because it shows that it is only the unit vectors that undergo the dot product operation. Since we assumed that the vector B

points along the positive x -axis with positive x -component

Bx, our answer can be zero, positive, or negative depending on the x -component of the vector A

. In Figure A.2.3, we show the three different cases.


Figure A.2.3 Dot product that is (a) positive, (b) zero or (c) negative. The result for the dot product can be generalized easily for arbitrary vectors x y z A A A= + +A i j k

(A.2.15) and x y z B B B= + +B i j k

(A.2.16) to yield x x y y z zA B A B A B = + +A B

(A.2.17) A.3 Cross Product (link) We shall now introduce our second vector operation, called the cross product that takes any two vectors and generates a new vector. The cross product is a type of multiplication law that turns our vector space (law for addition of vectors) into a vector algebra (laws for addition and multiplication of vectors). The first application of the cross product will be the physical concept of torque about a point

P which can be described mathematically by the cross product of a vector from

P to where the force acts, and the force vector. A.3.1 Definition: Cross Product Let A

and B

be two vectors. Since any two vectors form a plane, we define the angle

to be the angle between the vectors A

and B

as shown in Figure A.3.2.1. The magnitude of the cross product A B

of the vectors A

and B

is defined to be product of the magnitude of the vectors A

and B

with the sine of the angle

between the two vectors, sinAB =A B

(A.3.1)
http://web.mit.edu/viz/EM/visualizations/vectorfields/DotCrossProduct/CrossProduct/crossProd.htm


where A and B denote the magnitudes of A

and B

, respectively. The angle

between the vectors is limited to the values

0 insuring that

sin 0.

Figure A.3.1 Cross product geometry.

The direction of the cross product is defined as follows. The vectors A

and B

form a plane. Consider the direction perpendicular to this plane. There are two possibilities, as shown in Figure A.3.1. We shall choose one of these two for the direction of the cross product A B

using a convention that is commonly called the right-hand rule. A.3.2 Right-hand Rule for the Direction of Cross Product

The first step is to redraw the vectors A

and B

so that their tails are touching. Then draw an arc starting from the vector A

and finishing on the vector B

. Curl your right fingers the same way as the arc. Your right thumb points in the direction of the cross product

A B

(Figure A.3.2).

Figure A.3.2 Right-Hand Rule. You should remember that the direction of the cross product A B

is perpendicular to the plane formed by A

and B

. We can give a geometric interpretation to the magnitude of the cross product by writing the definition as


( )sinA B =A B

(A.3.2)

The vectors A

and B

form a parallelogram. The area of the parallelogram equals the height times the base, which is the magnitude of the cross product. In Figure A.3.3, two different representations of the height and base of a parallelogram are illustrated. As depicted in Figure A.3.3(a), the term sinB is the projection of the vector B

in the direction perpendicular to the vector A

. We could also write the magnitude of the cross product as ( sin )A B =A B

(A.3.3)

Now the term sinA is the projection of the vector A

in the direction perpendicular to the vector B

as shown in Figure A.3.3(b).

Figure A.3.3 Projection of vectors and the cross product The cross product of two vectors that are parallel (or anti-parallel) to each other is zero since the angle between the vectors is 0 (or ) and sin(0) 0= (or sin( ) 0 = ). Geometrically, two parallel vectors do not have any component perpendicular to their common direction. A.3.3 Properties of the Cross Product (1) The cross product is anti-commutative since changing the order of the vectors cross

product changes the direction of the cross product vector by the right hand rule: = A B B A

(A.3.4)

(2) The cross product between a vector cA

where c is a scalar and a vector B

is ( )c c = A B A B

(A.3.5) Similarly, ( )c c = A B A B

(A.3.6)


(3) The cross product between the sum of two vectors A

and B

with a vectorC

is ( )+ = + A B C A C B C

(A.3.7) Similarly, ( ) + = + A B C A B A C

(A.3.8) A.3.4 Vector Decomposition and the Cross Product We first calculate that the magnitude of cross product of the unit vector i with j :

| | | || | sin 12 = =

i j i j (A.3.9)

since the unit vector has magnitude | | | | 1= =i j and sin( / 2) 1 = . By the right hand rule, the direction of i j is in the +k as shown in Figure A.3.4. Thus =i j k .

