What does the gradient tell you about a scalar ¬eld? Two important

things: the magnitude of the gradient indicates how quickly the ¬eld is

changing over space, and the direction of the gradient indicates the dir-

ection in that the ¬eld is changing most quickly with distance.

Therefore, although the gradient operates on a scalar ¬eld, the result of

the gradient operation is a vector, with both magnitude and direction. Thus,

if the scalar ¬eld represents terrain height, the magnitude of the gradient at

any location tells you how steeply the ground is sloped at that location, and

the direction of the gradient points uphill along the steepest slope.

The de¬nition of the gradient of the scalar ¬eld w is

@w ^ @w ^ @w

˜

grad°wÞ ¼ rw ^ þj þk °CartesianÞ: °5:3Þ

i

@x @y @z

Thus, the x-component of the gradient of w indicates the slope of the

scalar ¬eld in the x-direction, the y-component indicates the slope in the

y-direction, and the z-component indicates the slope in the z-direction.

The square root of the sum of the squares of these components provides

the total steepness of the slope at the location at which the gradient is

taken.

In cylindrical and spherical coordinates, the gradient is

@w 1 @w @w

˜

rw ^ ^

þ™ þ^ °cylindricalÞ °5:4Þ

r z

@r r @u @z

and

@w ^1 @w 1 @w

˜

rw ^ ^

þh þ™ °sphericalÞ: °5:5Þ

r

@r r @h r sin h @u

120 A student™s guide to Maxwell™s Equations

˜˜ ˜

r; r; r · Some useful identities

Here is a quick review of the del differential operator and its three uses

relevant to Maxwell™s Equations:

Del:

˜ i@ j@ k@

r ^ þ^ þ ^

@x @y @z

Del (nabla) represents a multipurpose differential operator that can

operate on scalar or vector ¬elds and produce scalar or vector results.

Gradient:

@w ^ @w ^ @w

˜

rw ^ þj þk

i

@x @y @z

The gradient operates on a scalar ¬eld and produces a vector result

that indicates the rate of spatial change of the ¬eld at a point and the

direction of steepest increase from that point.

Divergence:

˜ A @Ax þ @Ay þ @Az

r˜

@x @y @z

The divergence operates on a vector ¬eld and produces a scalar result

that indicates the tendency of the ¬eld to ¬‚ow away from a point.

Curl:

@Az @Ay ^ @Ax @Az ^ @Ay @Ax ^

˜A

r·˜ À iþ À jþ À k

@y @z @z @x @x @y

The curl operates on a vector ¬eld and produces a vector result that

indicates the tendency of the ¬eld to circulate around a point and the

direction of the axis of greatest circulation.

121

From Maxwell™s Equations to the wave equation

Once you™re comfortable with the meaning of each of these operators,

you should be aware of several useful relations between them (note that

the following relations apply to ¬elds that are continuous and that have

continuous derivatives).

The curl of the gradient of any scalar ¬eld is zero.

˜˜

r · rw ¼ 0; °5:6Þ

which you may readily verify by taking the appropriate derivatives.

Another useful relation involves the divergence of the gradient of a

scalar ¬eld; this is called the Laplacian of the ¬eld:

@2w @2w @2w

˜˜

r rw ¼ r2 w ¼ 2 þ 2 þ 2 °CartesianÞ: °5:7Þ:

@x @y @z

The usefulness of these relations can be illustrated by applying them to

the electric ¬eld as described by Maxwell™s Equations. Consider, for

example, the fact that the curl of the electrostatic ¬eld is zero (since

electric ¬eld lines diverge from positive charge and converge upon

negative charge, but do not circulate back upon themselves). Equation 5.6

indicates that as a curl-free (irrotational) ¬eld, the electrostatic ¬eld ˜

E

may be treated as the gradient of another quantity called the scalar

potential V:

˜

˜ ¼ À rV; °5:8Þ

E

where the minus sign is needed because the gradient points toward the

greatest increase in the scalar ¬eld, and by convention the electric force

on a positive charge is toward lower potential. Now apply the differential

form of Gauss™s law for electric ¬elds:

˜E q

r˜¼ ;

e0

which, combined with Equation 5.8, gives

q

r2 V ¼ À : °5:9Þ

e0

This is called Laplace™s equation, and it is often the best way to ¬nd the

electrostatic ¬eld when you are not able to construct a special Gaussian

surface. In such cases, it may be possible to solve Laplace™s Equation for

the electric potential V and then determine ˜ by taking the gradient of the

E

potential.

