Download 1 Chapter(1). Maxwell`s Equations (1.1) Introduction. The properties

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Chapter(1). Maxwell's Equations
(1.1) Introduction.
The properties of ordinary matter are a consequence of the forces
acting between charged particles. Extensive experimental
investigations have established the following properties of
electrical charges:
(1) There are two kinds of charges; these have been labled
positive charge and negative charge.
(2) Electrical charge is quantized. All particles so far observed
carry charges which are integer multiples of the charge on an
electron. In the MKS system of units, the charge on an electron
is e= -1.60x10-19 Coulombs. By definition, the electron carries
a negative charge and a proton carries a positive charge; the
charge on a proton is 1.60x10-19 Coulombs. No one knows why
charge comes in multiples of the electron charge.
(3) The quantum of positive charge and the quantum of negative
charge are equal to at least 1 part in 1020. This has been
determined from experiments designed to measure the net charge
on neutral atoms.
(4) In any closed system charge is conserved; ie. the algebraic
sum of all positive charges plus all negative charges does not
change with time. This does not mean that individual charged
particles are conserved. For example, a positron, which carries
a positive charge of 1.60x10-19 Coulombs, can interact with an
electron, which carries a negative charge of 1.60x10-19
Coulombs, in such a way that the electron and positron disappear
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and two neutral particles called photons are produced. The total
charge before and after this transformation occurs remains
exactly the same, namely zero. The individual charged particles
have disappeared but the total charge has been conserved.
(5) Charged particles set up a disturbance in space which can be
described by two vector fields; an electric field, E, and a
magnetic field, B. Since these are vector fields they are
characterized by a direction and a magnitude. Each of these
fields at any point in space can be described by its components
along three mutually perpendicular axes. For example, with
respect to a rectangular cartesian system of axes, xyz (fig.(1),
the electric field can be resolved into the three components
Ex(x,y,z,t), Ey(x,y,z,t), and Ez(x,y,z,t) where the magnitude of
2 2 2
the electric field is given by Ep=
Ex+Ey+Ez . The components

√
of these fields depend upon the orientation of the co-ordinate
system used to describe them, however the magnitude of each
field must be independent of the orientation of the co-ordinate
system.
(6) The fields E and B are real physical objects which can carry
energy, momentum, and angular momentum from one place to
another.
(7) The electromagnetic forces on a charged particle, q, can be
obtained from a knowledge of the fields E, B generated at the
position of q by all other charges. The force is given by
F= q[E + (vxB)]
(1.1)
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z
Ez
E
P
Ey
Ex
y
x
Fig.(1.1) An electric field at any point in space, P, can be
specified by its three components Ex, Ey, Ez. The magnitude
2 2 2
of the electric field is calculated from E =
Ex+Ey+Ez .

