Special Relativity and Electromagnetism

(Draft of Lecture Notes by A. Recknagel – September 2008)

Contents

I Electromagnetism

1. Basic facts from physics; fields2. Maxwell’s equations3. Static fields4. Time-dependent fields

II Special Relativity

5. Some basics; principles of relativity, constancy of speed of light6. Derivation of standard Lorentz transformations7. Some consequences of Lorentz transformations8. Lorentz group and Lorentz tensors9. Aspects of relativistic particle mechanics10. Relativistic formulation of electrodynamics

These notes are not quite complete and may well contain typos; additional pictures willbe provided in a separate handout. The odd remark ‘see hand-written notes’ refers to anolder, less organised set of notes.

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I ELECTROMAGNETISM

1. Basic facts from physics; fields

Although electromagnetic phenomena are ubiquitous, their origin is somewhat ‘hidden’to the human senses. Therefore, it took quite some time to isolate and identify simplephenomena, to order and explain them scientifically. ‘Physical investigation’ started about600 B.C. – already Thales found that rubbing a piece of amber with a woollen cloth resultsin strange properties: it becomes electrically charged. And the Greeks discovered thatpieces of loadstone can attract each other and also iron, and that they have the tendencyto point to the earth’s poles when suspended freely. (Note, in passing, that the Greekword for amber is ‘elektron’, and that the Greeks found their loadstones near the townof Magnesia in Asia Minor.) The theoretical description of electromagnetism came to a‘conclusion’ only in 1864, when Maxwell wrote down his equations – at King’s!We will not follow the historical route here but start from Maxwell’s equations as givenand derive specific electromagnetic phenomena from them. But first we need to introducesome basic physical quantities and to recall some mathematical notions.

1.1 Electric charge and currents; magnets

From experiments, one extracts the following facts:

• Physical objects can carry electric charge (in the following usually denoted q or Q).One can build devices that measure the electric charge carried by a body.

• The electric charge of a body is independent of other fundamental properties knownfrom mechanics, like mass. One needs a new physical unit to describe it, called theCoulomb, abbreviation 1 Cb.

• Electric charge is additive: Bringing two bodies with charges q1 and q2 together yieldsa body of charge q1 + q2. In this way, one also finds that charge can be positive ornegative (annihilation of charge); bodies with q = 0 are called (electrically) neutral.One also finds that the total charge of all bodies involved in some physical process isconserved during that process, for any process.By measuring charges with very high precision, one discovers that electric chargecomes in integer multiples of the so-called elementary charge qelem ≈ 1.6 × 10

−19Cb– a fact one can at best explain in elementary particle physics (although some basicquestions are still open). qelem is very small compared to typical charges occurring inmacroscopic physics, so the discreteness does not show in every day’s life.

For the theoretical description, we will use two different kinds of mathematical idealisa-tions:

• point charges, also called test charges: point-like objects carrying some amount q ofelectric charge; cf. the point masses from Newtonian mechanics. (If we talk of ‘testcharges’, we have in mind that q is ‘rather small’, since we want to use a test chargeto probe the electromagnetic properties of its surroundings – with the help of forcesacting on charges, see below – but without altering these surroundings (significantly)

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by the very presence of the additional test charge. (And this is approximately true inpractice if the test charge is very small.)

• continuous charge distributions in time and space, i.e. in R×R3: We write ρ(t, ~x) forthe charge density at a point ~x ∈ R3 at time t ∈ R; this function ρ is real-valued. Wehere ignore (idealisation!) the discrete nature of electric charge and imagine that itcan be distributed (‘smeared’) continuously over space. Given a charge density ρ(t, ~x),then for any volume V ⊂ R3, we have

∫

V

ρ(t, ~x) d3x = total charge contained in V at time t.

(We will see later that even point charges can be written in the form of a chargedensity ρ(t, ~x), using the delta distribution.)

Another important fact: moving electric charges constitute electric currents. Given acharge distribution ρ(t, ~x) where the ‘constituent’ at (t, ~x) moves with velocity ~v(t, ~x), wehave a current density

~(t, ~x) =

j1(t, ~x)j2(t, ~x)j3(t, ~x)

= ρ(t, ~x) · ~v(t, ~x) . (1.1)

This is our first example of a (three-dimensional) vector field, i.e. a map that assigns a(three-dimensional) vector to each point (t, ~x) ∈ R × R3. The physical unit of a currentdensity is 1 Cbs m2 =: 1

Am2 , where ‘1 A’ is the abbreviation of ‘1 Ampère’ (and ‘s’ is for second,

‘m’ for meter).While the charge density defined above is a volume density (charge q per volume), thecurrent density is an area density: The electric current IS flowing through a surface S isgiven by the surface integral

IS =

∫

S

~(t, ~x) · d~S(t, ~x) . (1.2)

We will often deal with surface integrals as above; in general, the oriented infinitesimalsurface element d~S depends on the point ~x on the surface and is defined by

d~S(t, ~x) := ~n(t, ~x) dS

with dS being an ordinary infinitesimal surface element (just like d2x) and ~n(t, ~x) is a unitvector attached to ~x and orthogonal to the surface; see the separate set of figures. So theintegrand of (1.2) is the function ~ · ~n (a scalar product of 3-vectors) evaluated on points(t, ~x) in the surface S.We will restrict ourselves to smooth surfaces, meaning that the normal vector field ~n(t, ~x)various smoothly as ~x runs through the surface. As a consequence, ~n(t, ~x) is uniquelydetermined by S up to an overall sign, the orientation of the surface. We will work withoriented surfaces, where the sign has been chosen as part of the definition of the surface.Finally, the time dependence allows to consider also surfaces that change with time.

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With these definitions, the electric current IS in (1.2) has unit 1Cbs and tells how much

charge flows through the surface S per unit time. (Electric fuses, e.g., are marked with theelectric current (in Ampère) they can withstand; they blast as soon as too high a current –too much charge per time – flows through them, thus protecting a more precious householddevice from a similar fate.)The scalar product of ~ and ~n in the integral accounts for the fact that the angle betweena surface element and the velocity of charge carriers matters: Charge carriers that moveparallel to the surface won’t penetrate it and thus won’t induce electric current throughthe surface. If we have a wire with cross section area a, carrying an electric current I, thenthe current density ~ has magnitude |~| = I/a and and is parallel to the wire.

Apart from electric charges and currents, there are also magnets in nature. (That magneticphenomena are closely related to electric ones took scientists over 2000 years to figure out.)Let us record that

• magnets always come with a ‘north’ and ‘south pole’ (in contrast to charges, wherepositive and negative ones can be separated);

• magnets are influenced by electric currents (i.e. the latter exert forces on the former).

1.2 From forces to electric and magnetic fields

Electric fields: One finds experimentally that

• two charged bodies (carrying non-zero charges q1 and q2) exert forces onto each other;the force is attractive if the two charges have opposite sign, the force is repulsive ifthe two charges have equal sign. Quantitatively, in the case of two test charges (i.e.point-like charged bodies) positioned at ~x1 and ~x2 one finds the force

~F12(~x) = const ·q1 q2|~x|3

~x (1.3)

where ~x is the vector between the two charges (i.e. ~x = ~x1 − ~x2) and where |~x|is the length of this distance vector (i.e. simply the distance between the two pointcharges). The force law given in (1.3) is called Coulomb force law; it is of fundamentalimportance in electromagnetism.

• If one replaces the point charge q2 by an arbitrary charge distribution described byρ(t, ~x), then one can still measure the (more complicated) force ~F (t, ~x1) acting on thepoint charge q1 at (t, ~x1). One finds that this force is always proportional to q1, and

so it seems a good idea to define the electric field ~E(t, ~x1) by

~E(t, ~x1) :=1

q1~F (t, ~x1) . (1.4)

~E is the electric field induced or generated by the charge distribution ρ. (Just like ~

from above, ~E is a vector-valued map from R × R3, hence ‘field’.) ~E is independentof q1, it can be viewed as describing how the presence of charges (above: q2 or ρ(t, ~x))alters properties in the surrounding space. If other charges (like the test charge q1above) are brought into a region of space with ~E 6= 0, they experience a force ~F = q1 ~E

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– which is why we could use a ‘test charge’ to probe the properties of the spacesurrounding a given charge distribution and to define ~E as in (1.4) in the first place.We will later see how the field induced by a charge distribution can be calculated fromthe latter – and vice versa. But we will also see that there are other ways to generatea non-vanishing electric fields, even if no charges are present.

As a first application of (1.4), we note that the electric field induced by a single pointcharge q2 fixed at the origin is

~E(t, ~x) = const ·q2|~x|3

~x

with the same constant of proportionality as in (1.3); this is called the Coulomb field.

Magnetic fields: Observing the behaviour of magnetic ‘test needles’ (tiny compass needles,say), one finds that there are forces between two magnets and between magnets and electriccurrents. Analogous to the electric case, one can say that a magnet alters the propertiesof its ambient space in such a way that forces act on other magnets (or electric currents)brought into that region. This new property of the ambient space is encoded in assigninga magnetic field ~B(t, ~x) to its points.

• The direction of the vector ~B(t, ~x) assigned to a point (t, ~x) in R×R3 is the directioninto which (the north pole of) a magnetic test needle points, when it is brought intothe magnetic field at time t and position ~x.

• The magnitude of ~B is defined with the help of the following additional fact: A testcharge q that moves with velocity ~v through a magnetic field ~B experiences a forcewhose magnitude is proportional to q and to |~v|, and which is orthogonal to ~v and to~B; therefore one defines

~F =: q ~v × ~B , (1.5)

where × is the vector product, to fix the magnitude | ~B| of the magnetic field.

(With this definition, the physical unit of ~B is 1 sm ·(the unit of~E) and is called ‘1

Tesla’; in some textbooks, you will find an additional factor 1/c on the right handside of their definition (1.5), since different conventions are used; c denotes the speedof light.)As mentioned above, magnetic fields also exert forces on electric currents (e.g., a pieceof wire that carries an electric current experiences a force in a magnetic field). Thisfollows directly from (1.5) since electric currents consist of moving electric charges.