122 A student™s guide to Maxwell™s Equations

2˜

r2˜ ¼ v12 @ A

A The wave equation

@t2

With the differential form of Maxwell™s Equations and several vector

operator identities in hand, the trip to the wave equation is a short

one. Begin by taking the curl of both sides of the differential form of

Faraday™s law

!

˜ BÞ

˜ @°r · ˜

@B

˜ ˜ EÞ ˜

r · °r · ˜ ¼ r · À ¼À : °5:10Þ

@t @t

Notice that the curl and time derivatives have been interchanged in the

¬nal term; as in previous sections, the ¬elds are assumed to be suf¬ciently

smooth to permit this.

Another useful vector operator identity says that the curl of the curl of

any vector ¬eld equals the gradient of the divergence of the ¬eld minus

the Laplacian of the ¬eld:

˜ ˜ AÞ ˜ ˜ AÞ

r · °r · ˜ ¼ r°r ˜ À r2˜ °5:11Þ

A:

This relation uses a vector version of the Laplacian operator that is

constructed by applying the Laplacian to the components of a vector

¬eld:

@ 2 A x @ 2 Ay @ 2 Az

r2˜ ¼ þ þ2 °CartesianÞ: °5:12Þ

A

@x2 @y2 @z

Thus,

˜ BÞ

@°r · ˜

˜ ˜ EÞ ˜ ˜ EÞ

r · °r · ˜ ¼ r°r ˜ À r2˜ ¼ À : °5:13Þ

E

@t

However, you know the curl of the magnetic ¬eld from the differential

form of the Ampere“Maxwell law:

!

@˜

E

˜B

r · ˜ ¼ l0 ˜ þ e 0 :

J

@t

So,

‚À ÁÃ

@ l0 ˜ þ e0 °@˜

J E=@tÞ

˜ ˜ EÞ ˜ ˜ EÞ

r · °r · ˜ ¼ r°r ˜ À r2˜ ¼ À :

E

@t

123

From Maxwell™s Equations to the wave equation

This looks dif¬cult, but one simpli¬cation can be achieved using Gauss™s

law for electric ¬elds:

˜E q

r˜¼ ;

e0

which means

‚À ÁÃ

@ l0 ˜ þ e0 °@˜

q J E=@tÞ

˜ ˜ EÞ ˜

r · °r · ˜ ¼ r 2˜

Àr E ¼À

e0 @t

@˜ @ 2˜

J E

¼ Àl0 À l 0 e0 2 :

@t @t

Gathering terms containing the electric ¬eld on the left side of this

equation gives

@ 2˜ ˜ q @˜

E J

r 2 ˜ À l0 e 0 2 ¼ r þ l0 :

E

@t e0 @t

In a charge- and current-free region, q = 0 and ˜ ¼ 0, so

J

@ 2˜

E

2˜

r E ¼ l0 e0 2 : °5:14Þ

@t

This is a linear, second-order, homogeneous partial differential equation

that describes an electric ¬eld that travels from one location to another “

in short, a propagating wave. Here is a quick reminder of the meaning of

each of the characteristics of the wave equation:

Linear: The time and space derivatives of the wave function (˜ in this

E

case) appear to the ¬rst power and without cross terms.

Second-order: The highest derivative present is the second derivative.

Homogeneous: All terms involve the wave function or its derivatives, so

no forcing or source terms are present.

Partial: The wave function is a function of multiple variables (space and

time in this case).

A similar analysis beginning with the curl of both sides of the Ampere“

Maxwell law leads to

@ 2˜

B

2˜

r B ¼ l0 e0 2 ; °5:15Þ

@t

which is identical in form to the wave equation for the electric ¬eld.