√
4
where v is the particle velocity. (Notice that B has the units of
an electric field divided by a velocity). Formula (1.1) applies
to a spinless particle. In actual fact the situation is more
complicated because most particles carry an intrinsic magnetic
moment associated with its intrinsic angular momentum (spin). In
the rest system of the particle its magnetic moment is subject
to a torque due to the presence of the field B, and to a force
due to spatial gradients of B. These magnetic forces will be
discussed later. For the present we shall discuss only spinless
charged particles, and we shall ignore the fact that real
charged particles are more complicated.
(8) Electric and magnetic fields obey the rules of superposition.
Given a system of charges which would by themselves produce the
fields E1, B1; given a second system of charges which would by
themselves produce the fields E2, B2; then together the two
systems of charges produce the total fields
E = E1 + E2, and B = B1 + B2.
(1.2)
This rule enormously simplifies the calculation of electric and
magnetic fields because it can be carried out particle by
particle and the total field obtained as the vector sum of all
the partial fields due to the individual charges.
(9) In the co-ordinate system in which a charged particle is
stationary with respect to the observer the electric and
magnetic fields which it generates are very simple, see
fig.(1.2):
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q
R
rq
P
rp
E
Origin O
R = rp-rq
E =
q R
4π ε0  R3
Volts/meter
B = 0 .
Fig.(1.2) The field due to a point charge of q Coulombs which is
stationary with respect to the observer.
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q
R
E = (
)( ) Volts/meter.
4π ε0 R3
(1.3a)
B = 0 Webers/meter2.
(1.3b)
Equation (1.3a) is called Coulomb's law. The amplitude of the
electric field decreases with distance from the charge like the
square of the distance ie. ~1/R2 where the exponent is equal to
two within 1 part in 1010. The MKS system of units has been used
to write eqns(1.3) in which the electric field is measured in
Volts/m and the charge is measured in Coulombs. A current of 1
Ampère at some point in a circuit consists of an amount of
charge equal to 1 Coulomb passing that point each second.
Distances in (1.3) are measured in meters. The factor of
proportionality is
1
= 10-7 x c2 = 8.987x109 meters/Farad,
4π ε0
(1.4)
where c= 2.9979x108 m/sec is the velocity of light in vacuum. The
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size of
is purely the consequence of the historical
4π ε0
definitions of the Volt and the Ampère. A second system of units
which is very commonly used in the current magnetism literature
is the CGS system in which distances are measured in
centimeters, mass is measured in grams, and time is measured in
seconds. In the CGS system the unit of charge- the statCoulombhas been chosen to make Coulomb's law very simple. In the CGS
system the field due to a stationary point charge is given by
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R
E = q  3 statVolts/cm,
R 
B = 0
(1.4)
Gauss.
The price that is paid for the simplicity of eqn.(1.4) is that
the conventional engineering units for the current and
potential, Ampères and Volts, cannot be used. The scaling
factors between MKS and CGS electrical units involve the
magnitude of the velocity of light, c. For example, in the CGS
system the charge on a proton is ep= 4.803x10-10 esu whereas in
the MKS system it is ep= 1.602x10-19 Coulombs. The ratio of
these two numbers is
ep
ep
esu
= 2.9979x109 .
(1.5)
MKS
(10) Consider a co-ordinate system in which a spinless charged
particle moves with respect to the observer with a constant
velocity v which is much smaller than the speed of light in
vacuum, c: ie. (v/c)<<1. The electric and magnetic fields
generated by such a slowly moving charge are given by
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E(R,t) =
B(R,t) =
q R
4π ε0  R3
1
(v x E)
c2
Volts/m
Webers/m2.
(1.6a)
(1.6b)
These expressions are correct to order (v/c)2. R is the vector
drawn from the position of the charged particle at the time of
observation to the position of the observer. Note that the moving
charge generates both an electric and a magnetic field. The above
fields can be used to calculate the force on a particle q2 located at
R: F2 (t)= q2[E(t) + (v2 (t)×B(t))] Newtons. The particle q of
eqns.(1.6) generates the fields that exert forces on the particle q2.
Equations (1.6) are simplified versions of the general
expressions for the electric and magnetic fields generated by a
spinless point charge moving in an arbritrary fashion: see " The
Feynman Lectures on Physics", VolumeII, page 21-1 (R.P.Feynman,
R.B.Leighton, and M.Sands, Addison-Wesley, Reading, Mass., 1964).
These expressions are
E =
q
4π ε0
2
 R

  3 +  R d  R3 + 12 d 2 R 
  R 
 c dt R 
c dt  R  Retarded
R
cB = R
x E .
Retarded
(1.7a)
(1.7b)
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R(tr)
Point of observation
at the time t
P
Position of the particle at
the retarded time
R
tr =  t - c


Position of the particle
at the time of observation, t
Trajectory of the
Spinless Charged
Particle
Ep(t) =
q  R
R d R
1 d2 R 
( 3) + (c)dt( 3) + 2
( )

4π ε0  R
R
c dt2 R  t =(t-R)
r
c

1  R 
Bp(t) = c   R  x Ep(t)
 

 tr
Fig.(1.3) The electric and magnetic fields generated at the
observation point, P, at the time t depend upon the
position, velocity, and acceleration of the charged

R
particle at the retarded time tr=  t - c .


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See fig.(1.3). Expression (1.7) is complicated by the fact that it's

