ELECTROMAGNETISM

Department of Physics and Astronomy, Faculty of Science,
University of Manchester

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Grant, I. S. (Ian S.)
Electromagnetism / I. S. Grant, W. R. Phillips.—2nd ed.
p. cm.—(The Manchester physics series)
ISBN 978 0 471 92711 2—ISBN 978 0 471 92712 9 (pbk.)
1. Electromagnetism. I. Phillips, W. R. (William Robert)
II. Title. III. Series.
QC760.G76 1990
537—dc20 90–35818
CIP

Grant, I. S.
Electromagnetism.—2nd ed.
1. Electromagnetism
I. Title II. Phillips, W. R. III. Series
537

The Manchester Physics Series is a series of textbooks at first degree level. It grew out of our experience at the Department of Physics and Astronomy at Manchester University, widely shared elsewhere, that many textbooks contain much more material than can be accommodated in a typical undergraduate course; and that this material is only rarely so arranged as to allow the definition of a shorter self-contained course. In planning these books we have had two objectives. One was to produce short books: so that lecturers should find them attractive for undergraduate courses; so that students should not be frightened off by their encyclopaedic size or their price. To achieve this, we have been very selective in the choice of topics, with the emphasis on the basic physics together with some instructive, stimulating and useful applications. Our second objective was to produce books which allow courses of different lengths and difficulty to be selected, with emphasis on different applications. To achieve such flexibility we have encouraged authors to use flow diagrams showing the logical connections between different chapters and to put some topics in starred sections. These cover more advanced and alternative material which is not required for the understanding of latter parts of each volume.

Although these books were conceived as a series, each of them is self-contained and can be used independently of the others. Several of them are suitable for wider use in other sciences. Each Author’s Preface gives details about the level, prerequisites, etc., of his volume.

The Manchester Physics Series has been very successful with total sales of more than a quarter of a million copies. We are extremely grateful to the many students and colleagues, at Manchester and elsewhere, for helpful criticisms and stimulating comments. Our particular thanks go to the authors for all the work they have done, for the many new ideas they have contributed, and for discussing patiently, and often accepting, the suggestions of the editors.

Finally, we would like to thank our publishers, John Wiley & Sons Ltd, for their enthusiastic and continued commitment to the Manchester Physics Series.

The basic content of undergraduate courses in electromagnetism does not change rapidly, and the range of topics covered in the second edition of this book is almost the same as in the first edition. We have made a few additions, for example by giving a fuller treatment of circuit analysis and by discussing the dispersion of electromagnetic waves. Some material which now seems outdated has been removed, and illustrative examples have been modernized.

We have made many small changes in presentation which we hope will make the argument clearer to readers. We gratefully acknowledge the help of all those who have suggested ways of improving the text. We are particularly indebted to Dr. R. Mackintosh and his colleagues at the Open University for a host of detailed suggestions. The adoption of this book as the text for the new ‘third-level’ Open University course on electricity and magnetism led to a careful scrutiny of the first edition by the course team. This has resulted, we believe, in changes which make the book more useful for students. Any errors or obscurities which remain are our responsibility.

This book is based on lectures on classical electromagnetism given at Manchester University. The level of difficulty is suitable for honours physics students at a British University or physics majors at an American University. A-level or high school physics and calculus are assumed, and the reader is expected to have some elementary knowledge of vectors. Electromagnetism is often one of the first branches of physics in which students find that they really need to make use of vector calculus. Until one is used to them, vectors are difficult, and we have accordingly treated them rather cautiously to begin with. Brief descriptions of the properties of the differential vector operators are given at their first appearance. These descriptions are not intended to be a substitute for a proper mathematical text, but to remind the reader what div, grad and curl are all about, and to set them in the context of electromagnetism. The distinction between macroscopic and microscopic electric and magnetic fields is fully discussed at an early stage in the book. It is our experience that students do get confused about the fields E and D, or B and H. We think that the best way to help them overcome their difficulties is to give a proper explanation of the origin of these fields in terms of microscopic charge distributions or circulating currents.