Figure A.3.4 Cross product of i j

We note that the same rule applies for the unit vectors in the y and z directions, , = =j k i k i j (A.3.10) Note that by the anti-commutatively property (1) of the cross product, , = = j i k i k j (A.3.11) The cross product of the unit vector i with itself is zero because the two unit vectors are parallel to each other, ( sin(0) 0= ),


| | | || | sin(0) 0 = =i i i i (A.3.12) The cross product of the unit vector j with itself and the unit vector k with itself, are also zero for the same reason. 0 0 , = =j j k k (A.3.13)

With these properties in mind we can now develop an algebraic expression for the cross product in terms of components. Lets choose a Cartesian coordinate system with the vector B

pointing along the positive x-axis with positive x-component

Bx. Then the vectors A

and B

can be written as x y z A A A= + +A i j k

(A.3.14) and xB=B i

(A.3.15)

respectively. The cross product in vector components is ( )x y z x A A A B = + + A B i j k i

(A.3.16)

This becomes, using properties (3) and (2),

( ) ( ) ( ) ( ) ( ) ( )

x x y x z x

x x y x z x

y x z x

A B A B A B

A B A B A B

A B A B

= + +

= + +

= +

A B i i j i k i

i i j i k i

k j

(A.3.17)

The vector component expression for the cross product easily generalizes for arbitrary vectors x y z A A A= + +A i j k

(A.3.18) and

x y z B B B= + +B i j k

(A.3.19)

to yield ( ) ( ) ( )y z z y z x x z x y y xA B A B A B A B A B A B = + + A B i j k

. (A.3.20)

B. Coordinate Systems

B. Coordinate Systems (link) B.1 Cartesian Coordinates A coordinate system consists of four basic elements:

(1) Choice of origin

(2) Choice of axes

(3) Choice of positive direction for each axis

(4) Choice of unit vectors for each axis

We illustrate these elements below using Cartesian coordinates. (1) Choice of Origin Choose an origin

O . If you are given an object, then your choice of origin may coincide with a special point in the body. For example, you may choose the mid-point of a straight piece of wire. (2) Choice of Axis Now we shall choose a set of axes. The simplest set of axes are known as the Cartesian axes, x -axis, y -axis, and the z -axis. Once again, we adapt our choices to the physical object. For example, we select the x -axis so that the wire lies on the x -axis, as shown in Figure B.1.1:

Figure B.1.1 A wire lying along the x-axis of Cartesian coordinates.
http://web.mit.edu/viz/EM/visualizations/vectorfields/CoordinateSystems/coordsystems.htm

B. Coordinate Systems 1

Then each point

P in space our

S can be assigned a triplet of values

(xP, yP,zP ), the Cartesian coordinates of the point

P . The ranges of these values are:

< xP < +,

< yP < +,

< zP < + . The collection of points that have the same the coordinate Py is called a level surface. Suppose we ask what collection of points in our space

S have the same value of Py y= . This is the set of points { }( , , ) such that

Py PS x y z S y y= = . This set

PyS is a plane, the

-x z plane (Figure B.1.2), called a level set for constant Py . Thus, the y -coordinate of any point actually describes a plane of points perpendicular to the y -axis.

Figure B.1.2 Level surface set for constant value Py .

(3) Choice of Positive Direction Our third choice is an assignment of positive direction for each coordinate axis. We shall denote this choice by the symbol + along the positive axis. Conventionally, Cartesian coordinates are drawn with the -x y plane corresponding to the plane of the paper. The horizontal direction from left to right is taken as the positive x -axis, and the vertical direction from bottom to top is taken as the positive y -axis. In physics problems we are free to choose our axes and positive directions any way that we decide best fits a given problem. Problems that are very difficult using the conventional choices may turn out to be much easier to solve by making a thoughtful choice of axes. The endpoints of the wire now have the coordinates ( / 2,0,0)a and ( / 2,0,0)a . (4) Choice of Unit Vectors We now associate to each point

P in space, a set of three unit directions vectors ( )P P P , ,i j k . A unit vector has magnitude one: 1P =i , 1P =j , and 1P =k . We

assign the direction of Pi to point in the direction of the increasing x -coordinate at the


point

P . We define the directions for Pj and Pk in the direction of the increasing y -coordinate and z -coordinate respectively. (Figure B.1.3).

Figure B.1.3 Choice of unit vectors.

