1

Section 2: Maxwell’s equations

Electromotive force

We start the discussion of time-dependent magnetic and electric fields by introducing the concept of the

electromotive force.

Consider a typical electric circuit. There are two forces involved in driving current around a circuit: the

source, s

f , which is ordinarily confined to one portion of the loop (a battery, say), and the electrostatic

force, E, which serves to smooth out the flow and communicate the influence of the source to distant parts

of the circuit. Therefore, the total force per unit charge is a circuit is

s

= +f f E . (2.1)

The physical agency responsible for s

f , can be any one of many different things: in a battery it’s a

chemical force; in a piezoelectric crystal mechanical pressure is converted into an electrical impulse; in a

thermocouple it’s a temperature gradient that does the job; in a photoelectric cell it’s light. Whatever the

mechanism, its net effect is determined by the line integral of f around the circuit:

s

d d= ⋅ = ⋅∫ ∫f l f l� �E . (2.2)

The latter equality is because 0d⋅ =∫ E l� for electrostatic fields, and it doesn’t matter whether you use f

or s

f . The quantity E is called the electromotive force, or emf, of the circuit. It’s a lousy term, since this is

not a force at all – it’s the integral of a force per unit charge.

Within an ideal source of emf (a resistanceless battery, for instance), the net force on the charges is zero,

so E = – fs. The potential difference between the terminals (a and b) is therefore

b b

s s

a a

d d d∆Φ = − ⋅ = ⋅ = ⋅ =∫ ∫ ∫E l f l f l� E . (2.3)

We can extend the integral to the entire loop because fs = 0 outside the source. The function of a battery,

then, is to establish and maintain a voltage difference equal to the electromotive force. The resulting

electrostatic field drives current around the rest of the circuit (notice, however, that inside the battery fs

drives current in the direction opposite to E). Because it's the line integral of fs, E can be interpreted as the

work done, per unit charge, by the source.

Fig. 2.1

One possible sources of electromotive force in a circuit is a generator. Generators exploit motional emf,

which arise when you move a wire through a magnetic field. Figure 2.1 shows a primitive model for a

generator. In the shaded region there is a uniform magnetic field B, pointing into the page, and the resistor

R represents whatever it is we're trying to drive current through. If the entire loop is pulled to the right

2

with speed v, the charges in segment experience a Lorentz force whose vertical component qvB drives

current around the loop, in the clockwise direction. The emf is

vmag

d Bh= ⋅ =∫ f l�E . (2.4)

There is a nice way to represent the emf generated in a moving loop. Let F be the flux of B through the

loop:

F da= ⋅∫B n . (2.5)

For rectangular loop in Fig.2.1,

F Bhx= . (2.6)

As the loop moves, the flux decreases

vdF dx

Bh Bhdt dt

= = − . (2.7)

The minus sign accounts for the fact that dx/dt is negative. But this is precisely the emf given by eq.(2.4);

evidently the emf generated in the loop is minus the rate of change of flux through the loop:

dF

dt= −E . (2.8)

This is the flux rule for motional emf. Apart from its delightful simplicity, it has the virtue of applying to

nonrectangular loops moving in arbitrary directions through noniniform magnetic fields; in fact, the loop

need not even maintain a fixed shape. Now we prove this statement.

Fig. 2.2

Figure 2.2 shows a loop of wire at time t and also a short time dt later. Suppose we compute the flux at

time t, using surface S, and the flux at time t + dt, using the surface consisting of S plus the “ribbon” that

connects the new position of the loop to the old. The change in flux, then, is

( ) ( )ribbon

dF F t dt F t da= + − = ⋅∫ B n . (2.9)

Focus our attention on point P: in time dt it moves to P'. Let v be the velocity of the wire, and u the

velocity of a charge down the wire; w = v + u is the resultant velocity of a charge at P. The infinitesimal

element of area on the ribbon can be written as

( )da d dt= ×n v l . (2.10)

(see inset in Fig. 2.2). Therefore

3

( )dF

ddt

= ⋅ ×∫ B v l� . (2.11)

Since w = v + u and u is parallel to dl, we can also write this as

( ) ( )dF

d ddt

= ⋅ × = − × ⋅∫ ∫B w l w B l� � . (2.12)

But ( ×w B ) is the Lorentz force per unit charge, magf , so

mag

dFd

dt= − ⋅∫ f l� (2.13)

and the integral of f mag is the emf

dF

dt= −E . (2.14)

Faraday’s law

The first quantitative observations relating time-dependent electric and magnetic fields were made by

Faraday (in 1831) in experiments on the behavior of currents in circuits placed in time-varying magnetic

fields. The three of these experiments (with some violence to history) can be characterized as follows. A

transient current is induced in a circuit if

(a) the circuit is moved through a magnetic field (Fig. 2.3a);

(b) a magnetic field is moved into or out of the circuit (Fig. 2.3b);

(c) the strength of the magnetic field is changed (Fig. 2.3c).

Fig. 2.3

The first experiment, of course, is an example of motional emf, conveniently expressed by the flux rule

(2.14). It is not surprising that exactly the same emf arises in Experiment 2 – all that really matters is the

relative motion of the magnet and the loop. Indeed, in the light of special relativity is has to be so. But

Faraday knew nothing of relativity, and in classical electrodynamics this simple reciprocity has

remarkable implications. For if the loop moves, it’s a magnetic force that sets up the emf, but if the loop

is stationary, the force cannot be magnetic – stationary charges experience no magnetic forces. Electric

fields could produce the force, and Faraday made an ingenious prediction: A changing magnetic field

induces an electric field.

Faraday attributed the transient current flow to a changing magnetic flux linked by the circuit. The

changing flux induces an electric field around the circuit, the line integral of which is called the

electromotive force,

4

d= ⋅∫ E l�E . (2.15)

The electromotive force causes a current flow, according to Ohm’s law. All the three experiments can

therefore be described by eq.(2.14) which takes the form

dF

ddt

⋅ = −∫ E l� . (2.16)

and is called the Faraday’s law. The negative sign in this equation is consistent with Lenz’s law.

According to Lenz’s law, the current induced around a closed loop is always such that the magnetic field

it produces tries to counteract the change in magnetic flux which generates the electromotive force.

Now let us consider the connection between Experiment 1 and Experiment 2 in a more detail. We know that when we are dealing with relative speeds that are small compared with the velocity of light, physical laws should be invariant under Galilean transformations. That is, physical phenomena are the same when viewed by two observers moving with a constant velocity v relative to one another, provided the coordinates in space and time are related by the Galilean transformation, t′ = −r r v , t t′ = . Faraday’s observations suggest that the same current is induced in a circuit whether it is moved while the magnetic field is stationary (Experiment 1) or it is held fixed while the magnetic field is moved into or out of the circuit in the same relative manner (Experiment 2).

Let us now consider Faraday’s law for a moving circuit and see the consequences of Galilean invariance. Expressing (2.16) in terms of the integrals over ′E and B, we have

C S

dd da

dt′ ⋅ = − ⋅∫ ∫E l B n� . (2.17)

The induced electromotive force is proportional to the total time derivative of the flux – the flux can be changed by changing the magnetic induction or by changing the shape or orientation or position of the circuit. In form (2.17) we have a far-reaching generalization of Faraday's law. The circuit C can be thought of as any closed geometrical path in space, not necessarily coincident with an electric circuit. Then (2.17) becomes a relation between the fields themselves. It is important to note, however, that the electric field,

′E is the electric field at dl in the moving coordinate system in which dl is at rest, since it is that field that causes current to flow if a circuit is actually present.