The total force that acts on a particle with charge q if both electric and magnetic fields(or: ‘an electromagnetic field’) are present is summarised in the formula for the so-calledLorentz force

~F (t, ~x) = q(~E(t, ~x) + ~v(t, ~x) × ~B(t, ~x)

); (1.6)

here, ~x is the coordinate of the particle at time t; the velocity ~v of the particle may dependon t and ~x; the Lorentz force depends only on the values of ~E and ~B at the position of theparticle, not on the fields at other space-time positions (one says that the Lorentz force is‘local’).

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If instead of a charged particle, we consider a charge density ρ in an electromagnetic field,we are led to consider a Lorentz force density

~f(t, ~x) = ρ(t, ~x)(~E(t, ~x) + ~v(t, ~x) × ~B(t, ~x)

)

= ρ(t, ~x) ~E(t, ~x) + ~(t, ~x) × ~B(t, ~x) ;(1.7)

the total force ~FV acting on the portion of charge contained in the volume V is given by~FV =

∫V~f(t, ~x) d3x .

The Lorentz force specifies how charged particles are influenced by electromagnetic fields.Together with Maxwell’s equations, which tell how ~E and ~B are generated by charge distri-butions and how they spread through space-time, they allow to derive all electromagneticphenomena.

A note on the vector character of ~E and ~B: Whenever we have a physical quantity that isa vector, like a velocity or a force or like the electric and the magnetic field, we know thatit has several (three) components. Or we say ‘a vector is a quantity that has a magnitudeand a direction’ (length and direction of a vector arrow). We can specify the vector if wegive its three components (x-, y- and z-component in a Cartesian coordinate system).Vector quantities have a property which we usually do not state explicitly, but which ismore or less intuitively clear: If we use a different coordinate system to describe the com-ponents of a given (fixed) vector, then we have to use new components to describe thissame vector. E.g., think of a velocity vector, pointing to your left: if you turn by 180degrees, then that same velocity vector points, in your new coordinate system, to yourright. I.e., the components of the velocity have also been rotated by 180 degrees. Thisgeneralises to arbitrary rotations of the coordinate system and can be put into a formula:Let S and S′ two Cartesian coordinate systems for three-dimensional space, whose coordi-nate axes are related by a rotation. This means that there is a rotation matrix R ∈SO(3)such that for each point in space, its coordinates ~x taken with the coordinate system S andits coordinates ~x′ taken with S′ are related by ~x′ = R~x. Then any other vector quantity(like a velocity, force, electric or magnetic field) has the same transformation property; i.e.

if ~F and ~F ′ denote the components of that vector taken using the axes of S resp. thoseof S′, then we have ~F ′ = R~F . (In other words: the components of any 3-vector quantitybehave under rotations in the same way as the components of the position vector.)In the electromagnetism part of this lecture, we will usually just pick a convenient coordi-nate system and perform all computations there. In the special relativity part, however,the behaviour of physical quantities under changes of coordinate systems will play a centralrole. Moreover, we will have larger, and less intuitive transformation group to contend withthan just the group of rotations in three-dimensional space (namely the Lorentz group).We will then take the above characterisation of vectors (more precisely: of ‘3-vectors’, sinceour present examples all have three components) and generalise it to a definition of whatwe mean by a ‘vector with respect to transformations from the Lorentz group’. It will inparticular turn out that the vectors needed in special relativity have four components.

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1.3 Visualising fields

Both ~E and ~B are vector-valued fields in 1 + 3 dimensions, i.e. maps

~E :

R × R3 −→ R3

(t, ~x) 7−→ ~E(t, ~x) =

E1(t, ~x)E2(t, ~x)E3(t, ~x)

;

likewise for the current density ~ and for some other quantities we will meet later. Thecharge density ρ is a function from a four-dimensional into a one-dimensional space (hence‘function’),

ρ :

{R × R3 −→ R(t, ~x) 7−→ ρ(t, ~x)

.

(In physics, maps from Rd to R are also called ‘scalar fields’.)

Working with such objects instead of ordinary functions f : R −→ R of a single variabledoes not pose great difficulties at an abstract level, because we have the tools of multi-dimensional calculus available. However, we often would like to have a rough ‘picture’ offunctions and fields, as well. This is no problem for functions of a single variable: We cansimply draw the graph of f in a two-dimensional diagram, with the argument x of f puton the x-axis and with the value f(x) depicted in the vertical y-direction. In principle (upto inaccuracies of the drawing, and the finite thickness of the pen), such a graph containscomplete information about the function f .

For (scalar or vector) fields from Rd to RD with d or D greater than 1, we can no longerdraw a graph on a sheet of paper (since the dimension of the paper is less than d+D), soone has to think of other possibilities to visualise such fields at least partially.For scalar fields F : R2 −→ R, one can draw equipotential sets into the plane R2. Thesesets EF (s) are the subsets of R

2 where F takes a given value s, i.e.

EF (s) := { ~x ∈ Rd | F (~x) = s } .

This is well-defined for all s ∈ R, the image space of the function.Here, the abstract definition was already given for arbitrary dimension d of the space whereF is defined. For d = 1 (an ordinary function), EF (s) can be empty (if s is not a valuetaken by F ), it can consist of several points (all those xn ∈ R such that F (xn) = s – this isthe generic case), or it can coincide with all of R if F happens to be the constant functiontaking the value s. The generic case for d = 2 is that the equipotential sets are lines: E.g.,consider the function

F0 :

{R2 −→ R(t, ~x) 7−→ x21 + x

22

;

this specific function just gives the Euclidian distance of the point ~x = (x1, x2) from theorigin in R2. Thus, all its values are positive, so the equipotential sets are

EF0(s) empty for s < 0

EF0(0) = { (0, 0) } ⊂ R2

EF0(s) the circle of radius s around the origin in R2 for s > 0

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One can produce a rough picture of F0 by drawing the sets EF0(s) for some selected valuesof s into the plane R2, labelling each EF0(s) by the value of s. In the present case, thiswould give a series of circles around the origin, labelled by their radius. If we take thefunction F1(~x) := 1/F0(~x) instead, the equipotential sets would still be circles (for s > 0),but in the picture they would be labelled by their inverse radius squared – thus providinga the visual impression of the fact that F1(~x) decreases as we go away from the origin.Note that we cannot draw all equipotential sets at once, this would simply fill the sheet ofpaper. Rather we have to select a discrete set of s-values, so our drawing does not containcomplete information about F , not even in principle. Still, this method of visualisingfunctions from higher-dimensional spaces is used widely – examples are height lines ongeographic maps, or lines of constant pressure or temperature in weather charts.

If we try to picture the analogous functions for d = 3, i.e. the functions G0(~x) = x21+x

22+x

23

or G1(~x) = 1/G0(~x), we run into difficulties because now the equipotential sets (for s > 0)are spheres around the origin in R3, and our sheet of paper simply can’t accommodate thethree-dimensional space. One can at best give a perspective drawing in two dimensions,or draw two-dimensional projections of the equipotential sets (i.e. say that the sheet ofpaper stands for the first two coordinates of R3, or the last two, etc). For our two examplefunctions, using projections to two dimensions would again yield pictures of circles.

Figure 1.1: Pictures of a constant vector field, a Coulomb-like vector field, and of a

tornado field

Things become even more awkward if we try to provide a picture of a vector-valued field~F : Rd −→ RD with D > 1. Drawing the graph of F would mean to draw, for each~x ∈ Rd, the D-dimensional vector ~F (~x) – which can only be done on a (d+D)-dimensional‘sheet’ of paper.

Let us restrict to d = D = 2 first. There are two main methods to visualise a vector field~F : R2 −→ R2. For one thing, one can select a discrete set of points ~xn in the base space(being our sheet of paper) and attach the arrow ~F (~xn) to the point ~xn in the drawingplane. This sketch at least tells the direction and the strength (length of the arrow) ofthe vector field in each of the selected points. Three examples are depicted above. On theleft, a constant vector field is shown, where the same image vector (here pointing verticallyupward) is attached to each point ~xn (here chosen from a regular lattice). The picture inthe middle is (a two-dimensional analogue of) the Coulomb field or the Coulomb force; thearrows are longer closer to the origin, meaning that the vector field is stronger there. Thepicture on the right is typical for the velocity field of a tornado; in that context, the arrow

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attached to the point ~xn indicates the wind velocity at that point.

If the dimensions d or D of the base resp. the target space are greater than 2, we againhave to be content with two-dimensional projections of arguments and image vectors.

Pictures of vector fields containing image vectors like above give a reasonably good im-pression of the strength of ~F at the selected points (via the length of the arrows), butthey convey a rather ragged view of continuous changes of the vector field. This aspectis captured more convincingly by the second method to visualise vector fields, namely bydrawing field lines. Here, one proceeds as follows: Pick a point ~x0 in the base space R

d,attach ~F (~x0) to that point and move infinitesimally from ~x0 in direction of ~F (~x0); at thisneighbouring point, do the same thing – ‘and so on’. In this way, one constructs pictureswith lines in Rd such that, at each point ~x on the line, the tangent to the line is in directionof ~F (~x). For the three examples from above, the field line pictures look as follows:

Figure 1.2: Field line pictures for the vector fields from above

Each line is marked by a little arrow indicating the direction of the vector field. Suchpictures give a good impression of smooth changes of ~F . Information about the strengthof ~F is lost, at first, but is often encoded indirectly in the density of lines (higher densityindicating stronger field).