B.1.1 Infinitesimal Line Element Consider a small infinitesimal displacement d s between two points P1 and P2 (Figure B.1.4). In Cartesian coordinates this vector can be decomposed into d dx dy dz= + +s i j k (B.1.1)

Figure B.1.4 Displacement between two points

B.1.2 Infinitesimal Area Element An infinitesimal area element of the surface of a small cube (Figure B.1.5) is given by ( )( )dA dx dy= (B.1.2)


Figure B.1.5 Area element for one face of a small cube Area elements are actually vectors where the direction of the vector dA

is perpendicular to the plane defined by the area. Since there is a choice of direction, we shall choose the area vector to always point outwards from a closed surface, defined by the right-hand rule. So for the above, the infinitesimal area vector is d dx dy=A k

(B.1.3)

B.1.3 Infinitesimal Volume Element An infinitesimal volume element (Figure B.1.6) in Cartesian coordinates is given by dV dx dy dz= (B.1.4)

Figure B.1.6 Volume element in Cartesian coordinates.

B.2 Cylindrical Coordinates We first choose an origin and an axis we call the z -axis with unit vector z pointing in the increasing z-direction. The level surface of points such that Pz z= define a plane. We shall choose coordinates for a point P in the plane Pz z= as follows.


The coordinate measures the distance from the z -axis to the point P . Its value ranges from 0 < . In Figure B.2.1 we draw a few contours that have constant values of . These level contours are circles. On the other hand, if z were not restricted to Pz z= , as in Figure B.2.1, the level surfaces for constant values of would be cylinders coaxial with the z-axis.

Figure B.2.1 Level surfaces for the coordinate . Our second coordinate measures an angular distance along the circle. We need to choose some reference point to define the angular coordinate. We choose a reference ray, a horizontal ray starting from the origin and extending to + along the horizontal direction to the right. (In a typical Cartesian coordinate system, our reference ray is the positive x-direction). We define the angle coordinate for the point P as follows. We draw a ray from the origin to P . We define as the angle in the counterclockwise direction between our horizontal reference ray and the ray from the origin to the point P , (see Figure B.2.2):

Figure B.2.2 The angular coordinate All the other points that lie on a ray from the origin to infinity passing through P have the same value of . For any arbitrary point, can take on values from 0 2 < . In Figure B.2.3 we depict other level surfaces for the angular coordinate.


Figure B.2.3 Level surfaces for the angle coordinate.

The coordinates ( , ) in the plane Pz z= are called plane polar coordinates. We choose two unit vectors in the plane at the point P as follows. We choose to point in the direction of increasing , radially away from the z-axis. We choose to point in the direction of increasing . This unit vector points in the counterclockwise direction, tangent to the circle. Our complete coordinate system is shown in Figure B.2.4. This coordinate system is called a cylindrical coordinate system. Essentially we have chosen two directions, radial and tangential in the plane and a perpendicular direction to the plane.

Figure B.2.4 Cylindrical coordinates

When referring to any arbitrary point in the plane, we write the unit vectors as and , keeping in mind that they may change in direction as we move around the plane, keeping z unchanged. If we need to make a reference to this time changing property, we will write the unit vectors as explicit functions of time, ( )t and ( )t . If you are given polar coordinates ( , ) of a point in the plane, the Cartesian coordinates ( , )x y can be determined from the coordinate transformations: cos , sinx y = = (B.2.1)


Conversely, if you are given the Cartesian coordinates ( , )x y , the polar coordinates ( , ) may be represented as 2 2 1 2 1( ) , tan ( / )x y y x = + + = (B.2.2) Note that 0 so you always need to take the positive square root. Note also that tan tan( ) = + . Suppose that 0 2 , then 0x and 0y . Then the point ( , )x y will correspond to the angle + . The unit vectors also are related by the coordinate transformations cos sin , sin cos = + = + i j i j (B.2.3) Similarly, cos sin , sin cos = = +i j (B.2.4) The crucial difference between cylindrical coordinates and Cartesian coordinates involves the choice of unit vectors. Suppose we consider a different point 1P in the plane. The

unit vectors in Cartesian coordinates 1 1 ( , )i j at the point 1P have the same magnitude

and point in the same direction as the unit vectors 2 2 ( , )i j at 2P . Any two vectors that are equal in magnitude and point in the same direction are equal; therefore 1 2 1 2 ,= =i i j j (B.2.5)

A Cartesian coordinate system is the unique coordinate system in which the set of unit vectors at different points in space are equal. In polar coordinates, the unit vectors at two different points are not equal because they point in different directions. We show this in Figure B.2.5.