If the circuit C is moving with a velocity v in some direction, the total time derivative in (2.17) must take into account this motion. The flux through the circuit may change because (a) the flux changes with time at a point, or (b) the translation of the circuit changes the location of the boundary. It is easy to show that the result for the total time derivative of flux through the moving circuit is

( )S S C

dda da d

dt t

∂⋅ = ⋅ + × ⋅∂∫ ∫ ∫B

B n n B v l� . (2.18)

Indeed, taking into account that the convective derivative is given by

d

dt t

∂= + ⋅∇∂

v , (2.19)

we can write

( ) ( ) ( )d

dt t t

∂ ∂= + ⋅∇ = + ∇ × × + ∇ ⋅∂ ∂

B B Bv B B v v B , (2.20)

where v is treated as a fixed vector in the differentiation. The last term is zero due to 0∇ ⋅ =B . Use of Stokes’s theorem on the second term yields (2.18).

Equation (2.17) can now be written in the form,

5

( )C S

d dat

∂′ − × ⋅ = − ⋅ ∂∫ ∫B

E v B l n� . (2.21)

This is an equivalent statement of Faraday’s law applied to the moving circuit C. But we can choose to interpret it differently. We can think of the circuit C and surface S as instantaneously at a certain position in space in the laboratory. Applying Faraday's law (2.17) to that fixed circuit, we find

C S

dd da

dt⋅ = − ⋅∫ ∫E l B n� . (2.22)

where E is now the electric field in the laboratory. The assumption of Galilean invariance implies that the left-hand sides of (2.21) and (2.22) must be equal. This means that the electric field ′E in the moving coordinate system of the circuit is

( )′ = + ×E E v B . (2.23)

Because we considered a Galilean transformation, the result (2.23) is an approximation valid only for speeds small compared to the speed of light. Faraday’s law is no approximation, however.

Faraday's law can be put in differential form by use of Stokes’s theorem. The transformation of the electromotive force integral into a surface integral leads to

0dat

∂ ∇ × + ⋅ = ∂ ∫

BE n . (2.24)

Since the circuit C and bounding surface S are arbitrary, the integrand must vanish at all points in space. Thus the differential form of Faraday's law is

t

∂∇× = −∂B

E . (2.25)

We note that this is the time-dependent generalization of the statement, 0∇ × =E , for electrostatic fields.

Energy in the Magnetic Field

In discussing steady-state magnetic fields we avoided the question of field energy and energy density. The

reason was that the creation of a steady-state configuration of currents and associated magnetic fields

involves an initial transient period during which the currents and fields are brought from zero to the final

values. For such time-varying fields there are induced electromotive forces that cause the sources of

current to do work. Since the energy in the field is by definition the total work done to establish it, we

must consider these contributions.

Suppose for a moment that we have only a single circuit with a constant current I flowing in it. If the flux

through the circuit changes, an electromotive force E is induced around it. To keep the current constant,

the sources of current must do work. The work done on a unit positive charge, against the emf, in one trip

around a circuit is –E (the minus sign records the fact that this is the work done against the emf, not the

work done by emf). The amount of charge per unit time passing down the wire is I. So the total work

done per unit time is

dW dF

I Idt dt

= − =E . (2.26)

This is in addition to ohmic losses in the circuit, which are not to be included in the magnetic-energy

content. Thus, if the flux change through a circuit carrying a current I is δF, the work done by the sources

is

W I Fδ δ= . (2.27)

6

We can express the increment of work done against the induced emf in terms of the change in magnetic

field through the loop:

( )S S C C

F da da d dδ δ δ δ δ= ⋅ = ∇ × ⋅ = ⋅ = ⋅∫ ∫ ∫ ∫B n A n A l A l� � , (2.28)

where C is the curve bounding surface S. Thus,

( )C C

W I d dlδ δ δ= ⋅ = ⋅∫ ∫A l A I� � . (2.29)

In this form the generalization to the volume currents is obvious:

( ) 3

V

W d rδ δ= ⋅∫ A J . (2.30)

An expression involving the magnetic fields rather than J and δA can be obtained by using the Ampere’s

law ∇ × =H J . Then

( ) 3

V

W d rδ δ= ⋅ ∇ ×∫ A H . (2.31)

Using the vector identity

( ) ( ) ( )∇ ⋅ × = ⋅ ∇ × − ⋅ ∇ ×v w w v v w . (2.32)

Eq.(2.31) can be transformed to

( ) ( ) 3

V

W d rδ δ δ= ⋅ ∇ × + ∇ ⋅ × ∫ H A H A . (2.33)

If the field distribution is assumed to be localized and we integrate over all space, the second integral

vanishes. With the definition of B in terms of A, the energy increment can be written:

3W d rδ δ= ⋅∫H B . (2.34)

This relation is the magnetic equivalent of the electrostatic equation (4.86). In its present form it is

applicable to all magnetic media, including ferromagnetic substances. If we assume that the medium is

para- or diamagnetic, so that a linear relation exists between H and B (i.e. B=µH), then

( )1

2δ δ⋅ = ⋅H B H B . (2.35)

If we now bring the fields up from zero to their final values, the total magnetic energy will be

31

2W d r= ⋅∫H B . (2.36)

In view of this result, we can say that the energy is “stored in the magnetic field”, in the amount of

½ ⋅H B per unit volume.

Eq.(2.36) is the magnetic analog of the respective electrostatic equation. The magnetic equivalent of the

equation where the electrostatic energy is expressed in terms of the charge density and the potential can

be obtained from eq. (2.30) by assuming linear relation between J and A. Then we find the magnetic

energy to be

( ) 31

2W d r= ⋅∫ A J . (2.37)

7

Self- and Mutual Inductances

The concept of self- and mutual inductances are useful for systems of current-carrying circuits. Imagine a

system of N distinct current-carrying circuits, with Ii being the total current carrying by the i-th circuit. The

circuits are not necessarily thin wires but are assumed for the present to be non-permeable. It appears that

the total energy of the system of currents (2.37) can be expressed as

1 1

1

2

N N N

i i ij i j

i i j i

W L I M I I= = >

= +∑ ∑∑ . (2.38)

where Li is the self-inductance of the i-th circuit and Mij is the mutual inductance between the i-th and j-th

circuits. To establish this result, we first use the expression for the vector potential

30 ( )( )

4d r

µπ

′ ′=′−∫

J rA r

r r. (2.39)

to convert (2.37) to

3 30 ( ) ( )

8W d r d r

µπ

′⋅′=′−∫ ∫

J r J r

r r . (2.40)

The integrals can now be broken up into sums of separate integrals over each circuit:

3 30

1 1

( ) ( )

8i j

N Ni j

i j

i jC C i j

W d r d rµπ = =

′⋅′=

′−∑ ∑∫ ∫J r J r

r r . (2.41)

In the sums there are terms with i = j and terms with i j≠ . The former define the first sum in (2.38) and

the latter, the second. Evidently, the coefficients Li and Mij are given by

3 30 ( ) ( )

4i i

i ii i i

i i iC C

L d r d rI

µπ

′⋅′=′−∫ ∫

J r J r

r r . (2.42)

3 30( ) ( )

4i j

i j

ij i j

i j C C i j

M d r d rI I

µπ

′⋅′=

′−∫ ∫J r J r

r r . (2.43)

Note that the coefficients of mutual inductance Mij are symmetric in i and j.