In the case of an electric field induced by some electric charge Q, field lines have a verydirect physical interpretation: They give the trajectories of a positive test charge that isbrought near the given charge Q and then let loose. Such a test charge will then moveunder the influence of the force exerted on it by Q. (Note that the test charge is thoughtof as point-like, while the given charge Q in fact can be any configuration of point chargesor a continuous charge density.) Because of this connection between trajectories of testcharges near a given charge configuration Q on the one hand and field lines of the electricfield ~EQ induced by Q on the other, one can often ‘deduce’ the symmetry properties of ~EQby simply ‘staring at the charge configuration’. E.g., if Q is also a point charge, it is clearthat the electric field induced by it must be symmetric under rotations around the pointwhere Q sits, so one expects a spherically symmetric field line picture. More precisely,we have observed that charges attract or repel each other, so the test charge would eithermove directly away from or directly towards Q. Furthermore, the force between Q andthe test charge falls of with the distance squared between the two – so we conclude thatindeed the middle pictures from above must be how the Coulomb field roughly looks (intwo-dimensional projection, of course).

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The field lines of a magnetic fields can be explored with the help of a compass needle (a‘test needle’): At any point in Rd, the needle is tangent to the magnetic field line passingthrough that point.

In electromagnetism, we will encounter three different types of field lines: Some are closedlike the eddies of the tornado field, some field lines are infinite without beginning or endlike those of a constant vector field, and some field lines start and/or end at some finitepoint in the base space, like those of the Coulomb field depicted in the middle. (Pointsfrom which field lines emanate are called sources, those where they end sinks.) We will see

that for the electric field ~E all options are available, while the third one is impossible formagnetic fields ~B. (Starting and end points of electromagnetic field lines are interpretedas charges, and there are no magnetic charges.)

1.4 Line and surface integrals

Looking at field line pictures, one ‘sees’ that there are obvious qualititative differencesbetween a tornado field, a constant field and a field with sinks and sources. To describethem quantitatively, one can use integrals.

Let ~K : R3 −→ R3 be a vector field and ~γ : [0, 1] −→ R3 be a closed curve, which weassume to be smooth and oriented; ‘closed’ means ~γ(0) = ~γ(1). Denote by C ⊂ R3 theimage of the interval [0, 1] under ~γ.

The line integral of ~K over the curve C is defined as

∮

C

~K · d~x :=

∫ 1

0

~K(~γ(s)

)·d

ds~γ(s) ds . (1.8)

Such closed line integrals measure the ‘vorticity’ of the vector field ~K (a property we

naturally associate with tornado fields). To see this, imagine ~K has a closed field line

along a curve C. Then everywhere along C, ~K is tangential to C, by definition of a fieldline. But the tangent of C in a point ~γ(s) is just d~γ(s)/ds, so if C is a closed field line of~K then the integrand of (1.8) is (d~γ(s)/ds)2, up to a positive factor, thus the integrand iseverywhere positive, thus the integral is positive.On the other hand, if ~K is a constant vector field (i.e. rather far from being a tornadofield), one can show with the help of Stokes’ theorem (see Section 2.2) that (1.8) vanishes.

The other type of integral used to characterise vector field quantitatively are surface inte-grals. If S ⊂ R3 is a smooth and oriented surface, the integral

∫

S

~K(~x) · d~S(~x) , (1.9)

the ingredients of which were already defined around eq. (1.2), is called the flux of ~Kthrough the surface S. (Compare to the electric current, which is a flux of electric chargesthrough some area.)As for the case of line integrals, the most interesting case occurs when the surface S isclosed (has no boundary; e.g. a sphere). This is the case whenever S is itself the boundarysurface ∂V of some volume V in R3 (e.g. a ball). Assume for simplicity that this is the

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case, i.e. S = ∂V , and choose the orientation of S such that the normal vectors ~n pointout of V everywhere. Then the flux through S measures ‘how many more field lines leavethe volume V (through the boundary S) than enter V (through the boundary S)’.Thinking back to the pictures of field lines, we see that surface integrals over a closedsurface tell us something about whether ~K has sources or sinks inside V . (If there are

no sources or sinks, e.g. for a constant ~K, then all fluxes through a closed surface vanish,intuitively ‘because as many field lines come out as go in’.)

2. Maxwell’s equations

We will now write down Maxwell’s equations in two forms and give a general discussionof some of their properties. Maxwell’s equations can be viewed as equations of motion ofelectrodynamics; instead of (generalised) coordinates qa(t) and their canonical momentapa(t) familiar from classical mechanics, the fundamental ‘variables’ in the equations of

motion are now the fields ~E(t, ~x) and ~B(t, ~x).

Together with the equation of motion m~̈x = ~FLorentz for a charged particle moving throughan electromagnetic field, Maxwell’s equations provide a complete description of any (clas-sical) electromagnetic phenomenon. (As soon as quantum effects play a role, one has toreformulate the theory.)

The first two sections of this chapter give Maxwell’s equations in two different forms,as integral and as differential equations. Then we will collect some general observationsand some remarks on their interpretation, while Section 2.4 deals with the continuityequation, which can be derived from Maswell’s equations and describes conservation of(net) electric charge. In Section 2.5, we will rewrite Maxwell’s equations in terms of new(fewer) field variables, the so-called scalar and vector potentials, which can be done becausethe equations are invariant under ‘gauge symmetry transformations’.

2.1 Integral form of Maxwell’s equations

We denote by ~E(t, ~x) the electric field and by ~B(t, ~x) the magnetic field induced by a chargedensity ρ(t, ~x) and a current density ~(t, ~x). We introduce a constant c ≈ 3.0 × 108m s−1

with the unit of a velocity (one finds in measurments that c coincides with the speedof light in vacuum – which as we will see later is not a coincidence); and we introduceanother constant ǫ0 = (4π × 10

7N A−2)−1c−2, often called the ‘dielectricity constant (ofthe vacuum)’. (The ‘N’ in that definition means ‘Newton’, the unit of a force.)Let furthermore C be a closed (oriented) curve in R3 and SC ⊂ R

3 a surface with (oriented)boundary ∂SC = C. Let V ⊂ R

3 be some volume with boundary given by the closedsurface ∂V = SV , endowed with ‘outward orientation’. (All relative orientations of setsand their boundaries are as in Stokes’ theorem or the divergence theorem used in Section2.2. All these sets are supposed to be continuous, or even smooth for simplicity; else theyare arbitrary up to the relations spelt out.)

Maxwell’s equations in their integral form read:

∮

C

~E · d~x = −d

dt

∫

SC

~B · d~S (M1)

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∮

C

~B · d~x =1

c2d

dt

∫

SC

~E · d~S +1

ǫ0 c2

∫

SC

~ · d~S (M2)

∮

SV

~E · d~S =1

ǫ0

∫

V

ρ d3x (M3)

∮

SV

~B · d~S = 0 (M4)

For ease of notation, we have not made explicit the (t, ~x)-dependence of all the integrands

(including the oriented area element d~S(t, ~x)).Note that the left hand sides of Maxwell’s equation involve precisely those integrals of thevector fields ~E and ~B which we used to characterise vorticity and sources.Note also that the charge and current densities ρ and ~ appear only in (M2) and (M3);these are called ‘inhomogenous’ Maxwell’s equations, the other two the ‘homogenous’ ones.

Warning: Textbooks often use different conventions for the units of ~E and ~B etc. There-fore, Maxwell’s equations as found in some books may look slightly different (because ofadditional factors of c and the absence of ǫ0, e.g.) from those given above.

Moreover, you might encounter symbols ~D and ~H in addition to ~E and ~B in Maxwell’sequations as stated in textbooks. This distinction plays a role if one considers electrody-namics ‘in a medium’ (necessary e.g. if one wants to have an effective description of lightpropagation through a medium like glass). We will not discuss this issue here (see the

appendix for some remarks) and work with ~D = ~E and ~H = ~B.

2.2 Differential form of Maxwell’s equations

Because of the connection to vorticity and sources, the integral form of Maxwell’s equationsis rather intuitive in certain respects and sometimes even useful for computations. But itis rather far from the equations of motions we know from classical mechanics. The latterare (ordinary) differential equations, and we can indeed produce a differential form ofMaxwell’s equations (now partial differential equations) by exploiting two classic theoremsfrom calculus. Let K : U −→ R3 be a differentiable vector field

~K(~x) =

K1(~x)K3(~x)K3(~x)

defined on an open subset U ⊂ R3. Then one can define the curl of ~K by

curl ~K := ~∇× ~K =

∂yK3 − ∂zK2∂zK1 − ∂xK3∂xK2 − ∂yK1

(this is again a vector field on U) and the divergence of ~K by

div ~K := ~∇ · ~K = ∂xK1 + ∂yK2 + ∂zK3

12

(this is just a function on U , sometimes also referred to as a ‘scalar field’). Here ∂x =∂∂x

etc.

Stokes’ Theorem: Let U and ~K be as above, and let C ⊂ U be a smooth curve in Uand SC ⊂ U be an oriented surface whose boundary is C (i.e. ∂SC = C). We choose theorientation of SC according to the ‘right hand rule’: thumb in direction of normal vectorof SC , then the fingers point in direction of C.For any such C and SC , we have

∮

C

~K · d~x =

∫

SC

(~∇× ~K

)· d~S .

Gauss’ Divergence Theorem: Let U and ~K be as above, and let V ⊂ U be a volume withsmooth boundary SV = ∂V , such that also SV ⊂ U . We choose the orientation of SV suchthat the normal vectors point to the outside of (i.e. away from) V .For any such V and SV , we have

∮

SV

~K · d~S =

∫

V

~∇ · ~K d3x .