The logical arrangement of the chapters is summarized in a flow diagram on the inside of the front cover. Provided that one is prepared to accept Kirchhoff’s rules and the expressions for the e.m.f.s across components before discussing the laws on which they are based, the A.C. theory in Chapters 7 and 8 does not require any prior knowledge of the earlier chapters. Chapters 7 and 8 can therefore be used at the beginning of a course on electromagnetism. Sections of the book which are starred may be omitted at a first reading, since they do not contain material needed in order to understand later chapters.

We should like to thank the many colleagues and students who have helped with suggestions and criticisms during the preparation of this book; any errors which remain are our own responsibility. It is also a pleasure to thank Mrs Margaret King and Miss Elizabeth Rich for their rapid and accurate typing of the manuscript.

This flow diagram shows the main logical connections between chapters. Any specific chapter can be understood if all the chapters above it which are connected by a line have been covered. Chapters 7 and 8 on A. C. Theory may be tackled before the earlier chapters provided that Kirchhoff’s rules are assumed. Strictly speaking Kirchhoff’s rules depend on earlier material, this is indicated by the dotted line joining Chapters 6 and 7.

CHAPTER 1

Force and energy in electrostatics

The only laws of force which are known with great precision are the two laws describing the gravitational forces between different masses and the electrical forces between different charges. When two masses or two charges are stationary, then in either case the force between them is inversely proportional to the square of their separation. These inverse square laws were discovered long ago: Newton’s law of gravitation was proposed in 1665, and Coulomb’s law of electrostatics in 1785. This chapter is concerned with the application of Coulomb’s law to systems containing any number of stationary charges. Before studying this topic in detail, it is worth pausing for a moment to consider the consequences of the law in the whole of physics.

In order to make full use of our knowledge of a law of force, we must have a theory of mechanics, that is to say, a theory which describes the behaviour of an object under the action of a known force. Large objects which are moving at speeds small compared to the speed of light obey very closely the laws of classical Newtonian mechanics. For example, these laws and the gravitational force law together lead to accurate predictions of planetary motion. But classical mechanics does not apply at all to observations made on particles of atomic scale or on very fast-moving objects. Their behaviour can only be understood in terms of the ideas of quantum theory and of the special theory of relativity. These two theories have changed the framework of discussion in physics, and have made possible the spectacular advances of the twentieth century.

It is remarkable that while mechanics has undergone drastic amendment, Coulomb’s law has stood unchanged. Although the behaviour of atoms does not fit the framework of the old mechanics, when the Coulomb force is used with the theories of relativity and quantum mechanics, atomic interactions are explained with great precision in every instance when an accurate comparison has been made between experiment and theory. In principle, atomic physics and solid state physics, and for that matter the whole of chemistry, can be derived from Coulomb’s law. It is not feasible to derive everything in this way, but it should be borne in mind that atoms make up the world around us, and that its rich variety and complexity are governed by electrical forces.

1.1 ELECTRIC CHARGE

Most of this book applies electromagnetism to large-scale objects, where the atomic origin of the electrical forces is not immediately apparent. However, to emphasize this origin, we shall begin by consideration of atomic systems. The simplest atom of all is the hydrogen atom, which consists of a single proton with a single electron moving around it. The hydrogen atom is stable because the proton and the electron attract one another. In contrast, two electrons repel one another, and tend to fly apart, and similarly the force between two protons is repulsive*. These phenomena are described by saying that there are two different kinds of electric charge, and that like charges repel one another, whereas unlike charges are attracted together. The charge carried by the proton is called positive, and the charge carried by the electron negative.

The magnitude and direction of the force between two stationary particles, each carrying electric charge, is given by Coulomb’s law. The law summarizes four facts:

(i) Like charges repel, unlike charges attract.

(ii) The force acts along the line joining the two particles.

(iii) The force is proportional to the magnitude of each charge.

(iv) The force is inversely proportional to the square of the distance between the particles.

The vector F₂₁ in Figure 1.1 represents the force on particle 1 (carrying a charge q₁) exerted by the particle 2 (carrying charge q₂). The line from q₂ to q₁ is represented by the vector r₂₁, of length r₂₁: since the unit vector along the direction r₂₁ can be written r₂₁/r₂₁, Equation (1.1) is an inverse square law of force, although

appears in the denominator. Notice that the equation automatically accounts for the attractive or repulsive character of the force if q₁ and q₂ include the sign of the charge. When the charges q₁ and q₂ are both positive or both negative, the force on q₁ is along r₂₁, i.e. it is repulsive. On the other hand, when one charge is positive and the other negative, the force is in the direction opposite to r₂₁, i.e. it is attractive.