This form of the wave equation doesn™t just tell you that you have a

wave “ it provides the velocity of propagation as well. It is right there in

124 A student™s guide to Maxwell™s Equations

the constants multiplying the time derivative, because the general form of

the wave equation is this

1 @ 2˜

A

r2˜ ¼ ; °5:16Þ

A

v2 @t2

where v is the speed of propagation of the wave. Thus, for the electric and

magnetic ¬elds

1

¼ l0 e0 ;

v2

or

s¬¬¬¬¬¬¬¬¬

1

v¼ : °5:17Þ

l0 e 0

Inserting values for the magnetic permeability and electric permittivity of

free space,

s¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬

1

v¼ ;

½4p · 10À7 m kg=C2 ½8:8541878 · 10À12 C2 s2 =kg m3

or

q¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬

v ¼ 8:987552·1016 m2 =s2 ¼ 2:9979 · 108 m=s:

It was the agreement of the calculated velocity of propagation with the

measured speed of light that caused Maxwell to write, ˜˜light is an elec-

tromagnetic disturbance propagated through the ¬eld according to

electromagnetic laws.™™

Appendix: Maxwell™s Equations

in matter

Maxwell™s Equations as presented in Chapters 1“4 apply to electric and

magnetic ¬elds in matter as well as in free space. However, when you™re

dealing with ¬elds inside matter, remember the following points:

The enclosed charge in the integral form of Gauss™s law for electric

¬elds (and current density in the differential form) includes ALL

charge “ bound as well as free.

The enclosed current in the integral form of the Ampere“Maxwell law

(and volume current density in the differential form) includes ALL

currents “ bound and polarization as well as free.

Since the bound charge may be dif¬cult to determine, in this Appendix

you™ll ¬nd versions of the differential and integral forms of Gauss™s law

for electric ¬elds that depend only on the free charge. Likewise, you™ll ¬nd

versions of the differential and integral form of the Ampere“Maxwell law

that depend only on the free current.

What about Gauss™s law for magnetic ¬elds and Faraday™s law? Since

those laws don™t directly involve electric charge or current, there™s no

need to derive more “matter friendly” versions of them.

Gauss™s law for electric ¬elds: Within a dielectric material, positive and

negative charges may become slightly displaced when an electric ¬eld is

applied. When a positive charge Q is separated by distance s from an

equal negative charge ’Q, the electric “dipole moment” is given by

˜ ¼ Q˜ °A:1Þ

p s;

where ˜ is a vector directed from the negative to the positive charge with

s

magnitude equal to the distance between the charges. For a dielectric

125

126 A student™s guide to Maxwell™s Equations

material with N molecules per unit volume, the dipole moment per unit

volume is

˜ ¼ N˜ °A:2Þ

P p;

a quantity which is also called the “electric polarization” of the material.

If the polarization is uniform, bound charge appears only on the surface

of the material. But if the polarization varies from point to point within

the dielectric, there are accumulations of charge within the material, with

volume charge density given by

qb ¼ À˜ ˜

r P; °A:3Þ

where qb represents the volume density of bound charge (charge that™s

displaced by the electric ¬eld but does not move freely through the

material).

What is the relevance of this to Gauss™s law for electric ¬elds? Recall

that in the differential form of Gauss™s law, the divergence of the electric

¬eld is

˜ ˜ ¼ q ;

rE

e0

where q is the total charge density. Within matter, the total charge density

consists of both free and bound charge densities:

q ¼ qf þ qb ; °A:4Þ

where q is the total charge density, qf is the free charge density, and qb is

the bound charge density. Thus, Gauss™s law may be written as

˜ ˜ ¼ q ¼ qf þ qb :

rE °A:5Þ

e0 e0

Substituting the negative divergence of the polarization for the bound

charge and multiplying through by the permittivity of free space gives

˜ e0˜ ¼ qf þ qb ¼ qf À ˜ ˜

r r P; °A:6Þ

E

or

˜ e 0 ˜ þ ˜ ˜ ¼ qf :

r E rP °A:7Þ

Collecting terms within the divergence operator gives

˜ °e0˜ þ ˜ ¼ qf :

r °A:8Þ

E PÞ

127

Appendix

In this form of Gauss™s law, the term in parentheses is often written as a

vector called the “displacement,” which is de¬ned as

D ¼ e0 ˜ þ ˜

˜ °A:9Þ

E P:

Substituting this expression into equation (A.8) gives

˜ D ¼ qf ;

r˜ °A:10Þ

which is a version of the differential form of Gauss™s law that depends

only on the density of free charge.

Using the divergence theorem gives the integral form of Gauss™s law

for electric ¬elds in terms of the ¬‚ux of the displacement and enclosed free

charge:

I

˜n

D ^ da ¼ qfree; enc : °A:11Þ

S

What is the physical signi¬cance of the displacement ˜ In free space, the

D?

displacement is a vector ¬eld proportional to the electric ¬eld “ pointing in

the same direction as ˜ and with magnitude scaled by the vacuum permit-

E

tivity. But in polarizable matter, the displacement ¬eld may differ signi¬-

cantly from the electric ¬eld. You should note, for example, that the

displacement is not necessarily irrotational “ it will have curl if the polar-

ization does, as can be seen by taking the curl of both sides of Equation A.9.

˜

The usefulness of D comes about in situations for which the free charge

is known and for which symmetry considerations allow you to extract the

displacement from the integral of Equation A.11. In those cases, you may

be able to determine the electric ¬eld within a linear dielectric material

˜

by ¬nding D on the basis of the free charge and then dividing by the

permittivity of the medium to ¬nd the electric ¬eld.

The Ampere“Maxwell law: Just as applied electric ¬elds induce polar-

ization (electric dipole moment per unit volume) within dielectrics, applied

magnetic ¬elds induce “magnetization” (magnetic dipole moment per unit

volume) within magnetic materials. And just as bound electric charges act

as the source of additional electric ¬elds within the material, bound cur-

rents may act as the source of additional magnetic ¬elds. In that case, the

bound current density is given by the curl of the magnetization:

˜b ¼ ˜ · M:

r˜ °A:12Þ

J

where ˜b is the bound current density and M represents the magnetization

˜

J

of the material.

128 A student™s guide to Maxwell™s Equations

Another contribution to the current density within matter comes

from the time rate of change of the polarization, since any movement of

charge constitutes an electric current. The polarization current density is

given by

˜

˜P ¼ @ P : °A:13Þ

J

@t

Thus, the total current density includes not only the free current density,

but the bound and polarization current densities as well:

˜ ¼ ˜f þ ˜b þ ˜P : °A:14Þ

JJ J J

Thus, the Ampere“Maxwell law in differential form may be written as

!

˜

˜ · ˜ ¼ l Jf þ Jb þ JP þ e0 @ E :

˜˜˜

rB °A:15Þ

@t

0

Inserting the expressions for the bound and polarization current and

dividing by the permeability of free space

1 ˜ ˜ ˜ ˜ ˜ @˜ @˜

P E

r · B ¼ Jf þ r · M þ þ e0 : °A:16Þ

l0 @t @t

Gathering curl terms and time-derivative terms gives

˜ ˜ ˜

˜ · B À ˜ · M ¼ Jf þ @ P þ @°e0 EÞ :

r˜ ˜

r °A:17Þ

l0 @t @t

Moving the terms inside the curl and derivative operators makes this

!