We can apply these theorems to the integrals in Maxwell’s equations; that the integrandsalso depend on time does not cause problems. One arrives at Maxwell’s equations in theirdifferential variant:

~∇× ~E = −∂

∂t~B (M1)

~∇× ~B =1

c2∂

∂t~E +

1

ǫ0 c2~ (M2)

~∇ · ~E =1

ǫ0ρ (M3)

~∇ · ~B = 0 (M4)

Again, we have dropped the (t, ~x)-dependence. We spell out the derivation of the differ-ential form from the integral one by applying Stokes and Gauss for (M1) and (M3), theother two equations are dealt with analogously (see Problem Sheets). For (M1), we canfirst of all rewrite the rhs

∮

C

~E · d~x = −d

dt

∫

SC

~B · d~S = −

∫

SC

∂

∂t~B · d~S

if we assume that the surface SC does not change with time (so that ~B is the only part

that depends on time and we can pull the time-differential inside the integral; since ~B also

13

depends on ~x, the total time derivative becomes a partial derivative). The lhs can now berewritten with the help of Stokes, and we get

∫

SC

(~∇× ~E

)· d~S = −

∫

SC

∂

∂t~B · d~S .

This holds for all surfaces SC (which are constant in time), so we can infer (e.g. by takinga very small surface concentrated around a single point) that the integrands themselvesmust be equal – which implies (M1) in differential form.(Note that equality of the integrals of two functions over only one specific surface doesnot imply that the two functions are the same; to make this conclusion we need equalityto hole for all SC . As a simply one-dimensional analogue, take e.g. the integral of thefunction f(x) = x over the interval [−1, 1]; the integral is zero, just as the integral of thezero function – but f(x) 6= 0. And if one picks another integration interval, f(x) canindeed yield a non-zero integral.)

In the case of (M3), we have

1

ǫ0

∫

V

ρ d3x =

∮

SV

~E · d~S =

∫

V

~∇ · ~E d3x

by applying the divergence theorem to the ~E-integral. Again, the first and the last termmust be equal for arbitrary volumes V , so we can infer (e.g. by taking a very small volumeV concentrated around a single point) that the integrands must coincide – which implies(M3) in differential form.

Side remark: In general, the integral form is slightly stronger than the differential form:It depends on the topology of U whether we can also derive the integral form of MEs fromtheir differential form; for U = R3 (meaning in particular that our vector fields are definedeverywhere), this is true.

2.3 General remarks on Maxwell’s equations, and interpretation

Maxwell’s equations constitute the equations of motion of the theory of electromagneticfields; the fields ~E(t, ~x) and ~B(t, ~x) are dynamical variables that generalise the point par-ticle coordinates qa(t) familiar from classical mechanics. More specifically, a field can bethought of as a generalisation of particles to an object with infinitely many degrees offreedom:A single point particle in R3 is described by ~q(t) and has 3 degrees of freedom. A collec-tion of N point particles is described by coordinates ~qi(t) with i = 1, . . . , N , and has 3Ndegrees of freedom. We can take the limit N → ∞, so that the label i can take infinitelymany values – and we can even imagine that i is taken from a continuous set, and arriveat the ‘label structure’ of vector fields like ~E(t, ~x) = ~E~x(t), where ~x replaces the label i.To push this analogy a little further, note that the typical interaction between the N par-ticles in such a collection is given by a sum of potentials V (~qi(t)− ~qj(t)) depending on thedistance of two particles, and that typically the interaction is strongest when two particlesare close to each other, while interactions between particles that are far away from each

14

other can often be ignored. Letting the ‘number’ of particles become continuous, but keep-ing this last property, we arrive at the idea that the only noticeable interaction is between~q~x(t) and an infinitely close ~q~x

′(t); if ~q~x(t) depends continously on ~x, this means that theonly noticeable interaction is between ~q~x(t) and ~q~x ′(t) at an infinitely close ‘label’ ~x

′. Thusit is not so surprising that partial derivatives (with respect to ~x) of the fields appear inMaxwell’s equations alongside the time derivatives familiar from Newtonian mechanics.Note that these remarks are only meant as analogies, to make it more plausible thatMaxwell’s equations are really equations of motion. In many problems of quantum fieldtheory, one would very much like to be able to make these ideas work quantitatively (key-word: field theories as ‘continuum limit’ of discrete systems), but this is a formidable task.What one can do rather easily, on the other hand, is to show that both Newton’s equa-tions of motion and Maxwell’s equations are Euler-Lagrange equations for the variation ofLangrangians; see the textbooks.

Typical questions in electromagnetism are: How does a charged particle move through agiven electromagnetic field, treating ~E and ~B as fixed? This is answered analysing theequation of motion m~̈x = ~FLorentz of the particle governed by the Lorentz force. (Thecharge of the ‘test particle’ should be small so that we can ignore the fields that are gen-erated by the moving test charge, otherwise this ‘back-reaction’ of the test charge wouldchange the given fields.) This problem is essentially familiar from classical mechanics.The ‘complementary problem’ is: Given a prescribed distribution of charges and electriccurrents (described by the charge density ρ and the current density ~), what is the electro-

magnetic field ( ~E, ~B) generated by them? This is analysed by solving Maxwell’s equationsfor given ρ and ~, and will be the main topic of this course.In the most general situation, one has to take the back-reaction of a moving charges on thefields into account and has to consider Maxwell’s equations and the equations of motionwith ~FLorentz at the same time, as coupled differential equations. This type of problemscan typically not be solved exactly, and we will not try to tackle them here.

Maxwell’s equations have an important property which simplifies life considerably: Theyare linear partial differential equations:If ~Ei, ~Bi solve MEs for given ρi, ~i, with i = 1, 2, then ~E := α~E1+β ~E2 and ~B := α~B1+β ~B2satisfy MEs for ρ := αρ1 + βρ2 and ~ := α~1 + β~2, where α and β are arbitrary numbers.One also says that Maxwell’s equations satisfy the superposition principle.Likewise, the Lorentz force depends linearly on ~E and ~B.

Differential equations that are non-linear are much harder to solve. But the linearity ofMEs also plays an enormous role for purposes of everyday life, e.g. by making it possibleto ‘superpose’ electromagnetic waves without them disturbing each other (transmission ofmany phone calls or radio programmes through a single cable or through the air). (More tothe point, linearity is responsible for the fact that only innocent bystanders are disturbed,but not the person with the mobile.)

We will furthermore see that Maxwell’s equations have non-trivial solutions even if ρ = 0and ~ = 0. This e.g. accounts for the fact that electromagnetic waves (light) can propagatethrough the vaccum.

15

Interpretation of (M3) and (M4): We use the integral forms in this discussion. Recall from

Section 1.4 that the integral∮SV

~K · d~S measures whether ‘more’ field lines of ~K enter thevolume V than it, i.e. whether field lines end inside V , i.e. whether there are sources orsinks of ~K within V . Therefore:

(M3) Field lines of the electric field ~E can end on electric charges only, since the integralover SV is non-zero precisely iff the charge density inside V is non-zero; this holdsfor all V , in particular for very small ones, which are concentrated tightly arounda point).The third of Maxwell’s equations, also called Gauss’ law, in particular implies thatelectric charges generate electric fields: If there’s some charge, the rhs of (M3) isnon-zero (for a suitable V ), thus the electric field can’t vanish.

(Note: This does not imply that ~E(t, ~x) = 0 at points (t, ~x) where there is nocharge – see the remark above about the existence of non-trivial solutions to MEswith ρ = 0, or look at the Coulomb field, which is non-zero everywhere even ifthere’s only a single point charge around. Gauss’ law merely says that no field lineof ~E can end in a point where no charge sits.)

(M4) says that there simply are no sources or sinks for the magnetic field. No field lines

of ~B can start or end within a finite volume V . (In other words, all magnetic fieldsare such that there field lines are closed or extend to infinity.)

So one could criticise that ~E and ~B enter Maxwell’s equations in a somewhat ‘asymmetric’way and that ME are not ‘quite as beautiful’ as they could be. Indeed, this led Diracto propose that magnetic charges do exist (‘magnetic monopoles’), only have not beendiscovered in nature, yet.

Interpretation of (M1) and (M2): Recall from Section 1.4 that the integral∮C~K · d~x

measures whether ~K has closed field lines.

(M1) says that ~E with closed field lines can be generated iff the flux∫SC

~B · d~S of ~Bthrough a surface SC bounded by the curve C is non-zero and varies with time(without the help of electric charges, since ρ doesn’t appear in this ME).(M1) is also called Faraday’s law of induction. He discovered that, if one bringsa magnet near an open loop of wire and wiggles the magnet around a bit, thenan electric voltage is induced between the ends of the loop: Wiggling the magnetchanges the ~B-field that penetrates the surface (= SC) bounded by the loop (= C),and the voltage is due to an electric field induced along the wire. This was one ofthe first controllable experiments that established that electric and magnetic fieldsare related. Faraday also found the quantitative relation (M1).

Note that (M1) does not require ~B to change with time, only its flux. Thus one can

generate an electric field also by leaving ~B constant but changing the area of SC thatis penetrated by ~B. This principle is used in bicycle dynamos, where SC rotates (aswheels turn) relative to the constant magnetic field induced by an ordinary magnet;see Problem Sheets.

(M2) has two terms on the rhs. The first one is analogous to the situation described by

(M1), with ~E and ~B interchanged (and some constants put in). Its interpretationis therefore that a non-zero magnetic field with closed field lines is generated by an

16

electric flux∫SC

~E · d~S through SC , which varies with time.

That this works was postulated by Maxwell on theoretical grounds (without thisfirst term on the rhs of (M2), one cannot derive the continuity equation from MEs,cf. the next section and the Problem Sheets); no experimental evidence for this factwas known at the time.The second term on the rhs of (M2) was based on experiments. It means that anon-zero magnetic field with closed field lines is generated by letting an electriccurrent flow through a surface SC . Orientations again follow the right hand rule:~B is in direction of the four fingers if the current ~ flows in direction of the thumb.This fact was discovered before Maxwell by Ampère; therefore, (M2) in the absence

of ~E, i.e. the equation ∮

C

~B · d~x =1

ǫ0 c2

∫

SC

~ · d~S (2.1)

is also called Ampère’s law.The first term on the rhs of (M2) is often referred to as the flux of ‘Maxwell’s

displacement current ~Max :=∂∂t~E through SC ; this displacement current has of

course a totally different nature than the ordinary electric current ~ (which arisesfrom moving electric charges), but its role in (M2) is analogous to that of ~.