These general expressions for self- and mutual inductance are the rigorous versions of the more elementary

definitions in terms of flux linkage. To establish the connection, consider the expression for mutual

inductance (for which the ambiguities in the definition of flux linkage for self-inductance are absent). The

integral over 3

jd r′ times 0 / 4µ π is just the expression (2.39) for the vector potential ( )j iA r at position i

r in

the i-th circuit caused by the current Ij, flowing in the j-th circuit:

30( )

( )4

j

j

j i j

C i j

d rµπ

′′=

′−∫J r

A rr r

. (2.44)

So that eq. (2.43) can be written in form

31( ) ( )

i

ij j i i i

i j C

M d rI I

= ⋅∫ A r J r . (2.45)

If the i-th circuit is imagined to be negligible in cross section compared to the overall scale of both circuits,

we can write the integrand 3( )i i

d rJ r for the integration over the volume of the i-th circuit as

8

3d r J dad=J l� , where da is a locally defined element of cross-sectional area and dl is a directed longitudinal

differential in the sense of current flow. With the vector potential sensibly constant in the cross-sectional

integral at a fixed position along the circuit, the mutual inductance becomes

( )1 1 1 1

i i i i i

ij ij ij ij ij

i j i j j jC C S C S

M J dad d J da d daI I I I I I

= ⋅ = ⋅ = ⋅ = ∇ × ⋅∫ ∫ ∫ ∫ ∫A l A l A l A n� �� � . (2.46)

where Aij is the vector potential caused by the j-th circuit at the integration point on the i-th and the factor Ii,

comes from the integral over the cross section

i

i

S

I J da= ∫ � . Stokes’s theorem has been used to obtain the

second form. Since the curl of A is the magnetic field B, the area integral is just the magnetic-flux linkage

i

ij ij

S

F da= ⋅∫B n . (2.47)

Thus the mutual inductance is finally

1

ij ij

j

M FI

= , (2.48)

where Fij is the magnetic flux from circuit j linked within circuit i. For self-inductance, the physical argument

is the same, but the ambiguity in the meaning of the self-flux linkage Fii requires a return to the rigorous

expression (2.42) based on the magnetic energy.

Another way to represent mutual inductance from linear current circuits is to use eq. (2.43) directly. Similar

to the arguments used in the derivation of eq. (2.46) we can integrate over the cross-sectional area of the

circuits and write

0

4i j

i j

ij

C C i j

d dM

µπ

⋅=

−∫ ∫l l

r r . (2.49)

This is the Neumann formula; it involves a double line integral - one integration around loop i and the

other around loop j. It's not very useful for practical calculations, but it reveals that mutual inductance is a

purely geometrical quantity, which depends only on the size, shape, and relative position of the two loops.

Example: A small loop of wire of radius a lies at distance z above the center

of a large loop of radius b, as shown in the figure. The planes of the two loops

are parallel, and perpendicular to the common axis. Find the mutual

inductances and confirm that M12 = M21.

We use formula (2.49) to calculate the mutual inductance. The z-component

of the field produced by the loop of radius b is given by

( )2

0 0 0

3/ 22 2 2 2

sin

4 2b

z

C

I dl I bB

b z b z

µ θ µπ

= =+ +

∫ . (2.50)

Hence, the flux of the magnetic field crossing the area of loop a is

( )2 2

0

3/ 22 22

aS

I a bF da

b z

µ π= ⋅ =

+∫ B n . (2.51)

Similarly, we can calculate the flux of a magnetic field produced by loop a and crossing the area of loop

b. The magnetic field can be evaluated within the dipole approximation because the loop a is assumed to

be small. We find:

a

b

z

θ0

9

( )2 2

0 0 0

3 2 3

sin ˆ ˆˆ( ) ( ) 2cos sin4 4 4

II a a

r r r

µ µ µπ θ φ θ θπ π

×= ∇ × = ∇ × = ∇ × = +m rB r A r r

θ . (2.52)

By integrating over a spherical surface bounded by loop b and centered at loop a we obtain:

( )02 2 2 2

20 0 0

3/ 23 2 20

ˆ 2cos sin 4 cos sin4 4 2

bS

I I Ia a a bF da r d d d

r r b z

θµ µ µ πθ θ θ φ π θ θ θ= ⋅ = = =+

∫ ∫ ∫B r , (2.53)

which is identical to eq. (2.51). Therefore the mutual inductance is

( )2 2

012 21 3/ 2

2 22

a bM M

b z

µ π= =

+ . (2.54)

Maxwell Equations

All the electromagnetism laws discussed in preceding sections can be summarized in four equations

Coulomb’s law (Gauss’s law): ρ∇ ⋅ =D (2.55)

Ampere’s law: ∇× =H J (2.56)

Absence of free magnetic poles: 0∇ ⋅ =B (2.57)

Faraday’s law: 0t

∂∇× + =∂B

E (2.58)

These equations represent the state of electromagnetic theory before Maxwell. It appears that there is a

fatal inconsistency in these equations. All the equations except the Faraday’s law were derived from

steady-state observations. However, there is no a priopi reason to expect that the static equations will hold

unchanged for time dependent fields.

The inconsistency has to do with the rule that divergence of curl is always zero. If you apply the

divergence to eq. (2.58), everything works out:

( ) ( ) 0t t

∂ ∂∇ ⋅ ∇ × = −∇ ⋅ = − ∇ ⋅ =∂ ∂B

E B (2.59)

The left side is zero because divergence of curl is zero; the right side is zero by virtue of equation (2.57).

But when we do the same thing to eq. (2.56), we get into trouble:

( )∇ ⋅ ∇ × = ∇ ⋅H J (2.60)

the left side must be zero, but the right side, in general, is not. For steady currents, the divergence of J is

zero, but evidently when we go beyond magnetostatics Ampere’s law cannot be right.

There's another way to see that Ampere's law is bound to fail for nonsteady currents. Suppose we're in the

process of charging up a capacitor (Fig. 2.4). In integral form, Ampere’s law reads

encl

d I⋅ =∫ H l� (2.61)

where encl

I is the total current enclosed by the loop. If we apply it to the amperian loop shown in Fig.2.4

the result appears to be dependent on the choice of the surface bounded by the loop. For the surface lying

in the plane of the loop – the wire punctures this surface, so the enclosed current is encl

I I= . However, for

the balloon-shaped surface, no current passes through this surface, and we conclude that 0encl

I = ! We

never had this problem in magnetostatics because the conflict arises only when charge is piling up

10

somewhere (in this case, on the capacitor plates). But for non-steady currents (such as this one) "the

current enclosed by a loop" is an ill-defined notion, since it depends entirely on what surface you use.

Fig. 2.4

Of course, we had no right to expect Ampere's law to hold outside of magnetostatics; after all, we derived

it from the Biot-Savart law. However, in Maxwell's time there was no experimental reason to doubt that

Ampere's law was of wider validity. The flaw was a purely theoretical one, and Maxwell fixed it by

purely theoretical arguments.