Figure B.2.5 Unit vectors at two different points in polar coordinates.


B.2.1 Infinitesimal Line Element Consider a small infinitesimal displacement d s between two points P1 and P2 (Figure B.2.6). This vector can be decomposed into

Figure B.2.6 displacement vector d s between two points

d d d dz = + +s k (B.2.6)

B.2.2 Infinitesimal Area Element Consider an infinitesimal area element on the surface of a cylinder of radius (Figure B.2.7).

Figure B.2.7 Area element for a cylinder The area of this element has magnitude dA d dz = (B.2.7)


Area elements are actually vectors where the direction of the vector dA

points perpendicular to the plane defined by the area. Since there is a choice of direction, we shall choose the area vector to always point outwards from a closed surface. So for the surface of the cylinder, the infinitesimal area vector is d d dz =A

(B.2.8) Consider an infinitesimal area element on the surface of a disc (Figure B.2.8) in the

-x y plane.

Figure B.2.8 Area element for a disc. This area element is given by the vector d d d =A k

(B.2.9)

B.2.3 Infinitesimal Volume Element An infinitesimal volume element (Figure B.2.9) is given by dV d d dz = (B.2.10)

Figure B.2.9 Volume element for a cylinder.


B.3 Spherical Coordinates We first choose an origin. Then we choose a coordinate, r , that measures the radial distance from the origin to the point P . The coordinate r ranges in value from 0 r < . The set of points that have constant value for r are spheres (level surfaces). Any point on the sphere can be defined by two angles ( , ) and r. We will define these angles with respect to a choice of Cartesian coordinates ( , , )x y z . The angle is defined to be the angle between the positive z -axis and the ray from the origin to the point P . Note that the values of only range from 0 . The angle is defined (in a similar fashion to polar coordinates) as the angle in the between the positive x -axis and the projection in the -x y plane of the ray from the origin to the point P . The coordinate angle can take on values from 0 2 < . The spherical coordinates ( , , )r for the point P are shown in Figure B.3.1. We choose the unit vectors ( , , )r at the point P as follows. Let r point radially away from the origin, and point tangent to a circle in the positive direction in the plane formed by the z -axis and the ray from the origin to the point P . Note that points in the direction of increasing . We choose to point in the direction of increasing . This unit vector points tangent to a circle in the xy plane centered on the z -axis. These unit vectors are also shown in Figure B.3.1.

Figure B.3.1 Spherical coordinates

If you are given spherical coordinates ( , , )r of a point in the plane, the Cartesian coordinates ( , , )x y z can be determined from the coordinate transformations

sin cossin sincos

x ry rz r

===

(B.3.1)


Conversely, if you are given the Cartesian coordinates ( , , )x y z , the spherical coordinates ( , , )r can be determined from the coordinate transformations

2 2 2 1 2

12 2 2 1 2

1

( )

cos( )

tan ( )

r x y z

zx y z

y x

= + + +

= + + =

(B.3.2)

The unit vectors also are related by the coordinate transformations

sin cos sin sin cos cos cos cos sin sin

sin cos

= + +

= +

= +

r i j k

i j k

i j

(B.3.3)

These results can be understood by considering the projection of ( , )r into the unit vectors ( , ) k , where is the unit vector from cylindrical coordinates (Figure B.3.2),

Figure B.3.2 Cylindrical and spherical unit vectors

sin cos

cos sin

= +

=

r k

k (B.3.4)

We can use the vector decomposition of into the Cartesian unit vectors ( , )i j : cos sin = + i j (B.3.5)


To find the inverse transformations we can use the fact that sin cos = + r (B.3.6) to express

cos sin sin cos

=

= +

i

j (B.3.7)

as

cos sin cos cos sin sin sin sin cos cos

= +

= + +

i r

j r (B.3.8)

The unit vector k can be decomposed directly into ( , )r with the result that cos sin = k r (B.3.9)

B.3.1 Infinitesimal Line Element Consider a small infinitesimal displacement d s between two points (Figure B.3.3). This vector can be decomposed into sind dr rd r d= + +s r (B.3.10)

Figure B.3.3 Infinitesimal displacement vector d s .