To complete the statement of the force law, we must decide what units to use, and hence determine the constant of proportionality in Equation (1.1). We shall use SI (Systeme International) units, which are favoured by most physicists and engineers applying electromagnetism to problems involving large-scale objects. A different system of units, called the Gaussian system, is frequently used in atomic physics and solid state physics, and it is an unfortunate necessity for students to become reasonably familiar with both systems. (The two systems of units are discussed in Appendix A.) In SI units, Coulomb’s law is written as

The magnitude of the unit of charge, which is called the coulomb, is actually defined in terms of magnetic forces, and we shall leave discussion of the definition until Chapter 4. The factor 4π in the constant of proportionality in Coulomb’s law is introduced in order to simplify some important equations which we shall meet later. The constant ε₀, which is called the permittivity of free space, has the value

Figure 1.2. How electrostatic forces are added when there are more than two charges.

The value of ε₀ is not determined experimentally, but has been defined in a way which makes the SI system of units self-consistent: the relation between electrical units and other units is discussed in Appendix A.

Electrostatic forces are two-body forces, which means that the force between any pair of charges is unaltered by the presence of other charges in their neighbourhood*. In a system containing many charges, the electrostatic force between each pair is given by Coulomb’s law. To find the total force on any one particle, one simply makes a vector sum of the forces it experiences due to all the others separately. This rule is illustrated in Figure 1.2 for a system of three charges. The forces on q₁ due to the presence of the charges q₂ and q₃ are F₂₁ and F₃₁ and the total force on q₁ is

In general, the force F_j on a charge q_j due to a number of other charges q_i is

The symbol i ≠ j under the summation signs indicates that the summation for charge i is over all the other charges j, but of course not including charge i itself. This equation can be written in another way in terms of the position vectors of the charges with respect to a fixed origin O. If the position vectors of the charges q₁, q₂ … q_i … are r₁, r₂ … r_i … then the vector joining charges i and j is r_ij = r_j – r_i. The total force on q_j is thus

A trivial example of the application of Equation (1.3) is in working out the electrostatic forces exerted by atomic nuclei containing many protons on the electrons surrounding them. Nuclei are much smaller than atoms, and for this purpose can be regarded as point charges. Equation (1.3) then tells us that the attractive force between an electron and a nucleus containing Z protons is Z times as great as that between an electron and a single proton.

It turns out that apart from the sign, the charge carried by electrons and protons is the same, and has the magnitude e = 1.602 × 10⁻¹⁹ coulombs*: the charge on the proton is +e, that on the electron is – e. The strength of atomic interactions is governed by the size of the electronic charge e. Although e is a very small number when expressed in coulombs, this does not imply that electrostatic forces are feeble. On the contrary, they are immensely strong. For example, electrostatic forces are responsible for the great strength of solids under compression. When neighbouring atoms are close together, their electron clouds begin to overlap, and the mutual repulsion of these clouds opposes any compressing force.

Another example of the strength of the electrostatic force acting on the atomic scale is given by the experiment which led Rutherford to propose the nuclear model of the atom. He found that when swiftly moving α-particles are allowed to collide with gold atoms, they are sometimes deflected through 180°, implying that a strong force is at work. The force is just the electrostatic repulsion experienced by an α-particle when it chances to approach close to the nucleus of a gold atom. Let us calculate the magnitude of the force. An α-particle is a helium nucleus, containing two protons and carrying a charge + 2e, and the gold nucleus carries a charge + 79e. In Rutherford’s experiment, the α-particles were energetic enough to approach within 2 × 10⁻¹⁴ m of the nucleus (still well outside the range of the nuclear forces). Substituting in Equation (1.3), the repulsive force at this distance is

This force, acting within a single atom, is nearly as much as the weight of a mass of 10 kilogrammes!

