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example of a scalar ¬eld is the height of terrain above sea level.
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


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

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

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 Š½8:8541878 · 10À12 C2 s2 =kg m3 Š

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 :

˜·
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

˜

˜
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

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