The problem is on the right side of Eq.(2.60), which should be zero, but isn't. Applying the continuity

equation and Gauss's law, the offending term can be rewritten:

( )t t t

ρ∂ ∂ ∂∇ ⋅ = − = − ∇ ⋅ = −∇ ⋅∂ ∂ ∂

DJ D (2.62)

It occurs that if we were to combine t

∂∂D

with J, in Ampere's law, it would be just right to kill off the extra

divergence:

New formulation of Ampere’s law: t

∂∇× = +∂D

H J (2.63)

Such a modification changes nothing, as far as magnetostatics is concerned: when D is constant, we still

have ∇× =H J . In fact, Maxwell's term is hard to detect in ordinary electromagnetic experiments, that's

why Faraday and the others never discovered it in the laboratory. However, it plays a crucial role in the

propagation of electromagnetic waves, as we'll see below. Without it there would be no electromagnetic

radiation. It was Maxwell’s prediction that light is an electromagnetic wave phenomenon.

Maxwell eq. (2.63) suggests that just as a changing magnetic field induces an electric field (Faraday's

law), a changing electric field induces a magnetic field. Of course, theoretical convenience is only

suggestive – there might, after all, be other ways to fix Ampere's law. The real confirmation of Maxwell's

theory came in 1888 with Hertz's experiments on electromagnetic waves.

Maxwell called his extra term the displacement current:

d

t

∂=∂D

J (2.64)

Let's see now how the displacement current resolves the paradox of the charging capacitor (Fig. 3.3). If

the capacitor plates are very close together, then the electric displacement field between them is

Q

DA

σ= = (2.65)

where Q is the charge on the plate and A is its area. Thus, between the plates

11

D dQ I

t A t A

∂ = =∂ ∂

(2.66)

Now, Eq. 7.36 reads, in integral form,

encl

S

d I dt

∂⋅ = + ⋅∂∫ ∫D

H l a� (2.67)

If we choose the flat surface, then D = 0 and encl

I I= . If, on the other hand, we use the balloon-shaped

surface, then 0encl

I = , but d It

∂ ⋅ =∂∫D

a� . So we get the same answer for either surface, though in the first

case it comes from the genuine current and in the second from the displacement current.

The final set of Maxwell equations can be written as follows

Coulomb’s law (Gauss law): ρ∇ ⋅ =D (2.68)

Ampere’s law: t

∂∇× = +∂D

H J (2.69)

Absence of free magnetic poles: 0∇ ⋅ =B (2.70)

Faraday’s law: 0t

∂∇× + =∂B

E (2.71)

These equations combined with constitutive relations connecting E and B with D and H form the basis of

all classical electrodynamics.

Derivation of the Equations for Macroscopic Electromagnetism

Equations (2.68)-(2.71) are macroscopic equations which include quantities E, D, B, H, J and ρ averaged

over volume which include many atoms (molecules). In this section we will perform an accurate

derivation of these equations starting from a microscopic point. The derivation remains within a classical

framework even though atoms must be described quantum-mechanically.

We consider a microscopic world made up of electrons and nuclei. For dimensions large compared to

10-12

cm, the nuclei and electrons can be treated as point charges. We assume that the equations

controlling electromagnetic phenomena for these point charges are the microscopic Maxwell equations:

0

ηε

∇ ⋅ =e , (2.72)

0 0 0t

µ ε µ∂∇× − =∂e

b j , (2.73)

0∇ ⋅ =b , (2.74)

0t

∂∇× + =∂b

e . (2.75)

Here e and b are the microscopic electric and magnetic fields and η and j are microscopic charge and

current densities. There are no corresponding microscopic fields d and h because all the charges are

included in η and j.

Microscopic quantities fluctuate widely over atomic distances, and we would like to smooth out these

fluctuations to define corresponding macroscopic quantities. For example, the charge density η switches

from zero to infinity whenever a point charge is encountered, and we would like to replace this by a

12

quantity that varies smoothly to reflect only macroscopic changes of density. We need to define the

averaging procedure which allows us to average out all the microscopic fluctuations, giving smooth and

slowly varying macroscopic quantities, such as appear in macroscopic Maxwell equations.

One simple-minded way of smoothing out a microscopic quantity f is to average it over a volume V(r)

centered at some position r; this volume is imagined to be microscopically large (so that all microscopic

fluctuations will be averaged over) but macroscopically small (so that variations over relevant

macroscopic scales are not accidentally discarded). The averaging operation is an integration of f over the

volume V(r) For example, the integration can be performed over a sphere of radius R centered at point r.

This averaging procedure has the advantage of conceptual simplicity, the disadvantage is the abrupt

change of the integration at the boundary of the sphere. The latter will lead to the fine scale variation of

the averaged quantity as a singe atom or group of atoms move in or out the averaging volume.

In this averaging operation it is convenient to introduce a smoothing function ( )w r such that the special

average of function f is given by

3( , ) ( , ) ( )f t f t w d r′ ′ ′= −∫r r r r . (2.76)

The smooth test function, e.g. the Gaussian, eliminates such difficulties provided its scale is large

compared to atomic dimensions. Fortunately, the test function ( )w r does not need to be specified in

detail; all that are needed are general continuity and smoothness properties that permit a rapidly

converging Taylor series expansion of ( )w r over distances of atomic dimensions, as indicated

schematically in Fig.2.5.

Since space and time derivatives enter the Maxwell equations, we must consider these operations with

respect to averaging according to (2.76). Evidently, we have

3 3 3( , ) ( ) ( , )

( ) ( ) ( ) ( ) ( )i i i i i

f t f f tf w d r f w d r w d r

x x x x x

∂ ′∂ ∂ ∂ ∂′ ′ ′ ′ ′ ′ ′ ′= − = − − = − =′ ′∂ ∂ ∂ ∂ ∂∫ ∫ ∫

r r rr r r r r r r r , (2.77)

where in integrating by parts we took into account that the boundary term at infinity is zero because w(r)

vanishes there. For differentiation with respect to time we obviously have

( , ) ( , )f t f t

t t

∂ ∂=∂ ∂

r r, (2.78)

Thus, the operations of space and time differentiation commute with the averaging operation.

Fig. 2.5

Schematic diagram of test function w(r) used in the spatial averaging procedure. The extent

L of the plateau region, and also the extent ∆L of the region where w(r) falls to zero, are

both large compared to the interatomic distance a.

w

13

We can now consider the averaging of the microscopic Maxwell equations (2.72)-(2.75). The

macroscopic electric and magnetic field quantities E and B are defined as the averages of the microscopic

fields e and b:

( , ) ( , )t t=E r e r , (2.79)

( , ) ( , )t t=B r b r . (2.80)

The averages of the microscopic Maxwell equations (2.72)-(2.75) become the corresponding macroscopic

equations,

0

ηε

∇ ⋅ =E , (2.81)

0 0 0t

µ ε µ∂∇× − =∂E

B j , (2.82)

0∇ ⋅ =B , (2.83)

0t

∂∇× + =∂B

E . (2.84)

The next step is to examine the averaged quantities η and j in terms of free and bound charges.