Normally we do not notice that electrostatic forces are so powerful, because matter is usually electrically neutral, carrying equal amounts of positive and negative charge. Not only are large lumps of matter electrically neutral, but the positive charge on the nucleus of a single isolated atom is precisely cancelled out by the negative charge of the surrounding electrons. So far as we know the cancellation is exact, and it has been shown experimentally that the magnitude of the net residual charge on a neutral atom is less than 10⁻²⁰e. This is very remarkable, since apart from their electrical behaviour, protons and electrons are totally dissimilar particles. Many charged particles besides electrons and protons have been discovered by nuclear physicists, and all* share the property of carrying charges ±e. It follows that the total charge carried by any piece of matter must be an integral multiple of the electronic charge e. A situation like this one, in which a physical quantity is not allowed to have a continuous range of values, but is restricted to a set of definite discrete values, is referred to as a quantum phenomenon. No one knows why electric charge should obey this quantum rule; it is an experimental fact. Nevertheless, because the rule is universal in its application, we can be sure that the electronic charge is a physical quantity of fundamental importance.

1.2 THE ELECTRIC FIELD

We have seen that the total force on a charged particle is the vector sum of the forces exerted on it by all other charges. Usually there is an enormous number of charged particles present in real matter. When considering the forces acting on any one of them, it is helpful to distract attention from the multitude of sources contributing to the net force by introducing the concept of the electric field. If a charge q experiences a force F, then the ratio F/q is called the electric field at the point where q is located. The dimensions of electric field are [force] [charge]⁻¹, and in SI units the electric field is measured in newton coulomb⁻¹. (An equivalent unit, which will be explained when we come to deal with electrostatic energy, is the volt m⁻¹; 1 volt m⁻¹ ≡ 1 newton coulomb⁻¹.)

The electric field acting on q can be expressed in terms of the magnitudes of the other charges in the neighbourhood of q and their relative positions with respect to q. Let us assume that q is a test charge which can be put anywhere, and that its magnitude is very small, so that it exerts negligible forces on the other charges and can be moved about without altering their positions. Now we can evaluate the electric field at a point P, caused by an assembly of charges q_i by placing the test charge at P. In Figure 1.3 P has a position vector r, and the charges q_i have position vectors rf with respect to an origin at O. The force on the test charge is given by Equation (1.3) as

where (r – r_i) represents the vector joining q_i to the point P. The factor q is common to all terms in the sum, and it follows that

The magnitude q of the test charge does not appear on the right-hand side of this equation. We can therefore allow q to become vanishingly small—then we are quite sure that the presence of the test charge does not modify the position of the other charges. Let us call the electric field at the point P with position vector r E(r).

Equation (1.4) is the definition of the electric field E(r), and it contains no reference to a test charge q. The test charge was introduced because it illustrates that the electric field is a force per unit charge, and because it helps one to visualize the electric field if one imagines a test charge which can be moved around to sample the strength of the field at any position. The electric field E(r) is a function of position; just as the value of a function f(x) is determined by the argument x, so the value of E(r) is given by Equation (1.4) in terms of its argument, which is the position vector r. The function E(r) is itself a vector, specifying the direction as well as the magnitude of the force per unit charge on a point charge at r. The electric field is only the first of a number of functions of position which are useful in electromagnetism. Those functions of position which are themselves vectors are called vector fields. When discussing vector fields we shall frequently omit any reference to the argument, writing the electric field, for example, simply as ‘E’, leaving it to be understood that the field is a function of position. Whenever there is some uncertainty about the position at which the function is to be evaluated, we shall always write out the vector field in full, including the argument.

In Equation (1.4) each charge q_i appears just once on the right-hand side. If the charge q_i were the only charge present, the field would be

In other words, the total electric field is the sum of the electric fields due to each charge separately. This is an example of the Principle of Superposition. As applied to electrostatic fields, the Principle of Superposition states that if we have two electrostatic fields due to different groups of charged particles, the fields must be added together (superimposed) to find the field due to all the particles in both groups. This follows immediately from Equation (1.6). We shall find that the Principle of Superposition applies to all the different kinds of field which occur in electromagnetism.