R
terms must be evaluated at the retarded time, tr=  t - c , rather than


at the time of observation, t. This corresponds to the requirement
that changes in the motion of the particle can not be communicated to
the observer faster than is permitted by the speed of light in
vacuum. For a slowly moving particle, the first two terms of
eqn.(1.7) add together to give Coulomb's law in which the distance R
is evaluated at the time of observation rather than at the retarded
time; in other words, one can ignore time retardation if v/c is
small. The last term in eqn.(1.7) gives a field which is proportional
to the component of acceleration perpendicular to the position vector
R in the limit (v/c)<<1. This field decreases with distance like R-1
as opposed to the R-2 decrease of Coulomb's law. It is called the
radiation field and is given by the expression
Erad =
q
[a x R] x R
R
4π ε0
c2R3
tc
R
cB = R R x Erad .
tc
(1.8a)
(1.8b)
(1.2) Maxwell's Equations.
In principle, given the positions of a collection of charged
particles at each instant of time one could calculate the electric
and magnetic fields at each point in space and at each time from
eqns.(1.7). For ordinary matter this is clearly an impossible task.
Even a small volume of a solid or a liquid contains enormous numbers
of atoms. A cube one micron on a side (10-6m x10-6m x10-6m) contains
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~1011 atoms, for example. Each atom consists of a positively charged
nucleus surrounded by many negatively charged electrons, all of which
are in motion and which will, therefore, generate electric and
magnetic fields which fluctuate rapidly both in space and in time.
For most purposes one does not wish to know in great detail the space
and time variation of the fields. One usually wishes to know about
the average electric and magnetic fields. For example, the average
magnitude and direction of E over a time interval which is determined
by the instrument used to measure the field. Typically this might be
of order 10-6 seconds or more; a time which is very long compared
with the time for an electron to complete an orbit around the atomic
nucleus in an atom (10-16-10-21 seconds). Moreover, one is usually
interested in the value of the fields averaged over a volume which is
small compared with ~1 micron but large compared with atomic
dimensions, ~10-10 meters. In 1864 J.C.Maxwell proposed a system of
differential equations which can be used for calculating electric and
magnetic field distributions, and which automatically provide the
space and time averaged fields which are of practical interest. These
Maxwell's Equations for a macroscopic medium are as follows:
∂B
curlE = - ∂t
(1.9a)
divB = 0
(1.9b)

∂P 
∂E
curlB = µ0 Jf + curlM + ∂t  + ε0µ0∂t


(1.9c)
divE =
1
ε0
(ρ f
- divP)
(1.9d)
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where ε0µ0= 1/c2 and c is the velocity of light in vacuum. These
equations underlie all of electrical engineering and much of physics
and chemistry. They should be committed to memory.
In large part, this book is devoted to working out the
consequences of Maxwell's equations for special cases which provide
the required background and guidance for solving practical problems
in electricity and magnetism. In eqns.(1.9) ε0 is the permativity of
free space; it has already been introduced in connection with
Coulomb's law, eqn.(1.3). The constant µ0 is called the permeability
of free space. It has the defined value
µ0 = 4π x 10-7 Henries/meter.
(1.10)
Maxwell's equations as written above contain four new quantities
which must be defined: they are (1) Jf, the current density due to
the charges which are free to move in space, in units of Ampères/m2;
(2) ρf, the net density of charges in the material, in units of
Coulombs/m3; (3) M, the density of magnetic dipoles per unit volume
in units of Amps/m; (4) P, the density of electric dipoles per unit
volume in units of Coulombs/m2. In Maxwell's scheme these four
quantities become the sources which generate the electric and
magnetic fields. They are related to the space and time averages of
the position and velocities of the microscopic charges which make up
matter.
(a) The Definition of the Free Charge Density,
f.
Construct a small volume element, ∆V, around the particular point
in space specified by the position vector r. Add up all the
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charges contained in ∆V at a particular instant; let this amount
of charge be ∆Q(t). Average ∆Q(t) over a time interval which is
short compared with the measuring time of interest, but which is
long compared with times characteristic of the motion of
electrons around the atomic nuclei; let the resulting time
averaged charge be <∆Q(t)>. Then the free charge density is
defined to be
ρf(r,t) = <∆Q(t)>
Coulombs/m3.
∆V
(1.11)
The dimensions of the volume element ∆V is rather vague; it will
depend upon the scale of the spatial variation which is of
interest for a particular problem. It should be large compared
with atomic dimensions but small compared with the distance over
which ρf changes appreciably.
(b) Definition of the Free Current Density, Jf.
The free charge density, ρf(r,t), will in general change with
time as charge flows from one place to the other; one need only
think of charge flowing along a wire. The rate at which charge
flows across an element of area is described by the current
density, Jf(r,t). It is a vector because the charge flow is
associated with a direction. The components of the current
density vector can be measured by counting the rate of charge
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z
Area ∆Ax oriented with
its normal along x and
centered at r.
qi
y
x
Let <∆Q> be the amount of charge which passes through the small area
∆Ax in a small time interval ∆t. Then
 <∆Q> 
Jf)x =  ∆t ∆A  Ampères/meter2.