˜ ˜˜

˜ @°e0 E þ PÞ :

B˜

˜·

r À M ¼ Jf þ °A:18Þ

l0 @t

In this form of the Ampere“Maxwell law, the term in parentheses on the

left side is written as a vector sometimes called the “magnetic ¬eld

intensity” or “magnetic ¬eld strength” and de¬ned as

˜

B˜

˜

H ¼ À M: °A:19Þ

l0

˜˜

Thus, the differential form of the Ampere“Maxwell law in terms of H, D,

and the free current density is

129

Appendix

˜

˜ · H ¼ ˜free þ @ D :

r˜ J °A:20Þ

@t

Using Stokes™ theorem gives the integral form of the Ampere“Maxwell

law:

I Z

d

˜˜ ˜n

D ^ da

H dl ¼ Ifree; enc þ °A:21Þ

dt S

C

˜

What is the physical signi¬cance of the magnetic intensity H? In free

space, the intensity is a vector ¬eld proportional to the magnetic ¬eld “

pointing in the same direction as ˜ and with magnitude scaled by the

B

vacuum permeability. But just as D may differ from ˜ inside dielectric

˜ E

materials, H may differ signi¬cantly from ˜ in magnetic matter. For

˜ B

example, the magnetic intensity is not necessarily solenoidal “ it will have

divergence if the magnetization does, as can be seen by taking the

divergence of both sides of Equation A.19.

˜

As is the case for electric displacement, the usefulness of H comes

about in situations for which you know the free current and for which

symmetry considerations allow you to extract the magnetic intensity from

the integral of Equation A.21. In such cases, you may be able to deter-

˜

mine the magnetic ¬eld within a linear magnetic material by ¬nding H on

the basis of free current and then multiplying by the permeability of the

medium to ¬nd the magnetic ¬eld.

130 A student™s guide to Maxwell™s Equations

Here is a summary of the integral and differential forms of all of

Maxwell™s Equations in matter:

Gauss™s law for electric ¬elds:

I

˜n

D ^ da ¼ qfree; enc °integral formÞ;

S

˜ D ¼ qfree

r˜ °differential formÞ:

Gauss™s law for magnetic ¬elds:

I

˜ ^ da ¼ 0 °integral formÞ;

Bn

S

˜ ˜ ¼ 0

rB °differential formÞ:

Faraday™s law:

I Z

˜ d˜¼ À d ˜ ^ da °integral formÞ;

E l Bn

dt

C S

˜

˜ · ˜ ¼ À @B

rE °differential formÞ:

@t

Ampere“Maxwell law:

I Z

d

H d˜¼ Ifree; enc þ

˜ ˜n

D ^ da °integral formÞ;

l

dt S

C

˜

˜ · H ¼ ˜free þ @ D

˜J

r °differential formÞ:

@t

Further reading

If you™re looking for a comprehensive treatment of electricity and magnetism, you

have several excellent texts from which to choose. Here are some that you may

¬nd useful:

Cottingham W. N. and Greenwood D. A., Electricity and Magnetism. Cambridge

University Press, 1991; A concise survey of a wide range of topics in

electricity and magnetism.

Grif¬ths, D. J., Introduction to Electrodynamics. Prentice-Hall, New Jersey, 1989;

The standard undergraduate text at the intermediate level, with clear

explanations and informal style.

Jackson, J. D., Classical Electrodynamics. Wiley & Sons, New York, 1998; The

standard graduate text, but you must be solidly prepared before embarking.

Lorrain, P., Corson, D., and Lorrain, F., Electromagnetic Fields and Waves.

Freeman, New York, 1988; Another excellent intermediate-level text, with

detailed explanations supported by helpful diagrams.

Purcell, E. M., Electricity and Magnetism Berkeley Physics Course, Vol. 2.

McGraw-Hill, New York, 1965; Probably the best of the introductory-level

texts; elegantly written and carefully illustrated.

Wangsness, R. K., Electromagnetic Fields. Wiley, New York, 1986; Also a great

intermediate-level text, especially useful as preparation for Jackson.

And for a comprehensible introduction to vector operators, with many examples

drawn from electrostatics, check out:

Schey, H. M., Div, Grad, Curl, and All That. Norton, New York, 1997.