Now let us investigate the properties of the electric field in the neighbourhood of isolated point charges. From Equation (1.5), the magnitude of the field at a distance r from a positive point charge +q is q/4πε₀r², and the field points away from the charge. The field around a negative point charge – q has the same magnitude, but it points towards the charge. In Figure 1.4 the direction of the field around positive and negative charges is indicated by the arrowed lines. These continuous lines, everywhere following the direction of the field, are called lines of force or field lines. Lines of force begin on positive charges and end on negative charges, but they may also go to infinity without terminating, as in Figure 1.4. Notice that the lines of force are close together near the point charges where the field is strong, and far apart at large distances where the field is weak.

We can also draw diagrams of lines of force to illustrate the electric field when there are many charges present. Lines of force are continuous, except where they terminate on positive or negative charges, and they never cross one another, since the direction of the field is unique at every point. One can often get a rough idea of the field around a distribution of charges simply by sketching lines of force, and without doing any mathematics. For example, Figure 1.5 shows the lines of force near a pair of point charges of equal magnitude, one positive and one negative. Such a pair of equal and opposite charges is called an electric dipole. Very close to each charge, the field is almost the same as for isolated point charges, but the field lines starting off at the positive charge curve round to finish at the negative charge. The diagram gives an indication of the strength of the field as well as its direction, because lines of force are always densely packed in regions where the field is strong. The total number of lines on the diagram is not significant; if twice as many lines were drawn terminating on each charge, regions of relatively high or low density of lines of force would still correspond to regions of high or low field strength. The example of the electric dipole is an important one and later on we shall obtain a mathematical expression for the field at some distance from a dipole.

1.3 ELECTRIC FIELDS IN MATTER

1.3.1 The atomic charge density

So far electric fields have been dealt with in terms of idealized particles which are stationary and which carry point charges. This is not satisfactory if we want to discuss the electric field within an actual atom, since electrons are certainly not stationary point charges. Indeed, according to quantum mechanics the position of an electron cannot even be sharply defined. Instead of imagining the electron as a point, it is more realistic to regard its charge as being smeared out in a cloud around the nucleus. The electric field in and around a hydrogen atom, for example, behaves as though only part of the electronic charge is contained by any section of the electron cloud. Let us write ρ_el(r)δτ for the charge contained within a small volume δτ situated at the point with position vector r. The quantity ρ_el(r) is the charge density of the electron cloud, i.e. its charge per unit volume at r. The charge density is another function of position, but unlike the electric field it is a scalar quantity, fully specified by its magnitude at each point. Scalar functions of position such as the charge density are called scalar fields. As with the vector fields, we shall often omit the argument of scalar fields, writing the electron charge density just as ρ_el, for example. The total charge (– e) carried by the electron in a hydrogen atom is equal to the sum of the charges contained in all the small volumes δτ making up the electron cloud. In the limit as the volumes δτ become infinitesimal, the sum becomes an integral; integrating over a volume V large enough to contain the whole atom, we write

Here we have introduced a shorthand notation for the volume integral, which is really a triple integral over the three components needed to specify position vectors r within the volume V. In Cartesian coordinates dτ is dx dy dz, the volume of the rectangular box enclosed by sides of length dx dy dz. The volume of integration V must include the whole atom, but since ρ_el(r) is zero outside the atom, it can be made as large as we please without affecting the result. If we extend V to cover the whole of space, in Cartesian coordinates the integral becomes

In atoms more complex than hydrogen, the electron clouds of different electrons overlap, and a volume element may contain parts of the charge of more than one electron. Even the charge carried by the nucleus should be represented by a charge density since although the nucleus is small compared with the whole atom, it does have a definite size. Now we define the atomic charge density ρ_atomic(r) in a piece of matter as the net charge density at r, including positive contributions from nuclei. Where the charge densities associated with different particles overlap, they are added together to form the atomic charge density. The atomic charge density has large positive values inside the nucleus of each atom and is negative in the electron cloud. The volume integral ∫_V ρ_atomic(r)dτ represents the net charge within V: thus for a volume V containing an electrically neutral piece of matter, the value of the integral must be zero.