We distinguish between the charges that can move freely within the medium – the free charges – and the

charges that are tied to the molecules – the bound charges. We will consider the bound charges to be part

of the medium and remove them from the right-hand side of Maxwell's equations, which will then involve

the free charges only. The influence of the bound charges will thus be taken to the left-hand side of the

equations; it will be described by new macroscopic quantities, P (electric polarization of the medium) and

M (magnetization of the medium). From these we will define the new fields D and H.

Fig. 2.6

The left panel shows the free charges (small solid disks) and molecules (large

open circles) in the medium. The right panel shows the distribution of bound

charges within a given molecule.

Fig.2.6 shows the charge distribution in the medium. Let rf(t) be the position vector of the free charge qf

labeled by “f”, and let vf(t) = drf/dt be its velocity vector. Let rn be the position vector of the center of

mass of the molecule labeled by n; for simplicity we assume that the molecules do not move within the

medium. Finally, let rb(t) be the position vector of the bound charge qb labeled by “b” within a given

molecule, relative to the center of mass of this molecule, and let vb(t) = drb/dt be its velocity vector. The

density of free charge is

rf

rn

rn

rb

14

( , ) ( )f f f

f

t q tη δ = − ∑r r r , (2.85)

The density of bound charge within the molecule labeled by n is

[ ]( , ) ( )n b n b

b

t q tη δ= − −∑r r r r , (2.86)

We assume that each molecule is electrically neutral, so that 0b

b

q =∑ . The density of bound charge is

then

( , ) ( , )b n

n

t tη η=∑r r , (2.87)

and the total density of charge is

( , ) ( , ) ( , )f bt t tη η η= +r r r . (2.88)

We first calculate the macroscopic average of the molecular charge density. We have

[ ] [ ]3 3( , ) ( , ) ( ) ( ) ( ) ( )n n b n b b n b

b b

t t w d r q t w d r q w tη η δ′ ′ ′ ′ ′ ′= − = − − − = − −∑ ∑∫ ∫r r r r r r r r r r r r . (2.89)

The function ( )n b

w − −r r r is peaked at n b

= +r r r . Because the scale over which w varies is very large

compared with the intra-molecular displacement b

r , we can approximate w by expanding it in Taylor

series about n

−r r :

1

( ) ( ) ( ) ( ) ...2

i i j

n b n b n b b n

i iji i j

w w x w x x wdx dx dx

∂ ∂ ∂− − = − − − + − +∑ ∑r r r r r r r r r (2.90)

where to avoid confusing in notation we make the vectorial index i a superscript instead of subscript (i.e. i

bx denotes i component of vector

br ). Substituting this into the expression (2.89) yields

1

...2

i i j

n b b b i b b b i j

b b i b ij

q w q x w q x x wη = − ∇ + ∇ ∇ +∑ ∑∑ ∑∑ , (2.91)

where w now stands for ( )n

w −r r . We simplify this equation by recalling that each molecule is

electrically neutral, i.e. 0b

b

q =∑ , so that the first sum vanishes. In the second term we recognize the

expression for the molecular dipole moment,

( ) ( )i i

n b b

b

p t q x t=∑ . (2.92)

and in the third term we recognize the expression for the molecular quadrupole moment

( ) 3 ( ) ( )ij i j

n b b b

b

Q t q x t x t= ∑ . (2.93)

Note that here the quadrupole moment is not defined as a tracefree tensor. Combining these results gives

for the molecular charge density:

1

( , ) ( ) ( ) ( ) ( ) ...6

i ij

n n i n n i j n

i ij

t p t w Q t wη = − ∇ − + ∇ ∇ − +∑ ∑r r r r r . (2.94)

The macroscopic charge density is obtained by summing up over all molecules. This results in

15

1( , ) ( ) ( ) ( ) ( ) ...

6

1( ) ( ) ( ) ( ) ...

6

1( ) ( ) ( ) ( ) ...

6

i ij

b n i n n i j n

n i n ij

i ij

i n n i j n n

i n ij n

i ij

i n n i j n n

i n ij n

t p t w Q t w

p t w Q t w

p t Q t

η

δ δ

= − ∇ − + ∇ ∇ − + =

= − ∇ − + ∇ ∇ − + =

= − ∇ − + ∇ ∇ − +

∑∑ ∑∑

∑ ∑ ∑ ∑

∑ ∑ ∑ ∑

r r r r r

r r r r

r r r r

. (2.95)

because ( )n

w −r r is the macroscopic average of ( )n

δ −r r . The quantity

( , ) ( ) ( )i

i n n

n

P t p t δ≡ −∑r r r , (2.96)

is the macroscopic polarization. The quantity

( , ) ( ) ( )ij

ij n n

n

Q t Q t δ= −∑r r r (2.97)

is the macroscopic quadrupole density. They have a straightforward physical interpretation. Consider, for

example, Eq.(2.96). The quantity within the averaging brackets is the sum over all molecules of the

product of each molecule's dipole moment with a δ-function centered at the molecule. This is clearly the

microscopic density of molecular dipole moments, and Pi is its macroscopic average. So the vector P(r,t)

is called the macroscopic polarization, is nothing but the macroscopic average of the dipole-moment

density of the medium's molecules. Similarly, Qij(r,t) is the macroscopic average of the molecular

quadrupole-moment density. In terms of these quantities we have

1

( , ) ( , ) ( , ) ...6

b i i i j ij

i ij

t P t Q tη = − ∇ + ∇ ∇ +∑ ∑r r r (2.98)

For the macroscopic average of the bound charge density.

The average of the free-charge density is simply

( , ) ( , ) ( )f f f

f

t t q tρ η δ ≡ = − ∑r r r r , (2.99)

and the macroscopic average of the total charge density is therefore

1

( , ) ( , ) ( , ) ...6

i i i j ij

i ij

t P t Q tη ρ= − ∇ + ∇ ∇ +∑ ∑r r r . (2.100)

Substituting this into eq. (2.81) yields

0

1...

6i i i j ij

i j

E P Qε ρ

∇ + − ∇ + =

∑ ∑ . (2.101)

The vector within the brackets is the macroscopic displacement field D(r,t)

0

1...

6i i i j ij

j

D E P Qε= + − ∇ +∑ , (2.102)

and we arrive at

ρ∇ ⋅ =D . (2.103)

16

Notice that this macroscopic Maxwell equation involves only the density of free charges on the right-hand

side, and that apart from a factor of ε0 the effective electric field D is the electric field E augmented by the

electrical response of the medium, which is described by the molecular dipole-moment and quadrupole-

moment densities. In practice the term involving Qij in Eq.(2.102) is negligible, and we can write

D=ε0E+P.