B.3.2 Infinitesimal Area Element Consider an infinitesimal area element on the surface of a sphere of radius r (Figure B.3.4).

Figure B.3.4 Area element for a sphere.

The area of this element has magnitude 2( )( sin ) sindA rd r d r d d= = (B.3.11) points in the radially direction (outward from the surface of the sphere). So for the surface of the sphere, the infinitesimal area vector is 2 sind r d d =A r

(B.3.12)

B.3.3 Infinitesimal Volume Element An infinitesimal volume element (Figure B.3.5) is given by 2 sindV r d d dr = (B.3.13)

Figure B.3.5 Infinitesimal volume element.

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8.02SC Physics II: Electricity and MagnetismFall 2010 For information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/terms.
http://ocw.mit.eduhttp://ocw.mit.edu/terms

Module 3: Sections 2.1 through 2.8 Module 4: Sections 2.9 through 2.141

Table of Contents 2.1 Electric Charge......................................................................................................... 2-3

2.2 Coulomb's Law .................................................................................................... 2-3

2.2.1 Van de Graaff Generator (link)...................................................................... 2-4

2.3 Principle of Superposition.................................................................................... 2-5

Example 2.1: Three Charges .................................................................................... 2-5

2.4 Electric Field........................................................................................................ 2-7

2.4.1 Electric Field of Point Charges (link) ............................................................ 2-8

2.5 Electric Field Lines .............................................................................................. 2-9

2.6 Force on a Charged Particle in an Electric Field ............................................... 2-10

2.7 Electric Dipole ................................................................................................... 2-11

2.7.1 The Electric Field of a Dipole...................................................................... 2-122.7.2 Electric Dipole Animation (link) ................................................................. 2-13

2.8 Dipole in Electric Field...................................................................................... 2-13

2.8.1 Potential Energy of an Electric Dipole ........................................................ 2-14

2.9 Charge Density................................................................................................... 2-16

2.9.1 Volume Charge Density ............................................................................... 2-162.9.2 Surface Charge Density ............................................................................... 2-172.9.3 Line Charge Density .................................................................................... 2-17

2.10 Electric Fields due to Continuous Charge Distributions.................................... 2-18

Example 2.2: Electric Field on the Axis of a Rod ................................................. 2-18Example 2.3: Electric Field on the Perpendicular Bisector (link) ......................... 2-19Example 2.4: Electric Field on the Axis of a Ring (link) ...................................... 2-21Example 2.5: Electric Field Due to a Uniformly Charged Disk ............................ 2-23

2.11 Summary ............................................................................................................ 2-25

2.12 Problem-Solving Strategies ............................................................................... 2-27

2.13 Solved Problems ................................................................................................ 2-29

2.13.1 Hydrogen Atom ........................................................................................ 2-292.13.2 Millikan Oil-Drop Experiment ................................................................. 2-30

1 These notes are excerpted Introduction to Electricity and Magnetism by Sen-Ben Liao, Peter Dourmashkin, and John Belcher, Copyright 2004, ISBN 0-536-81207-1.

2-1

2.13.3 Charge Moving Perpendicularly to an Electric Field ............................... 2-322.13.4 Electric Field of a Dipole.......................................................................... 2-342.13.5 Electric Field of an Arc............................................................................. 2-362.13.6 Electric Field Off the Axis of a Finite Rod............................................... 2-37

2.14 Conceptual Questions ........................................................................................ 2-39

2-2

Coulombs Law

2.1 Electric Charge

There are two types of observed electric charge, which we designate as positive and negative. The convention was derived from Benjamin Franklins experiments. He rubbed a glass rod with silk and called the charges on the glass rod positive. He rubbed sealing wax with fur and called the charge on the sealing wax negative. Like charges repel and opposite charges attract each other. The unit of charge is called the Coulomb (C).

The smallest unit of free charge known in nature is the charge of an electron or proton, which has a magnitude of

e = 1.6021019 C (2.1.1)

Charge of any ordinary matter is quantized in integral multiples of e. An electron carries one unit of negative charge, e , while a proton carries one unit of positive charge, +e . In a closed system, the total amount of charge is conserved since charge can neither be created nor destroyed. A charge can, however, be transferred from one body to another.