Maxwell’s equations show that electric and magnetic fields are intimately linked to eachother; this will become even more manifest in a relativistic reformulation of electrodynam-ics, where one sees that both fields are just different components of a single object (theelectromagnetic field strength tensor with six independent components). We will also seethat what looks like an electric field seen from one inertial frame looks like (an electricplus) a magnetic field from another inertial frame that is in motion relative to the firstone.

2.4 The continuity equation, and charge conservation

The continuity equation is the simplest conservation law that follows from Maxwell’s equa-tions. As an example for a conservation law in classical mechanics, let us recall conservationof the total energy of a point particle (with mass m) moving in one dimension under the in-fluence of a force F (x) = − d

dxV (x). The particle obeys the equation of motion mẍ = F (x).

Then the total energy E := m2 ẋ2 + V (x) of the particle is conserved during the motion:

d

dtE =

m

22ẋ ẍ+

d

dxV (x) ẋ = ẋ [mẍ− F (x)] = 0

Note that we have to insert the equation of motion (here: Newton’s second law) in orderto show that the total energy is constant in time.

In electromagnetism, conservation of energy is more subtle to derive, we will first need todefine what the energy stored in electromagnetic fields is. But it is fairly easy to deriveconservation of electric charge from Maxwell’s equations. (In analogy to the mechanicsexample from above, it will be necessary to insert MEs, i.e. the equations of motion forelectromagnetic fields, during the derivation. The continuity equation is a conservation law

17

that holds because of the special kind of dynamics dictated by the equations of motion;there are also conservation laws that we expect to hold independently of the details ofmotion, like conservation of total momentum to be discussed in Chapter 9.)

Consider then the rate of change (in time) of the electric charge contained in some volumeV :

d

dt

(∫

V

ρ d3x

)(M3)= ǫ0

d

dt

∫

SV

~E · d~S (2.2)

We have used Maxwell’s third equation (i.e. one of the equations of motion of electrody-namics) to rewrite the total charge contained in V as a surface integral of the electric fieldover the surface SV that bounds V . Next we use (M2)

∮

C

~B · d~r =1

c2d

dt

∫

SC

~E · d~S +1

ǫ0 c2

∫

SC

~ · d~S

which holds for any curve C and any surface SC that is bounded by C. In particular, wecan take the limit where C approaches a point, i.e. where SC becomes a closed surface: Inthat limit, the lhs of (M2) vanishes (since C does), and we get

0 =1

c2d

dt

∫

Sclosed

~E · d~S +1

ǫ0 c2

∫

Sclosed

~ · d~S (2.3)

for any closed surface Sclosed. The surface SV is such a surface (since SV is the boundaryof V , it has no boundary itself), thus we can insert (2.3) into (2.2) and obtain

d

dt

(∫

V

ρ d3x

)= −

∫

SV

~ · d~S , (2.4)

which is called continuity equation. Its physical interpretation can be phrased as follows:Any change of the amount of electric charge contained in a volume V is entirely due totransport of charge (i.e. an electric current) through the boundary SV of that volume.Charge can move away but cannot not simply appear or disappear from or to nowhere. Inother words: The total electric charge is conserved.

(Recall here that initial remark about ‘generating’ electric charge by rubbing amber witha piece of wool: Actually, electric charge is just transferred from the wool to the amber,not generated from nothing. So no contradiction here.)

Just like for Maxwell’s equations themselves, we can pass to a differential variant of thecontinuity equation. Assume that V (thus also SV ) is constant in time (so we can pullthe time derivative under the volume integral). Since V is otherwise arbitrary, Gauss’Theorem applied to (2.4) yields

∂

∂tρ+ ~∇~ = 0 . (2.5)

This equation can also be derived directly from the differential form of Maxwell’s equations.The form of (2.5) (partial time derivative of a function plus divergence of a vector fieldequals zero) is typical for conservation laws in (relativistic) field theory.

18

2.5 Vector and scalar potentials, gauge invariance

We will now show that we can rewrite Maxwell’s equations using new field variables ϕ, ~Ainstead of ~E, ~B. While ~A is again a vector field, ϕ is a function (or a ‘scalar field’), sothat only 3+1 instead of 3+3 component functions remain. This reduction of unknownsis achieved by solving two of the four Maxwell’s equations, namely the homogenous ones(those not involving the charge or the current density).

Let us start with the differential form of (M4),

~∇ ~B = 0 :

Standard theorems from three-dimensional calculus guarantee that any vector field withzero divergence can be written as curl of another vector field,

~B(t, ~x) = ~∇× ~A(t, ~x) . (2.6)

(More precisely, the theorem asserts that if a vector field satisfies ~∇ ~B = 0 everywhere,then for each ~x ∈ R3 there exists a neighbourhood U of ~x such that there exists a vectorfield ~A such that for all ~x′ ∈ U we have ~B(~x′) = ~∇× ~A(t, ~x′). The time coordinate t justplays the role of a spectator here. To prove this theorem, it is sufficient to give an explicitformula for ~A: One can check that (2.6) is e.g. satisfied by

A1(t, ~x) =

∫ z

z0

B2(t, x, y, z′) dz′ −

∫ y

y0

B3(t, x, y′, z) dy′

A2(t, ~x) = −

∫ z

z0

B1(t, x, y, z′) dz′ ,

A3(t, ~x) = 0 ;

here, y0 and z0 are arbitrary as long as all the (x, y′, z′) to be integrated over are in U .)

If we insert (2.6) into the other homogenous ME (M1) and use that time and space deriva-tives commute, we get

~∇× ~E +∂

∂t~∇× ~A = 0 .

A similar theorem to the one used before states that for any vector field ~K whose curlvanishes everywhere, i.e. ~∇ × ~K = 0, there exists a neighbourhood U of each ~x and afunction ϕ defined on U such that ~K = −~∇ϕ: Curl-free vector fields are (locally) thegradients of functions. (Again, to prove the theorem, one just needs to give an explicitformula for ϕ: Try the line integral

ϕ(t, ~x) = −

∫ ~x

~x0

~K(t, ~x′) · d~x′

along an arbitrary curve from some ~x0 ∈ U to ~x. Due to Stokes’ theorem, the integral isindependent of the curve because ~∇× ~K = 0.)

19

Now the vector field ~E + ∂∂t~A does have vanishing curl, by virtue of (M1). Thus it can (at

least ‘locally’, i.e. in a neighbourhood of the space point ~x) be written as the gradient ofa function ϕ, and we get

~E = −~∇ϕ−∂

∂t~A . (2.7)

Since the electric and the magnetic fields are given as derivatives of ϕ and ~A, the latterare called scalar resp. vector potentials. (Moreover, we will see during the discussion ofstatic electric fields that ϕ indeed encodes a potential energy.)

We can now plug eqs. (2.6) and (2.7) into the (differential versions of the) inhomogeneousMaxwell equations (M2) and (M3), and apply the following rule for the curl of a curl,

~∇× (~∇× ~A) = ~∇(~∇ · ~A) −△ ~A;

here, △ ~A is the vector field with three components △A1, △A2, △A3, where

△ := ~∇2 =∂2

∂x2+

∂2

∂y2+

∂2

∂z2

denotes the Laplace operator.All in all, we are left with the following equations (four on the whole) for ϕ and ~A :

△ ~A− ~∇(~∇ ~A+

1

c2∂

∂tϕ)−

1

c2∂2

∂t2~A = −µ0 ~ (M̃2)

△ϕ+∂

∂t~∇ ~A = −

1

ǫ0ρ (M̃3)

We have also introduced a new constant µ0, sometimes called ‘magnetic permeability ofthe vacuum’, as the abbreviation

µ0 :=1

ǫ0 c2. (2.8)

Although we now only need to deal with four differential equations on four unknowncomponents, instead of the eight differential equations on six unknowns from before, theform of the new equations is rather messy. It turns out, however, that we can beautifythem by exploiting gauge invariance. To this end, let us study what happens if we changea given pair ϕ, ~A in a certain way, namely as

ϕ 7−→ ϕΛ := ϕ−∂

∂tΛ

~A 7−→ ~AΛ := ~A+ ~∇Λ

(2.9)

where Λ = Λ(t, ~x) is an arbitrary (differentiable) function on space-time. Transformationsof scalar and vector potentials of the form (2.9) are called gauge transformations. (Notethat the gauge transformations in electromagnetism form an abelian group: We have(ϕΛ1)Λ2 = ϕΛ1+Λ2 , and analogously for

~A.)

20

Inserting the new scalar and vector potentials ϕΛ and ~AΛ into eqs. (2.7) and (2.6), we seethat

~BΛ := ~∇× ~AΛ = ~B

since the curl of a gradient vanishes, and also that

~EΛ := −~∇ϕΛ −∂

∂t~AΛ = ~E .

In other words, the electric and magnetic derived from the original and from the gaugetransformed potentials are the same: ~E and ~B are invariant under gauge transformations,or gauge invariant for short. The possibility to transform ϕ and ~A according to (2.9)without changing the electromagnetic fields is called gauge freedom.