Now we need to average the current density. We define the microscopic current densities following the

definitions (2.85)-(2.87) of the respective charge densities. The current density of free charge is

( , ) ( ) ( )f f f f

f

t q t tδ = − ∑j r v r r . (2.104)

The current density of bound charge within the molecule labeled by n is

[ ]( , ) ( ) ( )n b b n b

b

t q t tδ= − −∑j r v r r r , (2.105)

The current density of bound charge is

( , ) ( , )b n

n

t t=∑j r j r , (2.106)

and the total current density is

( , ) ( , ) ( , )f bt t t= +j r j r j r . (2.107)

The macroscopic average of the molecular current density is

[ ] [ ]

3

3

( , ) ( , ) ( )

( ) ( ) ( ) ( ) ( ) .

n n

b b n b b b n b

b b

t t w d r

q t t w d r q t w tδ

′ ′ ′= − =

′ ′ ′= − − − = − −

∫∑ ∑∫

j r j r r r

v r r r r r v r r r (2.108)

Expanding w in Taylor series about n

−r r we obtain for the i component of the current density

v ( ) v ( ) ...i i i j

n b b n b b b j n

b b j

j q w q x w= − − ∇ − +∑ ∑ ∑r r r r . (2.109)

In the first term we recognize the time derivative of the molecular dipole moment

( )

( ) v ( )i

i inb b b b

b b

dp t dq x t q t

dt dt= =∑ ∑ . (2.110)

To put the second term in eq.(2.109) in a recognizable form we define the molecular moment by

1

( ) ( ) ( )2

n b b b

b

t q t t≡ ×∑m r v . (2.111)

Now consider a vector n

w∇ ×m . It can be written as

( ) ( ) ( )1 1

2 2n b b b b b b b b

b b

w q w q w w ∇ × = ∇ × × = ∇ ⋅ − ∇ ⋅ ∑ ∑m r v r v v r , (2.112)

where we used the vector identity ( ) ( ) ( )× × = ⋅ − ⋅a b c b a c c a b . The i component of the vector n

w∇ ×m

(2.112) is therefore

17

( ) ( )

( )

1 1v v v v

2 2

1 1v v v v

2 6

ii j i j i j i j

n b b b j b b j b b b b b j

b j j b j

i j i j i j ij i j

b b b b b j b b b j n j b b b j

b j b j j b j

w q x w x w q x x w

dq x x w q x w Q w q x w

dt

∇ × = ∇ − ∇ = − ∇ =

= + ∇ − ∇ = ∇ − ∇

∑ ∑ ∑ ∑∑

∑∑ ∑∑ ∑ ∑∑

m

(2.113)

Here we took into account the definition of the quadrupole moment (2.93), which gives

( )1( ) ( ) ( ) v v

3

ij i j i j i j

n b b b b b b b b

b b

d dQ t q x t x t q x x

dt dt= = +∑ ∑ . (2.114)

Combining eqs. (2.109), (2.110), and (2.113) we obtain

( )

( )

1...

6

1...,

6

i iji

i n n

n j n

j

ii ij

n n j n

j

dp dQj w w w

dt dt

dp w Q w w

dt

= − ∇ + ∇ × + =

= − ∇ + ∇ × +

∑

∑

m

m

(2.115)

where we used the identity ( )n n nw w w∇× = ∇ × + ∇ ×m m m and the fact that

nm is independent of r.

Summing over n yields

1( ) ( ) ( ) ...

6

1( ) ( ) ( ) ...

6

i

i i ij

b n n j n n n n

n j n n

i

i ij

n n j n n n n

n j n n

dj p w Q w w

dt

dp Q

dtδ δ δ

= − − ∇ − + ∇× − + =

− − ∇ − + ∇ × − +

∑ ∑ ∑ ∑

∑ ∑ ∑ ∑

r r r r m r r

r r r r m r r

(2.116)

Introducing the macroscopic magnetization

( , ) ( )n n

n

t δ= −∑M r m r r . (2.117)

As the macroscopic average of the magnetic moment density of the molecules in the medium, we finally

arrive at the expression for the macroscopic average of the density of bound current:

( )1...

6

i

b i j ij ij

dj P Q

dt

= − ∇ + ∇ × +

∑ M (2.118)

The average of the free-current density is simply

( , ) ( , ) ( )f f f f

f

t t q δ= = −∑J r j r v r r , (2.119)

and the macroscopic average of the total current density is therefore

( )0

d

dtε= + − + ∇ ×j J D E M , (2.120)

having used eq.(2.102). Substituting eq. (2.120) into eq. (2.82) we obtain

18

( )0 0 0 0

d

t dtµ ε µ ε∂ ∇× − = + − + ∇ × ∂

EB J D E M , (2.121)

or

0 tµ

∂∇× − − = ∂

B DM J . (2.122)

The vector in brackets is the macroscopic field 0µ

= −BH M and we arrive at

t

∂∇× − =∂D

H J . (2.123)

Notice that this macroscopic Maxwell equation involves only the density of free currents on the right-

hand side, and that apart from a factor of µ0, the effective magnetic field H is the magnetic field B

diminished by the magnetic response of the medium which is described by the molecular magnetic-

moment density.

In summary the macroscopic Maxwell equations take the familiar form

Coulomb’s law (Gauss law): ρ∇ ⋅ =D (2.124)

Ampere’s law: t

∂∇× − =∂D

H J (2.125)

Absence of free magnetic poles: 0∇ ⋅ =B (2.126)

Faraday’s law: 0t

∂∇× + =∂B

E (2.127)

Here ρ is the macroscopic free charge density and J is the macroscopic free current density. The fields D

and H are defined as follows

0ε= +D E P (2.128)

and

0

1

µ= −H B M , (2.129)

where

( ) ( )n n

n

t δ≡ −∑P p r r , (2.130)

is the macroscopic polarization (pn is the molecular dipole moment) and

( ) ( )n n

n

t δ= −∑M m r r , (2.131)

is the macroscopic magnetization (mn is the molecular magnetic moment).

Poynting's Theorem and Conservation of Energy

In this section we consider conservation of energy, often called Poynting’s theorem.

Suppose that we have some charge and current distribution which at time t produces fields E and B. In the

next instant dt the charges move around a bit. The question is how much work dw is done by the

electromagnetic forces acting on these charges in the interval dt? According to the Lorentz force law, the

19

work done on a charge q is

( )dw d q dt q dt= ⋅ = + × ⋅ = ⋅F l E v B v E v , (2.132)

which reflects the fact that the magnetic field does no work, since the magnetic force is perpendicular to

the velocity. In Eq.(2.132) the charge is 3q d rρ= which results is

( )3 3dw d rdt d rdtρ= ⋅ = ⋅E v E J , (2.133)

where ρ=J v is the current density. Therefore, the total rate of doing work by the fields in a finite

volume V is

( )3 3dW dwd r d r

dt dtV V

≡ = ⋅∫ ∫ E J . (2.134)

This power is delivered representing a conversion of electromagnetic energy into mechanical or thermal

energy. It must be balanced by a corresponding rate of decrease of energy in the electromagnetic field

within the volume V. To exhibit this conservation law explicitly, we use the Maxwell equations to express

(2.134) in other terms. Thus we use the Ampere’s law to eliminate J:

( ) ( )3 3

V V

dWd r d r

dt t

∂ = ⋅ = ⋅ ∇ × − ⋅ ∂ ∫ ∫

DE J E H E . (2.135)

If we now employ the vector identity (which comes prom the possibility of cyclic permutation in the

triple product),

( ) ( ) ( )∇ ⋅ × = ⋅ ∇× − ⋅ ∇ ×E H H E E H . (2.136)

and use Faraday's law, the right-hand side of (2.135) becomes

( ) ( ) ( )3 3

V V

dWd r d r

dt t t t

∂ ∂ ∂ = ⋅ ∇× − ∇ ⋅ × − ⋅ = − ∇ ⋅ × + ⋅ + ⋅ ∂ ∂ ∂ ∫ ∫

D D BH E E H E E H E H . (2.137)

Now we make the assumption that the macroscopic medium is linear in its electric and magnetic

properties. In this case the total electromagnetic energy density is given by

( )1

2em

u = ⋅ + ⋅E D B H (2.138)

and therefore Eq.(2.137) can be written as follows

( ) 3em

V

udWd r

dt t

∂ = − + ∇ ⋅ × ∂ ∫ E H . (2.139)

Using the divergence theorem for the term containing ( )∇ ⋅ ×E H we obtain

( ) ( )3 3

em

V V S

dW dd r u d r da

dt dt= ⋅ = − − × ⋅∫ ∫ ∫E J E H n� . (2.140)

where S is the surface bounding V. This is Poynting’s theorem. The first integral on the right is the total

energy stored in the fields. The second term evidently represents the rate at which energy is carried out of

V, across its boundary surface, by the electromagnetic fields. Poynting’s theorem says, then, that the work

done on the charges by the electromagnetic force is equal to the decrease in energy stored in the field, less

the energy that flowed out through the surface.