131

Index

Ampere, Andre-Marie 83 polarization 128

Ampere--Maxwell law 83--111 units of 105

differential form 101--9

applying 108--9 del cross (see curl)

expanded view 101 del dot (see divergence)

main idea 101 del operator (see nabla)

version involving only free dielectric constant 19

current 128 dielectrics 18--19

integral form 83--100 dipole moment, electric 125

applying 95--100 displacement 127

expanded view 84 physical significance of 127

main idea 84 usefulness of 127

usefulness of 84 displacement current 94

version involving only free units of 107

current 129 divergence 32--5

Ampere™s law 83 in Cartesian coordinates 33

locating regions of 32

Biot--Savart law 47 main idea 120

in non-Cartesian coordinates 35

capacitance 18--19 relationship to flux 32

of a parallel-plate capacitor 18 divergence theorem 114--15

charge, electric main idea 114

bound 18, 126 dot product 6

density 16 how to compute 6

enclosed 16--17 physical significance 6

relationship to flux 17

circulation 65 electric field

of electric field 68 definition of 3

of magnetic field 85--6 electrostatic vs. induced 1, 62--3

closed surface 7 equations for simple objects 5

curl 76--8 induced 62--3

in Cartesian coordinates 77 direction of 63

of electric field 79 units of 3, 62

locating regions of 76--7 electromotive force (emf) 68

of magnetic field 102--4 units of 68

main idea 120 enclosed current 89--90

in non-Cartesian coordinates 78

relationship to circulation 76 Faraday, Michael

current density 105--6 demonstration of induction 58

bound 127 refers to ˜˜field of force™™ 3

132

133

Index

Faraday™s law 58--82 inductance 88

differential form 75--81 insulators (see dielectrics)

applying 79--81 irrotational fields 78

expanded view 75

Kelvin--Stokes theorem 116

main idea 75

integral form 58--74

LaGrange, J. L. 114

applying 72--4

Laplacian operator 121

expanded view 60

vector version 122

main idea 59

Laplace™s equation 121

usefulness of 61

Lenz, Heinrich 71

field lines 3, 13--14

Lenz™s law 71

flux

line integral 64

electric 13--15

Lorentz equation 45

rate of change of 91--4

units of 13

magnetic field

magnetic 48--9

definition of 45

rate of change of 69--70

distinctions from electric field 45

units of (see webers)

equations for simple objects 47

as number of field lines 13--14

intensity 128

of a vector field 10--12

physical significance of 129

usefulness of 129

Gauss, C. F. 114

units of 45

Gauss™s law for electric fields 1--41

magnetic flux density 45

differential form 29--40

magnetic induction 45

applying 38--40

magnetic poles 43

expanded view 30

always in pairs 44

main idea 29

magnetization 127

usefulness of 30

Maxwell, James Clerk

version involving only free

coining of ˜˜convergence™™ 32

charge 127

coining of ˜˜curl™™ 76

integral form 1--28

definition of electric field 3

applying 20--8

electromagnetic theory 112

expanded view 2

use of magnetic vortex model 91

main idea 1

usefulness of 2

nabla 31

version involving only free charge 127

main idea 120

Gauss™s law for magnetic fields 43--57

differential form 53--6 Oersted, Hans Christian 83

applying 55--6 Ohm™s law 73

expanded view 53 open surface 7

main idea 53

Ostrogradsky, M. V. 114

integral form 43--52

applying 50--2 path integral 65--7

expanded view 44 permeability

main idea 44 of free space 87--8

usefulness of 44 relative 87--8

Gauss™s theorem 114 permittivity

gradient 119--20 of free space 18--19

in Cartesian coordinates 119 relative 19

in non-Cartesian coordinates 119 polarization, electric 126

main idea 120 relationship to bound charge 126

Green, G. 114

scalar field 119

Heaviside, Oliver 32 scalar potential 121

scalar product (see dot product)

induced current 68 solenoidal fields 54

direction of 71 special Amperian loop 86, 95--100

134 Index

special Gaussian surface 25 vacuum permittivity (see permittivity

speed of light 124 of free space)

Stokes, G. G. 116 vector cross product 45

Stokes™ theorem 116--19 vector field 10

main idea 116

surface integral 9 wave equation 122--4

for electric fields 113

Thompson, William 116 for magnetic fields 113

webers 48

unit normal vector 7 work done along a path 65--6

direction of 7