While electromagnetic fields have definite physical meaning (they are defined by measuringforces on charged particles), scalar and vector potentials have been introduced for math-ematical convenience rather; they are auxiliary fields without direct physical meaning bythemselves. Therefore, we are entitled to change ϕ and ~A as long as the physical quantities~E and ~B remain unchanged. This fact can be used to impose additional constraints, calledgauge conditions, on the potentials, in such a way that the equations (M̃2) and (M̃3)

become simpler. E.g., we can require that ϕ and ~A satisfy the so-called Lorentz gaugecondition

1

c2∂

∂tϕ+ ~∇ ~A = 0 . (2.10)

This condition is to be understood as follows: Given a pair ϕ and ~A of arbitrary potentials(which might not yet satisfy the gauge condition), find a function Λ such that ϕΛ and ~AΛsatisfy eq. (2.10). After inserting, this means that we need to to find a Λ such that

Λ =1

c2∂

∂tϕ+ ~∇ ~A . (2.11)

Here, := 1c2

∂2

∂t2−△ is the d’Alambert operator, also called wave operator. It was just

in order to avoid heavier notation that we have renamed the gauge-transformed potentialsϕΛ and ~AΛ as ϕ and ~A when stating the gauge condition (2.10).We remark without proof that a function Λ such that the gauge-transformed potentialssatisfy the Lorentz gauge can always be found – in fact, many such Λ, because the Lorentzgauge condition (2.10) does not fix Λ completely: Performing another gauge transforma-tion using a function Λ̃ with Λ̃ = 0 does not disturb the condition (2.10).The Lorentz gauge is just one possible gauge condition that can be imposed on the poten-tials. In certain situations, it may be more advantageous to impose the so-called ‘Coulombgauge’ ~∇ ~A = 0, or some other condition.

With the extra condition (2.10), Maxwell’s equations can be rewritten in terms of fourdifferential equations of the same (relatively nice) type, supplemented by one differentialequation on the potentials to ensure the Lorentz gauge condition:

ϕ =1

ǫ0ρ ,

~A = µ0 ~ ,

1

c2∂

∂tϕ+ ~∇ ~A = 0 .

(2.12)

21

As (2.11) shows, the gauge condition in fact amounts to solving a fifth differential equationof the same type as the first four.The d’Alambert operator will re-appear later in connection with electromagnetic waves.Let us anticipate that an equation of the form f = g describes the propagation of awave f travelling away from a ‘disturbance’ g with the speed of light c.

3. Static fields

In this chapter, we will study a special class of electromagnetic phenomena: those where thefields ~E and ~B and also the densities ρ and ~ are time-independent. This assumption rulesout propagation of electromagnetic fields (like em waves), but rather concerns questionslike ‘what is the electric field induced by a static charge configuration (described by at-independent charge density ρ(~x)), or the magnetic field induced by a stationary currentdistribution ~(~x)’. (The assumption of t-independence is another idealisation which innature is only approximately satisfied: if e.g. a charge is brought to some point in space,an electric field will spread outwards from there with light-speed, and strictly speaking wewould have to wait infinitely long until a stationary situation has been reached everywhere;nevertheless, many realistic problems can be treated as static ones for all practical purposes,and anyway we can study such situations as a purely mathematical problems.)

In static situations all time-derivatives vanish, so Maxwell’s equations in differential formsimplify to

~∇× ~E = 0 , ~∇ ~E =1

ǫ0ρ ,

~∇× ~B = µ0 ~ , ~∇ ~B = 0 .

In particular, the equations involving ~E decouple from those involving ~B, so that elec-tric and magnetic phenomena do not mix in static situations. The first two equationsgovern electrostatics, the last two are those describing magnetostatics. The same decou-pling occurs for the integral version of Maxwell’s equations, where all terms with timederivatives drop out for static situations (if we assume that all integration domains aretime-independent, too).

As before in the general (non-static) case, the homogeneous equations of electrostatics andmagnetostatics can be solved by introducing scalar and vector potentials. In the staticcase, we can leave out the term ∂

∂t~A from the electric field and get

~E = −~∇ϕ , ~B = ~∇× ~A

for electrostatics resp. magnetostatics. From the remaining inhomogeneous equations, alltime derivatives drop out again, and it remains to solve

△ϕ = −1

ǫ0ρ , △ ~A = −µ0~

in electrostatics resp. magnetostatics; the latter equation is supplemented by the gaugecondition ~∇ ~A = 0, cf. (2.12).

22

Partial differential equations of the type △f = g are called Poisson equations, and forthe particular case g = 0 one often uses the name Laplace equation.

3.1 Electrostatics

The typical task in electrostatics is ‘given a charge distribution (i.e. a charge density) ρ(~x)

that does not depend on time, use Maxwell’s equations to compute the electric field ~E(~x)– then also t-independent – that is induced by ρ(~x)’.

ρ may be given for all ~x ∈ R3 or only in a bounded region of space; ~E then has to becomputed for the whole space or only in that bounded region, too. In the latter case,boundary conditions for ~E along the boundary of the region have to be prescribed.In order to solve this type of problems one can

• either use integral form of MEs – particularly useful if the charge distribution hassymmetries that can be exploited so as to simplify integrations

• or work with the differential form of MEs, express the electric field through a scalarpotential and then solve the Poisson equation for ϕ; one important (and moregenerally applicable) method to solve such linear PDEs is using Green’s functions.

We will look at examples for both methods, which are not unrelated either: The Green’sfunction is computed by solving the integral version of Maxwell’s equations for a particu-larly simple charge configuration.

3.1.1 The Coulomb field

First, we compute the electric field induced by a single point charge q placed at the origin,using MEs in their integral form, resorting to simple physical considerations and exploitingthe symmetry of the situation. The relevant equation for electrostatics is (M3),

∮

SV

~E · d~S =1

ǫ0

∫

V

ρ d3x .

(For the static case, the ME (M1) just states that the integral of ~E over any closed curvevanishes.)The charge configuration under consideration is invariant under rotations about the origin:The point charge ‘looks the same’ from whatever direction we look at it, there is nodistinguished direction in space. (Contrast this e.g. to a charged straight wire: therethe direction of the wire is certainly distinguished from other directions.) Rotational

invariance implies that the strength | ~E(~x)| of the electric field induced by the point chargeonly depends on how far the point ~x is away from the charge (here: from the origin), i.e.we have

| ~E(~x)| = | ~E(~x ′)| if |~x| = |~x ′| .

From the physical definition of electric fields (as force acting on a test charge divided by

the amount of the test charge), we also know the direction of ~E(~x): Any test charge (i.e.a second point charge) at ~x will move on a straight line towards or away from the point

charge at the origin (depending on the sign of the test charge), thus in our situation ~E(~x)points in ±~x-direction, thus

~E(~x) = E(r)~x

r(3.1)

23

with r := |~x| and some (positive or negative) function E(r) that depends on the distancefrom the origin only.(Note: If one feels that these arguments are too ‘physical’ to be rigorous, one can still usethem as a guide towards a solution of Maxwell’s equation for the given charge distribution,and then apply mathematical theorems which assert uniqueness of the solution as longas suitable boundary conditions at infinity are satisfied; see section 3.1.2 for a few moredetails.)

It remains to determine this function E(r); this can be done by applying (M3) to a suitablychosen surface SV . ‘Suitable’ means ‘making the surface integral in (M3) easy to carryout’, and the right choice is again suggested by the symmetries: SV should be such thatthe inner product ~E(~x) · ~n(~x) of electric field and outward-pointing normal unit vector

at any point ~x on the surface is a simple function. (Recall d~S = ~ndS.) In our case,because of (3.1), a simple inner product arises for a sphere around the origin, where we

have ~n(~x) = ~xr, i.e. ~n(~x) is proportional to ~E(~x). (That this might be a suitable choice of

integration surface should be obvious from the rotational symmetry of the problem: spheresaround the origin are precisely the surfaces that are invariant under rotations about theorigin.)Let therefore SV = S0(r0) be a sphere of radius r0 around the origin. It bounds a ballV = B0(r0) of the same radius around the origin, and

1

ǫ0

∫

B0(r0)

ρ d3x =q

ǫ0, (3.2)

since the total charge contained in the ball is the point charge q. The lhs of (M3) can beevaluated as follows:

∮

S0(r0)

~E(~x) · dS(~x) =

∮

S0(r0)

E(r)~x

r·~x

rdS = E(r0)

∮

S0(r0)

dS = E(r0) 4π r20 (3.3)

We have used that E(r) is constant on a sphere, and the area of a sphere in the last step.Eq. (3.3) holds for arbitrary r0 > 0, and putting it together with (M3), (3.2) and (3.1),we arrive at the formula for the electric field triggered by a point charge q sitting at theorigin,

~E(~x) =1

4π ǫ0

q

r3~x , (3.4)

also known as the Coulomb field.This electric field can be written as a gradient ~E = −~∇ϕ of the scalar potential

ϕ(~x) =1

4π ǫ0

q

r(3.5)

called the Coulomb potential. Note that both Coulomb field and potential become singularat the origin.(For completeness: As a consequence of ~E = −~∇ϕ, we automatically have ~∇ × ~E = 0,and thus (M1) is satisfied due to Stokes’ theorem. Furthermore, although we used a very

special SV in our construction of the Coulomb field, one sees that ~E from (3.4) satisfies

24

~∇ · ~E(~x) = 0 for all ~x 6= 0, and thus (M3) holds for all surfaces by virtue of the divergencetheorem.)

3.1.2 Distributions, Green’s functions, and the general solution to (M3)

The Coulomb potential ϕ from (3.5) has some rather remarkable properties: Since △ϕ =1ǫ0ρ and since there is no charge away from the origin, we have

△

(−

1

4π

1

r

)= 0 for all r 6= 0 ,

which can of course also be checked directly by differentiation. On the other hand, the‘function’ △ 1

ris extremely singular at the origin, because its integral over R3 is non-zero:

∫

R3

△

(−

1

4π

1

r

)=

∫

V

△

(−

1

4π

1

r

)=

∫

B0(1)

△

(−

1

4π

1

r

)= 1

where V ⊂ R3 is any volume that contains the origin, and where B0(1) is the unit ball

around the origin. In the very last step, we have used △ϕ = ~∇ ~E and Gauss’ Theorem –or simply the result from the proceeding section.We are forced to conclude that the object δ0 := △

(− 14π

1r

)cannot be an ordinary function

(since it vanishes everywhere except for the origin and nevertheless has integral 1); indeed,δ0 is an example for what is called a ‘distribution’:

Let U ⊂ Rn be an open set and denote by S(U), or simply by S if there’s no danger ofconfusion, the set of all smooth functions f : U −→ R that have compact support. (Recallthat a function is smooth if we can differentiate it infinitely many times. The support of afunction is the closure of the set of all ~x ∈ Rn where f(~x) 6= 0. A subset A of Rn is compactif it is closed (i.e. ∂A ⊂ A) and bounded (i.e. A fits inside an open ball of sufficiently largeradius).)S is a vector space (an infinite-dimensional one). Moreover, since every functions in Svanishes outside a bounded set and is smooth, it (and its modulus) can be integrated overRn; this allows to introduce a ‘distance’ between two functions f, g in S as

d(f, g) :=

[ ∫

Rn

|f(~x) − g(~x)|2 dnx

] 12

.