20

The energy per unit time per unit area transported by EM fields is called the Poynting’s vector. It is given

by

= ×S E H . (2.141)

Specifically da⋅S n is the energy per unit time crossing the infinitesimal surface da – the energy flux – so

S is the energy flux density.

The work done per unit time per unit volume by the fields ( )⋅E J is a conversion of electromagnetic

energy into mechanical or heat energy. Since matter is ultimately composed of charged particles

(electrons and atomic nuclei), we can think of this rate of conversion as a rate of increase of mechanical

energy (kinetic, potential, etc.) of the charged particles per unit volume. We denote this energy by umech

Then, we can interpret Poynting's theorem for the microscopic fields (E, B) as a statement of conservation

of energy of the combined system of particles and fields.

( ) 3 3

mech

V V

dW dd r u d r

dt dt= ⋅ =∫ ∫E J . (2.142)

Then the energy conservation law takes the form

( ) 3

em mech

V S V

u u d r da dat

∂ + = − ⋅ = − ∇ ⋅∂∫ ∫ ∫S n S� . (2.143)

Since the volume V is arbitrary, this can be cast into the form of a differential continuity equation

( )em mechu u

t

∂ += −∇ ⋅

∂S . (2.144)

It is differential form of the Poyning’s theorem. Compare it with the continuity equation expressing

conservation of charge:

t

ρ∂ = −∇ ⋅∂

J . (2.145)

The change density is replaced by the Pointing vector. The latter represents the flow of energy in the same

way that J describes the flow of charge.

Maxwell’s Stress Tensor

Let’s calculate the total electromagnetic force on the charges in volume V:

( ) ( )3 3

V V

d r d rρ ρ= + × = + ×∫ ∫F E v B E J B . (2.146)

The force per unit volume is evidently

ρ= + ×f E J B . (2.147)

Now we eliminate ρ and J to write it in terms of fields alone. Using Maxwell’s equations we find:

( )0 0

0

1

tε ε

µ ∂= ∇ ⋅ + ∇× − × ∂

Ef E E B B . (2.148)

Now

( )t t t

∂ ∂ ∂× = × + ×∂ ∂ ∂

E BE B B E . (2.149)

and Faraday’s law says

21

t

∂ = −∇ ×∂B

E , (2.150)

so

( ) ( )t t

∂ ∂× = × + × ∇ ×∂ ∂E

B E B E E . (2.151)

Thus

( ) ( ) ( ) ( )0 0 0

0

1

tε ε ε

µ∂= ∇ ⋅ − × ∇ × − × − × ∇ ×∂

f E E B B E B E E . (2.152)

Using the ( ) ( ) ( )× × = ⋅ − ⋅a b c b a c c a b rule we can write �( ) ( ) ( )× ∇ × = ∇ ⋅ − ⋅ ∇a c a c a c and

consequently

( ) ( ) ( )21

2E× ∇ × = ∇ − ⋅∇E E E E , (2.153)

where the factor ½ is the consequence of differential nature of the ∇ vector:

( ) ( ) � � �2 ( ) ( ) 2 ( )E∇ = ∇ ⋅ = ∇ ⋅ + ∇ ⋅ = ∇ ⋅E E E E E E E E . (2.154)

Similar we have for B:

( ) ( ) ( )21

2B× ∇ × = ∇ − ⋅∇B B B B . (2.155)

We can therefore rewrite Eq.(2.152) as follows

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

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

2 2

0 0 0

0

2 2

0 0 0

0 0

1 1 1

2 2

1 1 1

2

B Et

E Bt

ε ε εµ

ε ε εµ µ

∂ = ∇ ⋅ − ∇ − ⋅∇ − × − ∇ − ⋅∇ = ∂

∂ ∇ ⋅ + ⋅∇ + ⋅∇ + ∇ ⋅ − ∇ + ∇ − × ∂

f E E B B E B E E

E E E E B B B B E B

. (2.156)

where we used 0.∇ ⋅ =B This equation can be simplified by introducing the Maxwell stress tensor:

2 2

0

0

1 1 1

2 2ij i j ij i j ij

T E E E B B Bε δ δµ

≡ − + −

. (2.157)

This indices i and j refer to the coordinates x, y, and z, so the stress tensor has a total of nine components.

The Kronecker delta, ijδ , is 1 if indices are the same and is zero otherwise. For example,

( ) ( )2 2 2 2 2 2

0

0

1 1

2 2xx x y z x y zT E E E B B Bε

µ= − − + − − , (2.158)

0

0

1xy x y x yT E E B Bε

µ= + , (2.159)

and so on. Because of two indices sometimes tensors are denoted by a double arrow T�

. One can form a

dot product of a tensor T�

and a vector a:

( ) i ijj

i

a T⋅ =∑a T�

, (2.160)

The resulting object, which has one remaining index is a vector. In particular, the divergence of T�

has j

component

22

( )

( ) ( ) ( ) ( )

2 2

0

0

2 2

0

0

2 2

0

0

1 1 1

2 2

1 1 1

2 2

1 1 1

2 2

i i j ij i i j ijj

i i

j i i i i j j j i i i i j j

i i i i

j j j j j j

E E E B B B

E E E E E B B B B B

E E E B B B

ε δ δµ

εµ

εµ

∇ ⋅ = ∇ − + ∇ − =

∇ + ∇ − ∇ + ∇ + ∇ − ∇ =

∇ ⋅ + ⋅∇ − ∇ + ∇ ⋅ + ⋅∇ − ∇

∑ ∑

∑ ∑ ∑ ∑

T

E E B B

�

(2.161)

The force per unit volume given by Eq. (2.156) can be written in much more simpler form:

0 0t

ε µ ∂= ∇ ⋅ −∂S

f T�

, (2.162)

where S is the Poynting vector. The total force on charges (eq. (2.146)) in volume V is

3 3

0 0 0 0

V S V

dd r da d r

t dtε µ ε µ∂ = ∇ ⋅ − = ⋅ − ∂

∫ ∫ ∫S

F T T n S� �

� , (2.163)

where the divergence theorem was used to convert the first term to a surface integral. In the static case

(or, more generally, whenever 3d r∫S is independent of time), the second term drops out, and the

electromagnetic force on the charge configuration can be expressed entirely in terms of the stress tensor at

the boundary. Physically, T�

is the force per unit area (or stress) acting on the surface. More precisely, Tij

is the force (per unit area) in the ith direction acting on an element of surface oriented in the jth direction

– “diagonal” elements (Txx, Tyy, Tzz) represent compression stress or pressures, and “off-diagonal”

elements (Txy, Txz, etc.) are shear stress.