1.3.2 The atomic electric field

The electric field due to an assembly of point charges q_i was found by making a vector sum of the fields due to each charge separately:

Applying the same procedure to a continuous charge density ρ_atomic, the sum becomes an integral, and the atomic electric field at a point P with position vector r is

Figure 1.6. The charges within each volume element dτ′ contributes to the electric field at P.

Here ρ_atomic(r′) dτ′ is the charge within a volume element dτ′ located at the point with position vector r′, as illustrated in Figure 1.6. Both sides of Equation (1.7) are vectors, and to work out the volume integral in practice it is necessary to integrate each component separately.

Although it is easy to write down this equation for the atomic field*, it is not very useful, because in practice one does not know the charge density ρ_atomic. Not only does the charge density within each atom have a complicated form, but even in a solid, atoms are always in thermal motion. The charge density ρ_atomic and the field E_atomic at a fixed point are therefore continually changing. However, although the exact atomic field is rarely known, it is possible to describe its main qualitative features. Each atom has an internal field generated by its own nucleus and electron cloud. This field is very strong near the concentration of positive charge in the atomic nucleus. Superimposed on the internal field there may be fields caused by other atoms and molecules. Not only do charged atoms and molecules generate external electric fields, but even outside neutral atoms and molecules there may be fields which are significant over distances several times as large as a typical atomic diameter. A hydrogen chloride molecule, for example, is electrically neutral, but in the chemical bond which holds it together part of the electron cloud of the hydrogen atom is transferred to the chlorine atom. The hydrogen atom is thus left with an excess of positive charge, while the chlorine acquires a negative charge. Lines of force leave the positively charged hydrogen and end on the negatively charged chlorine as sketched in Figure 1.7 (notice that the lines of force indicate that the electric field outside the molecule has a similar shape to the field at some distance from the dipole in Figure 1.5). Fields of this kind fall off with distance much more quickly than the fields outside charged atoms and molecules. Nevertheless the local average field acting on a particular atom is often dominated by the contributions from neutral neighbours.

Figure 1.7. The field around a hydrogen chloride molecule: the hydrogen atom has lost part of its electron cloud and carries a net positive charge.

1.3.3 The macroscopic electric field

Fortunately, when one is considering large systems, the details of the atomic field are not important, and all that is needed is an average field. The value of the field averaged over a region much bigger than a single atom, but still small compared to the size of the whole system, is called the macroscopic electric field (the word macroscopic means large-scale, and is used to make a contrast with the microscopic scale of the atomic field). In the macroscopic field all the sharp variations of the atomic field within each atom are smoothed out: the macroscopic field is the one which would exist if matter were continuous and had no atomic structure.

By comparison with the strong atomic fields near nuclei, macroscopic fields are of modest magnitude. In a good electrical insulator like porcelain, electrons are bound tightly to their atoms, and any local excesses of positive or negative charge can remain immobile for a long time, maintaining a macroscopic field. But if this field becomes too large catastrophic breakdown occurs. The largest fields which occur are about 10⁹ volts/metre. Static fields as great as this do not occur over extensive regions, and even aver distances of a few atomic diameters, they are found only in some special situations. *.

Net charges may be distributed throughout the volume of an insulator, or may be concentrated in a thin layer at a surface. Only surface charges occur in a material like copper which is a good electrical conductor. In a conductor, some electrons, which are referred to as conduction electrons, are free to move in the presence of a macroscopic electric field. It follows that in static equilibrium the macroscopic electric field within a conductor is everywhere zero. What happens when a conductor is placed in an electric field generated by outside charges? Consider a slab of conductor in a uniform field directed perpendicularly to the surface of the slab. When the field is first applied, a macroscopic field extends throughout the conductor. The conduction electrons move under the influence of this field, leaving net charges on the surface as shown in Figure 1.8. The surface charges generate a field inside the conductor which is opposed to the external field, and the surface charges build up until the macroscopic field within the bulk of the conductor is exactly zero. Charges appearing on the surface of a material in this way, due to the presence of an external field, are called induced charges.