With this distance, we can declare what it means that two functions ‘are close together’ inS, and what a small open neighbourhood of a function is, just like open neighbourhoodsin Rn itself; S is a metric space. We call S endowed with all these structures the space oftestfunctions on U .The space of distributions on Rn is now simply defined as the vector space of all continuouslinear functionals on S(Rn). (A linear functional on a vector space V is a linear map fromV to R; continuity is defined with the help of the distance introduced above.)

Examples: Each ordinary function g ∈ S also defines a distribution, which we call Dg here:Let f ∈ S, then we define the value of Dg applied to f as

Dg[f ] :=

∫

Rn

g(~x) f(~x) dnx ∈ R ; (3.6)

25

the integral exists since both g and f are smooth and have compact support. (For Dg to bewell-defined, it is not even necessary that g ∈ S; any continuous or even locally integrablefunction g defines a distribution by virtue of (3.6).) So the set of distributions containsthe set of continuous functions.Another type of distributions is given by Dirac’s delta distributions: For each point ~a ∈ Rn,one defines a distribution δ~a which assigns to a function f ∈ S the value

δ~a[f ] := f(~a) ;

i.e., the delta distribution δ~a simply evaluates functions at the point ~a. One can check thatδ~a is a continuous linear functional on S, but also that these delta distributions are notof the type Dg for any function g; they are truly new objects. Since distributions containfunctions and more, they are also called ‘generalised functions’.

However, delta distributions can be obtained as limits of distributions of the type Dg: E.g.,we can take a sequence gm of smooth functions that vanish outside a ball of radius 1/maround the origin, have a peak of height m at the origin and are such that their integralover Rn is 1. Then we have that Dgm → δ0 as m → ∞. (On the space of distributions,there is again a notion of distance, so limits makes sense.) Explicit examples for sequencesof functions gm (though not with compact support) are given in the Problem Sheets.With such an approximation of δ~a by functions in mind, one often speaks somewhat looselyof ‘delta functions’, and imagines that the ‘graph’ of δ~a is an extremely sharp peak (ofinfinite height) at ~a and zero elsewhere. ‘Consequently’, one also uses the integral notationfor the delta distribution applied to a function:

δ~a[f ] =

∫

Rn

δ(~x− ~a) f(~x) dnx , (3.7)

pretending that there is a ‘delta function’ δ(~x−~a), in analogy to (3.6). From the definingproperties of δ~a, it follows in particular that this generalised function has the property thatfor each volume V ⊂ Rn

∫

V

δ(~x− ~a) dnx ={

1 if ~a ∈ V ,0 otherwise .

(3.8)

Other basic properties of δ~a can be verified easily using the ‘function notation’ from (3.7);e.g. we have

δ(α~x) =1

|α|δ(~x) for all α ∈ R . (3.9)

An important fact (and a somewhat striking one in view of the ‘spiky graph’ of δ) aboutdistributions is that they are always differentiable (infinitely often at that): Let T be adistribution on S, then its nth partial derivative with respect to xi is defined to be thedistribution acting as follows on testfunctions f ∈ S:

(∂n

∂xniT

)[f ] := T

[(−1)n

∂n

∂xnif

](3.10)

26

This can of course be extended to arbitrary differential operators on U . The definition(3.10) is motivated by the special distributions Dg from (3.6) with differentiable g (applyintegration by parts).

Distributions, in particular Dirac’s delta distributions, are very useful objects. First of all,more on the notational side, they allow us to treat singular charge distributions like pointcharges on the same footing as continuous ones: The ‘function’

ρ(~x) := q δ(~x− ~x0) (3.11)

is the charge density describing a point charge q sitting at the point ~x0; using (3.8), one seesthat the total charge inside some volume V is q iff that volume (however small) containsthe point ~x0.The other use that we will make of distributions is more fundamental, and concerns Green’sfunctions. (Again, the term ‘function’ is employed somewhat loosely here, ‘generalisedfunction’ would be more accurate.) It is not difficult to see that the δ0 = △

(− 14π

1r

)from

before is really Dirac’s delta distribution for ~a being the origin of Rn. Thus, − 14π

1r

is whatis called the Green’s function of the Laplace operator in R3:

Let D be a linear partial differential operator acting on testfunctions f ∈ S, e.g.D = ∂∂x

, orD = △, or D = . A (generalised) function GD : R

n × Rn −→ R , (~x, ~x ′) 7→ GD(~x, ~x′) ,

is called Green’s function for D if

D~xGD(~x, ~x′) = δ(~x− ~x ′) . (3.12)

Here, the subscript of D merely indicates that the differential operator acts on the firstargument ~x of GD(~x, ~x

′) only, not on the second. Such notational problems disappear if weabandon the ‘physicists’ terminology’ used in (3.12) and use the language of distributionseverywhere:A Green’s function for D with respect to the point ~x ′ ∈ U is a distribution GD,~x ′ suchthat the following holds as an equality of distributions:

DGD,~x ′ = δ~x ′

The action of the differential operator D on the distribution GD,~x ′ is declared with thehelp of (3.10), and one seeks to have a GD,~x ′ for any ~x

′ ∈ U .Another name for Green’s functions is fundamental solutions of D (perhaps more appro-priate in view of their applications, see below, and since they are not always ordinaryfunctions).

What makes Green’s functions useful for our purposes is the following fact: Once theGreen’s function of a linear differential operator D is determined, one can immediatelywrite down solutions f to PDEs of the form Df(~x) = h(~x) for any given (test)functionh: Put

f(~x) =

∫

Rn

GD(~x, ~x′) h(~x ′) dnx′ (3.13)

(integration over the second argument ~x ′ of GD!), then we have

D~xf(~x) =

∫

Rn

D~xGD(~x, ~x′) h(~x ′) dnx′ =

∫

Rn

δ(~x− ~x ′) h(~x ′) dnx′ = h(~x) ;

27

In the first step, we have inserted (3.13) for f and used linearity of D to pull it under theintegral; in the next step, the defining property (3.12) of the Green’s function was used(and the fact that D acts on ~x only, not on ~x ′); in the last step, the defining property ofthe delta distribution was used (after applying (3.9) for α = −1).

(If one wants to avoid the function notation used in (3.13), one has to introduce the so-called convolution T ∗ f of a distribution T with a testfunction f ; then the solution f weare after is given as GD,~0 ∗ h. For any f ∈ S, denote by τ~x(f) the function

~y 7−→(τ~x(f)

)(~y) := f(~x− ~y) .

Then one defines the function T ∗ f via(T ∗ f

)(~x) := T

[τ~x(f)

]. If the distribution GD,~0 is

given as the integral of an ordinary function (as is the case for electrostatics), the expressionGD,~0∗h coincides with (3.13) from above. Obviously, it is here that the physicists’ notationhas its advantages . . . )

In electrostatics, the differential operator of interest is D = △, the Laplace operator inR3. Our calculation of the Coulomb field in Section 3.1.1 has shown that

△

(−

1

4π

1

|~x− ~x ′|

)= δ(~x− ~x ′)

i.e. that the function in brackets is the Green’s function of the Laplace operator in threedimensions. (The generalisation of our previous computation to a point charge sitting at~x ′ instead of the origin is very easy.) Following the general reasoning from above, wecan now write down a solution ϕ of the Poisson equation (i.e. the electrostatics version ofMaxwell’s third equation)

△ϕ(~x) = −1

ǫ0ρ(~x) (3.14)

for an arbitrary static charge distribution ρ:

The scalar potential induced by this charge density is

ϕ(~x) =1

4π ǫ0

∫

R3

ρ(~x ′)

|~x− ~x ′|d3x′ . (3.15)

The electric field induced by ρ is given as (minus) the gradient of the scalar potential, thus

~E(~x) =1

4π ǫ0

∫

R3

ρ(~x ′)~x− ~x ′

|~x− ~x ′|d3x′ . (3.16)

One can also introduce Green’s functions for linear partial differential operators acting onfunctions that are defined on a subset U ⊂ Rn only. Then, one typically has to prescribeboundary conditions for GD along the boundary of U . We will not go into the details ofsuch situations here; see the textbook by Jackson for excessive discussions of D = △ ondomains with boundaries. Let us merely quote the following results concerning uniquenessof solutions to the Poisson equation:

28

If ρ is given on all of R3, then ϕ from (3.15) is the unique solution of △ϕ = − 1ǫ0ρ up to

addition of a harmonic function ϕ̃, i.e. a function that satisfies △ϕ̃(~x) = 0 for all ~x ∈ R3.If we require that ϕ vanishes for |~x| → ∞, then the solution is unique.Let U ⊂ R3 be a domain with smooth boundary ∂U on which a unit normal vector field~n(~x) is defined (~x ∈ ∂U). If ρ is given on such a domain, then there is a unique solutionϕ to the Poisson equation on U if one prescribes the values of ϕ(~x) for all ~x ∈ ∂U , and a

unique solution up to adding a constant if one prescribes the values of ~n(~x) · ~∇ϕ(~x) for all~x ∈ ∂U . The first type of boundary conditions (where the potential itself is given on ∂U) iscalled Dirichlet boundary conditions, the second (where the electric field is given on ∂U) iscalled Neumann boundary conditions. Typical situations where one has to compute electricfields on bounded domains are electrostatics problems involving conducting surfaces (whichmake up ∂U); compare the remarks made in the lecture, and see Jackson for N examples.