Let us consider a simple example of the calculation of force between two equal point charges q separated

by distance 2a. The force can be calculated by constructing a plane equidistant from the two charges and

integrating the Maxwell’s stress tensor over this plane. If we assume that the charges lie at the z axis at +a

and –a, the plane is the xy plane, and we intend to calculate the force on the upper charge, then the normal

to the plane is ˆ= −n z . Since by symmetry the force have only non-vanishing z component, we need only

the zz component of the tensor:

2 2

0

1

2zz zzT E Eε = −

. (2.164)

The electric field at a point on the surface placed at distance s from the z axis is

2 2

0

1ˆ2 cos

4

q

s aθ

πε=

+E s . (2.165)

where angle θ is such that 2 2cos /s s aθ = + . It follows that 0z

E = and

22

2

2 2 3

02 ( )

q sE

s aπε

= + . (2.166)

Therefore we have

22 2

0 2 2 3 2 3

0 00 0

1 12

2 2 ( ) 4 2 ( )z

S

q s q uduF da sds

s a u aε π

πε πε

∞ ∞ = ⋅ = = + + ∫ ∫ ∫T n�

� , (2.167)

where we let 2u s≡ . The integral can easily be taken by parts

23

2 3 2 2 2 2 2 2

0 0 0 0

1 1 1 1 1 1

( ) 2 ( ) 2 ( ) 2 ( ) 2

udu duud

u a u a u a u a a

∞∞ ∞ ∞ = − = = − = + + + +

∫ ∫ ∫ . (2.168)

Finally we have

2

2

0

1

4 (2 )z

qF

aπε= , (2.169)

as expected.

Newton’s Third Law in Electrodynamics

Imagine a point charge q traveling in along the x axis at a constant speed v. Because it is moving, its electric field is not given by Coulomb’s law; nevertheless, E still points radially outward from the instantaneous position of the charge (Fig. 3.6a). Since, moreover, a moving point charge does not constitute a steady current, its magnetic field is not given by the Biot-Savart law. Nevertheless, it’s a fact that B still circles around the axis in a manner suggested by the right-hand rule (Fig. 3.6b).

Fig. 3.6

Now suppose this charge encounters an identical one, proceeding in at the same speed along the y axis

(Fig. 3.7). The electric force between them is repulsive, but how about the magnetic force?

Fig. 3.7

The magnetic field of q1 points into the page (at the position of q2), so the magnetic force on q2 is toward

the right, whereas the magnetic field of q2 is out of the page (at the position of q1), and the magnetic force

on q1 is upward. The electromagnetic force of q1 on q2 is equal but not opposite to the force of q2 on q1 in

violation of Newton's third law. In electrostatics and magnetostatics the third law holds, but in

electrodynamics it does not.

24

Since forces are associated with a change of the momentum in time, this fact implies that the momentum

of the particles is not conserved. Momentum conservation is rescued in electrodynamics by the realization

that the fields themselves carry momentum. In the case of the two point charges in Fig. 3.7, whatever

momentum is lost to the particles is gained by the fields. Only when the field momentum is added to the

mechanical momentum of the charges is momentum conservation restored.

Conservation of Momentum

According to Newton's second law, the force on an object is equal to the rate of change of its momentum:

mechd

dt= P

F , (2.170)

Equation (2.163) can therefore be written in the form

3

0 0mech

V S

d dd r da

dt dtε µ= − + ⋅∫ ∫

PS T n

�

� , (2.171)

where Pmech is the total (mechanical) momentum of the particles contained in the volume V. This

expression is similar in structure to Poynting's theorem, and it invites an analogous interpretation: The

first integral represents momentum stored in the electromagnetic fields themselves:

3

0 0em

V

d rε µ= ∫P S , (2.172)

while the second integral is the momentum per unit time flowing in through the surface. Equation (2.171)

the general statement of conservation of momentum in electrodynamics: Any increase in the total

momentum (mechanical plus electromagnetic) is equal to the momentum brought in by the fields:

( )mech em

S

dda

dt+ = ⋅∫P P T n

�

� . (2.173)

If V is all of space, then no momentum flows in or out, and Pmech+ Pem is constant.

As in the case of conservation of charge and conservation of energy, conservation of momentum can be

given a differential formulation. Let pmech be the density of mechanical momentum, and pem the density of

momentum in the fields

0 0emε µ=p S , (2.174)

Then Eq. (2.171) in differential form says

( )mech emt

∂ + = ∇ ⋅∂

p p T�

, (2.175)

Evidently, T�

is the momentum flux density, playing the role of J (current density) in the continuity

equation, or S (energy flux density) in Poynting’s theorem. Specifically, -Tij is the momentum in the i

direction crossing a surface oriented in the j direction, per unit area, per unit time. Notice that the

Poynting vector has appeared in two quite different roles: S itself is the energy per unit area, per unit time,

transported by the electromagnetic fields, while 0 0ε µ S is the momentum per unit volume stored in those

fields. Similarly, T�

plays a dual role: T�

itself is the electromagnetic stress (force per unit area) acting on

a surface, and T�

describes the flow of momentum (the momentum current density) transported by the

fields.

Example: A transverse plane wave is incident normally to vacuum on perfectly absorbing flat screen.

From the law of momentum conservation show that the pressure (called radiation pressure) exerted on the

25

screen is equal to the field energy per unit volume in the wave.

Solution: Conservation of momentum says that

( )mech em

S

dda

dt+ = ⋅∫P P T n

�

� , (2.176)

The i component of the force transmitted across S and acting on the particles and field inside V.

( )mech em ij jijS

dT n da

dt+ = ∑∫P P � . (2.177)

We take surface S to cover outside screen and going to right to contain thickness of the screen. Take the

direction of propagation in +z direction, with n as outward normal. Then, ij j

j

T n∑ is pressure (a force per

unit area) across S into volume V. Since ˆ= −n z , radiation pressure ij j iz

j

T n T= −∑

The Maxwell’s stress tensor is given by eq. (2.158)

2 2

0

0

1 1 1

2 2ij i j ij i j ij

T E E E B B Bε δ δµ

= − + −

. (2.178)

Since 0z z

E B= = for the plane wave propagating along the z direction:

2 2 2 2

0 0

0 0

1 1 1 1 1

2 2 2iz i z iz i z iz izT E E E B B B E Bε δ δ δ ε

µ µ = − + − = − +

. (2.179)

Since only zz component is non-zero the radiation pressure is

2 2

0

0

1 1

2zz emT E B uε

µ

− = + =

, (2.180)

where uem is the field energy density of the wave.

Angular Momentum

Thus we conclude that the electromagnetic fields carrier energy

2 2

0

0

1 1

2emu E Bε

µ

= +

, (2.181)

momentum

0 0emε µ=p S , (2.182)

and, in addition, angular momentum

[ ]0 0em em ε µ= × = ×l r p r S . (2.183)