We now relate the macroscopic field to the charges which generate it, taking into account both distributed volume charges and surface charges. First consider the volume charges. Imagine a volume δV, small compared with a macroscopic system, but still large enough to contain many atoms. The net charge within δV is ∫_δV ρ_atomic(r)dτ. Neighbouring regions of the same volume δV may not contain precisely the same net amount of charge. However, the relative differences between regions are small, provided that each region contains a large enough number of atoms. We can define the macroscopic charge density ρ(r) at the point with position vector r as the net charge per unit volume contained in a volume δV in the vicinity of r, i.e.

Figure 1.8. Induced surface charges appear on a slab of conductor when it is placed in an electric field.

The charge density ρ(r) is a function of position which varies smoothly inside matter, except at boundaries between media, where ρ may change discontinuously. Near a boundary there may also be a concentration of surface charges. Surface charges always occupy a very thin layer, no more than a few atoms thick, and for the purpose of discussing the macroscopic field they may be represented by an infinitesimally thin sheet of charge lying exactly on the boundary surface. A surface charge density σ(r) is defined, analogously to the volume density ρ(r), as the net charge per unit area on an area δS which is small on the macroscopic scale, but large enough to include many surface atoms. Consider a slab of volume δV just thick enough to enclose all the surface charge σ(r)δS associated with an area δS in the vicinity of a point on the surface with position vector r. The net charge within δV is a volume integral over the atomic charge density, and we have*

Macroscopic electric fields obey the principle of superposition just as do atomic fields. To find the macroscopic field at a point with position vector r, it is necessary to add the contributions from the net charges ρ(r′) dτ′ in all the volume elements dτ′ covering the whole of space, and the contributions from the surface charges σ(r)dS′ on the surface elements dS′ over all boundary surfaces. The field generated by the volume charge density ρ is given by a volume integral of the same form as Equation (1.7) for the atomic field, namely

Similarly the surface charge density σ generates a field given in shorthand notation as

The surface integral is a double integral, and for example if the rectangular surface in the x–y plane shown in Figure 1.9 carries a charge density σ (and has corners at x = ± a, y = ± b), its contribution to the field at a point with position vector r is

Figure 1.9. The vector addition of the fields due to all surface elements δx′ δy′ leads to a surface integral.

Finally, adding the terms from volume and surface charge densities, the macroscopic electric field is defined to be

1.4 GAUSS’ LAW

When the charge density has a distribution with a simple symmetry, the electric field can be evaluated easily by the application of Gauss’ law, an important result which relates the field on any closed surface to the net amount of charge enclosed within the surface. Using Gauss’ law we shall be able to prove, for example, that the field outside an isolated and spherically symmetrical charged ion is exactly the same as if all its charge were concentrated at the centre; an application to a macroscopic system is at the surface of a conductor, where Gauss’ law leads to an expression for the external electric field in terms of the surface charge density. Gauss’ law can be illustrated by considering a point charge q at the centre of a sphere of radius r. The area of the sphere is 4πr², and the electric field on its surface is everywhere of magnitude q/4πε₀r². The product of electric field and area is q/ε₀, i.e. it is independent of r. This is rather like what happens when a light bulb is placed at the centre of a spherical lampshade which is completely transparent. All the light from the bulb escapes, or in other words the total amount of light passing through the shade is independent of its radius. Obviously the total amount of light escaping is also independent of the shape of the shade; all the light gets out whether or not the shade is spherical. Gauss’ law is the analogous result in electrostatics applied to a quantity called the flux of the electric field; the total flux out of a surface enclosing a charge q is q/ε₀, whatever the shape of the surface. In the rest of this section we shall explain what is meant by flux, and then go on to prove Gauss’ law rigorously.

1.4.1 The flux of a vector field

Flux may be defined for any vector function of position, but it is most easily visualized in the flow of fluids. Imagine that a fluid is flowing at a speed V and that a small loop of wire is placed in the fluid. In Figure 1.10(a) the shaded area δS, which is bounded by the loop of wire, is perpendicular to the direction of flow of the fluid. The flux of fluid through the area δS is the rate of flow of fluid through the loop, which in this case is the product v δS of speed and area. Now suppose that the loop is rotated to the position shown in Figure 1.10(b), where the normal to the surface δS is at an angle Ψ to the direction of flow. Looking along the direction of flow, the projected area within the loop is now only δS_p = δS cos Ψ, and the flux, i.e. the rate of flow of fluid through the loop, is reduced to

Figure 1.10. A column of fluid of length v passes through the area δS every second. The volume of fluid passing is larger when v is perpendicular to δS, as in (a), than when it is at an angle as in (b).