2.2 Coulomb's Law

Consider a system of two point charges, q1 and q2 , separated by a distance r in vacuum. The force exerted by q1 on q2 is given by Coulomb's law:

1 2 F r

12 = ke q q

2 r (2.2.1)r

where ke is the Coulomb constant, and =r / r is a unit vector directed from q1 to q2 ,r r

as illustrated in Figure 2.2.1(a).

(a) (b)

Figure 2.2.1 Coulomb interaction between two charges

Note that electric force is a vector which has both magnitude and direction. In SI units, the Coulomb constant ke is given by

2-3

1 9 2 2= = 8.987510 N m / C (2.2.2)k e 40

where

0 = 1

9 2 2 = 8 85 . 10 12 C2 N m 2 (2.2.3)

4 (8.99 10 N m C )

is known as the permittivity of free space. Similarly, the force on q1 due to q2 is given r r

by F21 = F12 , as illustrated in Figure 2.2.1(b). This is consistent with Newton's third law.

As an example, consider a hydrogen atom in which the proton (nucleus) and the electron are separated by a distance r = 5.3 10 11 m . The electrostatic force between the two particles is approximately F = k e 2 / r2 = 8.2 108 N . On the other hand, one may show e e that the gravitational force is only Fg 3.610

47 N . Thus, gravitational effect can be neglected when dealing with electrostatic forces!

2.2.1 Van de Graaff Generator (link)

Consider Figure 2.2.2(a) below. The figure illustrates the repulsive force transmitted between two objects by their electric fields. The system consists of a charged metal sphere of a van de Graaff generator. This sphere is fixed in space and is not free to move. The other object is a small charged sphere that is free to move (we neglect the force of gravity on this sphere). According to Coulombs law, these two like charges repel each another. That is, the small sphere experiences a repulsive force away from the van de Graaff sphere.

Figure 2.2.2 (a) Two charges of the same sign that repel one another because of the stresses transmitted by electric fields. We use both the grass seeds representation and the field lines representation of the electric field of the two charges. (b) Two charges of opposite sign that attract one another because of the stresses transmitted by electric fields.

The animation depicts the motion of the small sphere and the electric fields in this situation. Note that to repeat the motion of the small sphere in the animation, we have

2-4

the small sphere bounce off of a small square fixed in space some distance from the van de Graaff generator.

Before we discuss this animation, consider Figure 2.2.2(b), which shows one frame of a movie of the interaction of two charges with opposite signs. Here the charge on the small sphere is opposite to that on the van de Graaff sphere. By Coulombs law, the two objects now attract one another, and the small sphere feels a force attracting it toward the van de Graaff. To repeat the motion of the small sphere in the animation, we have that charge bounce off of a square fixed in space near the van de Graaff.

The point of these two animations (link) is to underscore the fact that the Coulomb force between the two charges is not action at a distance. Rather, the stress is transmitted by direct contact from the van de Graaff to the immediately surrounding space, via the electric field of the charge on the van de Graaff. That stress is then transmitted from one element of space to a neighboring element, in a continuous manner, until it is transmitted to the region of space contiguous to the small sphere, and thus ultimately to the small sphere itself. Although the two spheres are not in direct contact with one another, they are in direct contact with a medium or mechanism that exists between them. The force between the small sphere and the van de Graaff is transmitted (at a finite speed) by stresses induced in the intervening space by their presence.

Michael Faraday invented field theory; drawing lines of force or field lines was his way of representing the fields. He also used his drawings of the lines of force to gain insight into the stresses that the fields transmit. He was the first to suggest that these fields, which exist continuously in the space between charged objects, transmit the stresses that result in forces between the objects.

2.3 Principle of Superposition

Coulombs law applies to any pair of point charges. When more than two charges are present, the net force on any one charge is simply the vector sum of the forces exerted on it by the other charges. For example, if three charges are present, the resultant force experienced by q3 due to q1 and q2 will be

r r rF3 = F13 + F23 (2.3.1)

The superposition principle is illustrated in the example below.

Example 2.1: Three Charges

Three charges are arranged as shown in Figure 2.3.1. Find the force on the charge q3 assuming that q1 = 6.010

6 C , q2 = q1 = 6.010 6 C , q3 = +3.010

6 C and a = 2.010 2 m .

2-5

Figure 2.3.1 A system of three charges

Solution:

Using the superposition principle, the force on q3 is

r r r 1 q q q q F = F + F = 1 3 r + 2 3 r3 13 23 40 r13

2 13 r23 2 23

In this case the second term will have a negative coefficient, since q2 is negative. The unit vectors r13 and r23 do not point in the same directions. In order to compute this sum, we can express each unit vector in terms of its Cartesian components and add the forces according to the principle of vector addition.