The uniqueness statements for Dirichlet and Neumann boundary conditions can be provenwith the help of Green’s theorem in a relatively elementary way, see the Problem Sheets.The same methods also show uniqueness of ϕ with △ϕ̃ = 0 on R3 as long as one requiresthat ϕ(~x) falls off faster than 1/r for large r = |~x|; to show uniqueness holds even for theweaker condition that ϕ(~x) merely vanishes at infinity, one has to dig more deeply in themathematical toolkit.

The important basic message contained in formulas (3.15) and (3.16) is that all electrostat-ics problems can be solved, at least in principle. (Finding a harmonic function such thatthe boundary conditions on ∂U are satisfied can be rather tricky, and analytic evaluationof the integrals may be very difficult if ρ is a complicated function, but they can at leastbe computed numerically.)

3.1.3 Electric field of a charged infinite straight line

The formular (3.16) holds in full generality, but often it is rather cumbersome to workout the intergral explicitly even for simple charge configurations. Using (M3) is in factsometimes quicker. So let us sketch the computation of a static electric field (withoutusing (3.16)) in another simple case: a straight line (which we take as z-axis) carrying aconstant line charge density λ. A line charge density has the unit 1Cbm ; and if λ is constant,a segment of the line of length h carries the charge hλ. Using the delta function in onedimension δ(1), we can describe this charge distribution by the (volume) charge density

ρ(~x) = λ δ(1)(x) δ(1)(y) ; (3.17)

the right hand side vanishes unless the x- and y-coordinates of ~x are zero; compare toexpression (3.11) for a point charge. (Note that the units work out: ρ has unit 1Cb

m3; λ has

unit 1Cbm ; and each δ(1) has unit 1m−1, since by definition

∫dx δ(1)(x) = 1.)

We want to determine the electric field induced by such a charged line. This can be donesolving differential equations in cylindrical coordinates or using Green’s functions, but letus work with the integral form of Maxwell’s equations here – which is relatively convenientbecause the configuration is still highly symmetric. We start by exploiting the symmetriesof the problem to determine ~E qualitatively, then we will fix the remainder by evaluating(M3) for a cleverly chosen volume V .

29

Symmetries: The configuration is invariant under translations in z-direction; this impliesthat ~E and also the scalar potential ϕ do not depend on z, i.e.

~E(~x) = ~E(x, y) .

The invariance under rotations around the z-axis has the following consequences: Themagnitude | ~E(~x)| of the electric field (and also ϕ) depend only on the distance ρrad of ~xfrom the line, i.e.

| ~E(~x)| = E(ρrad) , where ρrad =√x2 + y2 ,

for some function E(ρrad), which remains to be determined.Furthermore, from the rotational symmetry it is clear that a test charge q brought nearthe charged line will move straight towards the line (or straight away from it, depending

on the relative sign of q and λ). Since the field lines of ~E point along the force acting on

a test charge, this means that also ~E points orthogonally towards or away from the line,

~E(~x) = E(ρrad) ~eρrad(~x) where ~eρrad(~x) =1

ρrad

xy0

(3.18)

is the unit vector in radial direction (pointing away from the z-axis).

Gauss’ law (M3) allows to determine E(ρrad). While (M3) holds for all volumes V withboundary SV , the integrals become easy to evaluate only for certain volumes, namely thosewhich enjoy the same symmetry as the charge and field configuration. For a point charge,we used a ball around the charge (spherical symmetry), now we are in a situation withcylindrical symmetry and therefore try a cylinder around the z-axis (of height h and radiusR, say) for V .The volume integral of the charge density is easy to compute,

∫

V

ρ(~x) d3x = total charge inside V = (height of cylinder) · λ = hλ .

To compute the other integral in (M3), we need to know the outward-pointing unit normalvector field ~n(~x) on the cylinder surface: The latter consists of the top and bottom ‘lid’ ofthe cylinder – here ~n(~x) points in plus resp. minus z-direction – and of the bent mantle,where ~n(~x) = ~eρrad(~x); in an equation:

~n(~x) =

{~ez ~x ∈ top lid~eρ ~x ∈ mantle−~ez ~x ∈ bottom lid

Since we already know that ~E(~x) ∼ ~eρrad(~x), the integrand~E(~x) · ~n(~x) vanishes for ~x in

the top and bottom lids. For the remainder of the cylinder surface, we have

~E(~x) · ~n(~x) = E(ρrad) ~eρrad(~x) · ~eρrad(~x) = E(ρrad) .

30

Since ρrad = R = const for all points on the cylinder mantle, is easy to compute the fluxof ~E through the cylinder surface:

∮

SV

~E(~x) · ~n(~x) dS =

∫

mantle

E(ρrad) dS = E(R) · (area of the mantle) = E(R) 2π R h .

Equating this to 1/ǫ0 times the charge inside the cylinder gives E(R) = λ/(2πǫ0R), and

therefore (recalling that R was the distance of the point where we determined ~E from thez-axis) we obtain

~E(~x) =λ

2πǫ0 ρrad~eρrad(~x) =

λ

2πǫ0

1

x2 + y2

xy0

(3.19)

for the electric field induced by an infinitely long charged line.

Alternatively, we can try to work with differential equations and solve the Poisson equationfor the charge density (3.17),

△ϕ(~x) =λ

ǫ0δ(1)(x) δ(1)(y) . (3.20)

Because of the symmetry of the configuration, it is advantageous to use cylindrical coor-dinates (ρrad, θ, z) defined by

xyz

=

ρrad cos θρrad sin θ

z

.

In these coordinates, the Laplace operator reads (see Problem Sheets)

△ = ∂2ρ +1

ρ∂ρ +

1

ρ2∂2θ + ∂

2z , (3.21)

where ∂ρ =∂

∂ρrad, ∂θ =

∂∂θ

and ∂z =∂∂z

. Again we exploit the symmetries of the chargeconfiguration in question and try to find a solution ϕ that depends only on ρrad, but neitheron θ (because of rotational invariance) nor on z (because of translational invariance). Recallthat if the symmetry arguments do not appear satisfactory, one can consider this as anansatz first and afterwards invoke suitable uniqueness theorems about harmonic functions.So let us try ϕ(~x) = ϕ(ρrad) and solve the equation △ϕ = 0 for ρrad 6= 0. Applying (3.21)to such a ϕ, we only have to solve an ordinary differential equation:

△ϕ(ρrad) = 0 =⇒1

ρrad∂ρ

(ρrad ∂ρ ϕ

)= 0 =⇒ ∂ρ ϕ =

c1ρrad

=⇒ ϕ = c1 ln ρrad + c2

for some constants c1 6= 0 and c2; we can put c2 = 0 without loss of generality. The functionϕ is singular at ρrad = 0 as it should be, in order to produce the two-dimensional delta

31

distribution upon taking △. We can fix the value of c1 by applying the two-dimensionalversion of Gauss’ divergence theorem:

∫

S

~∇(2) · ~K d2x =

∮

∂S

~K · ~n dr

where ~K now is a two-component vector field, ~∇(2) denotes the two-dimensional gradient,and ~n is a unit vector field defined on the boundary ∂S of the two-dimensional set S ⊂ R2

such ~n is orthogonal to ∂S and points to the outside. This theorem follows immediatelyfrom the three-dimensional divergence theorem, applied to the special case of a cylindricalvolume V = S × I, where I is an interval, and of a three-component vector field that doesnot depend on z and whose third component vanishes.In our situation, we compute

∫

R2

△(2)ϕ d2x =

∫

disk0

△(2)ϕ d2x =

∮

circle0

~∇(2)ϕ · ~n dr

=

∮

circle0

ϕ′(ρ0)~eρ · ~eρ dr =c1ρ0

2π ρ0 = 2π c1 .

In the first step, we have used that the integrand vanishes everywhere except at ρrad = 0,so we can equally well integrate over a disk (of radius ρ0) around the origin. The next

step is the divergence theorem, since △ϕ = ~∇ · (~∇ϕ); the boundary of the disk is a circle,whose outward pointing normal unit vector is ~eρ as defined in (3.18). Since ϕ is constanton any circle around the origin, evaluation of the integral is straightforward.We see that in order to satisfy the Poisson equation (3.20), we have to choose c1 = λ/(2πǫ0),which indeed reproduces the electric field we computed before.In passing, our calculation has shown that a Green’s function of the Laplace operator intwo dimensions is

G△(2)(~x, ~x′) =

1

2πln |~x− ~x ′|

where now ~x and ~x ′ are points in R2.

3.1.4 Dipole field, and multipole expansion

In a sense, electric dipoles are the next complicated charged objects after point particles.Associated to them is an electric field with a ‘typical’ (for a dipole) coordinate dependence(which is somewhat more complicated than the coordinate dependence of the Coulombfield, which is typical for point charges). It will also turn out that one can write anarbitrary static electric field as a sum of a Coulomb field (also called ‘monopole field’ inthis context), a dipole field and further terms (so-called ‘multipole fields’); the specificfeatures of the original field are entirely contained in the coefficients of this expansion (the‘multipole expansion’). An analogous expansion exists for the electric potential.

We can introduce the dipole as a limit of a configuration of two opposite point charges,taking their separation to zero in a certain way. Assume we have a point charge q placed

at~d2 and a point charge −q at −

~d2 . By the superposition principle, the electric potential

ϕ of this charge configuration is the sum of two Coulomb potentials

ϕ(~x) =1

4πǫ0

q

| ~x−~d2 |

+1

4πǫ0

−q

| ~x+~d2 |

. (3.22)

32

Likewise, the electric field is a sum of two Coulomb fields. We can now ask how thepotential looks ‘far away’