This flux can be written in vector notation if we represent the surface by a vector δS, of magnitude δS and directed along the normal to the surface. The velocity vector v is then at an angle Ψ to δS, and the flux through δS is given by the scalar product

The flux has a sense as well as a magnitude, and in Figure 1.10(b) the sense is from the left-hand side of the loop to the right-hand side. If we choose to represent the surface by the vector (– δS), still normal to the surface but pointing from right to left in the opposite direction to δS, the scalar product v·(– δS) becomes negative. A negative flux from right to left is equivalent to a positive flux from left to right, but to avoid ambiguity the sense of surface vectors must always be carefully specified.

In an electrostatic field there is of course nothing actually flowing, but the flux of an electric field E through δS is defined in the same way as E·δS, the value of E being taken at the position of the surface δS. In terms of components the flux of E through δS can be expressed as

So far we have only defined the flux through a small flat surface. To find the flux of the electric field through a surface S of arbitrary shape, we first approximate the shape of S by dividing it up into a lot of small flat surfaces. The flat surfaces are represented by vectors such as δS in Figure 1.11, all pointing outwards from the same side of the whole surface. The flux of the field E through δS is E·δS, and the total flux through the surface made up of the mosaic of flat surfaces is the sum of the fluxes through each separately, i.e.

If the size of each of the flat surfaces is made smaller and smaller, the mosaic approaches more and more closely to the smooth surface S, and the sumapproaches a limiting value which is equal to the flux through S. The limit is a two-dimensional surface integral written as

Figure 1.11. Splitting up a surface S into a lot of little flat surfaces in order to work out the flux of E through S.

The subscript S under the integral sign indicates that the area of integration is the whole of the surface S.

1.4.2 The flux of the electric field out of a closed surface

Let us now evaluate the flux through a closed surface S for the electric field E generated by a point charge q enclosed by S. We shall use spherical polar coordinates referred to an origin at q, and concentrate attention for the moment on a surface element of area δS at the position with coordinates (r, θ, Φ). The surface element is represented by the vector δS directed along the outward normal to S. The element is of such a size and shape that its projection along the radius vector from q lies between the polar angles θ and (θ + δθ) and the azimuthal angles Φ and (Φ + δΦ). As illustrated in Figure 1.12, the projected area is

According to Equation (1.5), the field at δS is directed outwards along the radius vector and has a magnitude

Figure 1.12. The area δS_p subtends angles δθ and δφ at the origin. It is a distance r from the origin, so that it is a rectangle of sides rδθ and r sin θδφ, i.e. δS_P = r² sin δθ δφ.

where the element δΩ. is termed the solid angle subtended by δS_p at the origin. Using Equation (1.11), we can now transform the surface integral expression for the flux through the complete surface S into an integral over angular variables only. The range of variation required to cover the whole of the closed surface is θ = 0 to π and φ = 0 to 2π. In the limit, as δS tends to zero, the total flux through S is

In this derivation it has been assumed that the flux through dS is always positive. Sometimes, however, when S has a complicated shape, the radius vector from q may pass from outside to inside the enclosing surface, as in Figure 1.13. Where the field is directed inwards it makes a negative contribution to the net outward flux. Provided that the direction of dS is chosen to be along the outward normal to S, the scalar product E·dS always gives the correct sign for the flux out of dS. But Equation (1.11) should be modified to

Figure 1.13. The radius vector from a charge inside a surface always makes one more outward crossing than the number of inward crossings.

with the negative sign applying for those parts of the surface where the field is directed inwards. Notice that however many times a radius vector may cross the surface S, if the origin lies within S the radius vector must always make one more outward crossing than the number of inward crossings. It follows that no matter how many times the cone of solid angle dΩ, is cut by the surface S, the net flux leaving S within dΩ is (q/4πε₀) dΩ, and Equation (1.12) is correct for any surface whatever which encloses q.

Figure 1.14. For a charge outside the surface the number of outward crossings is the same as the number of inward crossings.

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