From the figure, we see that the unit vector r13 which points from q1 to q3 can be written as

r13 = cos i + sin j = 2 (i + j)

2

Similarly, the unit vector r23 = i points from q2 to q3 . Therefore, the total force is

q q F r

3 = 1

q q1 3

2 r13 + q q2

23 r23

= 1

q q1 32

2 ( i + j) + ( 12) 3 i 40 r13 r23 40 ( 2 ) 2a a

1 3= 1 q q 2 2 1

i + 2 j

40 a 4 4

upon adding the components. The magnitude of the total force is given by

2-6

1 q q 2 2

2 2

1 2

F3 = 1 3 1 + 40 a2 4 4

9 2 2 (6.0106 C)(3.0106 C)(9.010 N m / C ) (0.74) = 3.0 N= (2.0102 m) 2

The angle that the force makes with the positive x -axis is

= tan1 F3, y

= tan1

2 / 4

=151.3 F3,x +1 2 / 4

Note there are two solutions to this equation. The second solution = 28.7 is incorrect because it would indicate that the force has positive i and negative j components.

For a system of N charges, the net force experienced by the jth particle would be

r N rFj = Fij (2.3.2)

i=1 i j

r where Fij denotes the force between particles i and j . The superposition principle implies that the net force between any two charges is independent of the presence of other charges. This is true if the charges are in fixed positions.

2.4 Electric Field

The electrostatic force, like the gravitational force, is a force that acts at a distance, even when the objects are not in contact with one another. To justify such the notion we rationalize action at a distance by saying that one charge creates a field which in turn acts on the other charge.

An electric charge q produces an electric field everywhere. To quantify the strength of the field created by that charge, we can measure the force a positive test charge q0

rexperiences at some point. The electric field E is defined as:

r r FeE = lim (2.4.1)

q0 0 q0

We take q0 to be infinitesimally small so that the field q0 generates does not disturb the source charges. The analogy between the electric field and the gravitational field r r g = lim Fm / m0 is depicted in Figure 2.4.1.m0 0

2-7

r Figure 2.4.1 Analogy between the gravitational field g r and the electric field E .

From the field theory point of view, we say that the charge q creates an electric field r r rE which exerts a force Fe = q0E on a test charge q0 .

Using the definition of electric field given in Eq. (2.4.1) and the Coulombs law, the electric field at a distance r from a point charge q is given by

rE =

4 1

0 rq

2 r (2.4.2)

Using the superposition principle, the total electric field due to a group of charges is equal to the vector sum of the electric fields of individual charges:

E r

= E r

i = 1 q 2 i r (2.4.3)4 ri i 0 i 2.4.1 Electric Field of Point Charges (link)

Figure 2.4.2 shows one frame of animations of the electric field of a moving positive and negative point charge, assuming the speed of the charge is small compared to the speed of light.

Figure 2.4.2 The electric fields of (a) a moving positive charge (link), (b) a moving negative charge (link), when the speed of the charge is small compared to the speed of light.

2-8

2.5 Electric Field Lines

Electric field lines provide a convenient graphical representation of the electric field in space. The field lines for a positive and a negative charges are shown in Figure 2.5.1.

(a) (b)

Figure 2.5.1 Field lines for (a) positive and (b) negative charges.

Notice that the direction of field lines is radially outward for a positive charge and radially inward for a negative charge. For a pair of charges of equal magnitude but opposite sign (an electric dipole), the field lines are shown in Figure 2.5.2.

Figure 2.5.2 Field lines for an electric dipole.

The pattern of electric field lines can be obtained by considering the following:

(1) Symmetry: For every point above the line joining the two charges there is an equivalent point below it. Therefore, the pattern must be symmetrical about the line joining the two charges

(2) Near field: Very close to a charge, the field due to that charge predominates. Therefore, the lines are radial and spherically symmetric.

(3) Far field: Far from the system of charges, the pattern should look like that of a single point charge of value Q = i Qi . Thus, the lines should be radially outward, unless Q = 0 .

2-9

r r (4) Null point: This is a point at which = , and no field lines should pass through it. E 0

The properties of electric field l

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