x
It follows from this that one can also write Jf)x= ρfvx, where vx is
the x-component of the velocity associated with the motion of the
charge density distribution.
Fig.(1.4) The current density due to a flowing charge distribution.
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flow across a small area located at the position specified by r, and
whose normal is oriented parallel with one of the co-ordinate axes;
parallel with the x-axis, for example, ∆Ax, (see fig.(1.4).Now at
time t measure the net amount of charge, <∆Q>, which has passed
through ∆Ax in a small time interval, ∆t: positive charge which flows
in the direction from +x to -x is counted as a negative contribution;
negative charge wich flows from -x to +x also makes a negative
contribution. The x-component of the current density is given by
<∆Q>
Jf(r,t)x = ∆t∆A
x
Amps/m2.
(1.12)
The other two components of Jf are defined in a similar manner.
The time interval ∆t, and the dimensions of the elements of area, ∆A,
are supposed to be chosen so that they are large compared with atomic
times and atomic dimensions, but small compared with the time and
length scales appropriate for a particular problem.
Free charge density can be visualized as a kind of fluid flowing
from place to place with a certain velocity. In terms of this
velocity the free current density is given by
Jf(r,t) = ρf(r,t) v(r,t).
(1.13)
In the process of charge flow electrical charge can neither be
created nor destroyed. Because charge is conserved, it follows that
the rate at which charge is carried into a volume must be related to
the rate at which the net charge in the volume increases with time.
The mathematical expression of this charge conservation law is
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∂ρf(r,t)
= - divJf(r,t).
∂t
(1.14)
(1.3) Point Dipoles.
In order to discuss the definitions of the two vector functions
P(r,t) and M(r,t) it is first necessary to discuss the concepts of a
point electric dipole and a point magnetic dipole.
The Point Electric Dipole.
Most atoms in a substance are electrically neutral, ie. the charge on
the nucleus is compensated by the electrons moving around that
nucleus. When examined from a distance which is long compared with
atomic dimensions (~10-10m) the neutral atom produces no substantial
electric or magnetic field. However, if, on average, the centroid of
the negative charge distribution is displaced from the position of
the nucleus the Coulomb field of the nucleus will no longer cancel
the Coulomb field from the electrons. To fix ideas, think of a
stationary hydrogen atom consisting of a proton and an electron. The
electron moves so fast that on a human time scale its charge appears
to be located in a spherical cloud which is tightly distributed
around the nucleus (see fig.(1.5)). In the absence of an external
electric field the centroid of the electronic charge distribution
will coincide with the position of the nucleus. Under these
circumstances the time-averaged Coulomb fields of the nucleus and the
electron cancel each other when observed from distances that
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Nuclear Charge
q = + 1.6x10-19 Coulombs
Time-averaged electron.
A sperically symmetric ball
of negative charge
e = - 1.6x10-19 Coulombs.
Radius~ 5x10-11 meters.
Fig.(1.5a) A hydrogen atom in zero external electric field.
Center of the
negative charge
distribution.
Nuclear
Charge
E0
External Electric Field
Fig.(1.5b) A hydrogen atom in a non-zero external electric field. The
displacement of the electron cloud has been exagerated for
purposes of illustration.
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are large compared with 10-10m. If the atom is subjected to an
external electric field the nucleus is pulled one way and the
centroid of the electron cloud is pulled the other way (eqn.(1.1)):
there is an effective charge separation (see fig.(1.5(b))). The
Coulomb fields due to the nucleus and the electron no longer exactly
cancel.
Let us use the law of superposition to calculate the field which
arises when two point charges no longer coincide; refer to fig.(1.6).
The electric field at the point of observation, P, due to the
positive charge is given by
E+ =
q r
.
4π ε0  r3
The electric field at P due to the negative charge is given by
E- = -
q  r+d 
.
4π ε0  (|r+d|3)
Referring to fig.(1.6) one has
r = xux + yuy + zuz ,
and
r =
x2 + y2 + z2
√
.
For d oriented along the z-axis as shown in fig.(1.6),
(r + d) = xux + yuy + (z+d)uz,
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z
P(x,y,z)
r
(r + d)
+q
y
d
-q
x
r = xû x + yû y + zû z
d = dû z
ûx,ûy,ûz are unit vectors along the co-ordinate axes.
Fig.(1.6) Two charges equal in magnitude but opposite in sign are
separated by the distance d. By definition the dipole
moment of this pair of charges is p= qd where d is the
vector distance from the negative to the positive charge.
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so that
or
|r + d| = [x2 + y2 + (z+d)2]1/2
|r + d| = [x2 + y2 + z2 + 2zd + d2]1/2.
Upon dividing out r2 this gives
2zd+d2 1/2
|r + d| = r [1 + (
)] .
r2
From this expression one has
1
1
(2zd+d2) -3/2
=
[1
+
]
.
|r+d|3
r3
r2
This is so far exact. Now make use of the fact that (d/r) is very
small and use the binomial theorem to expand the radical; it is
sufficient to keep only terms linear in (d/r). The result is
1
1
3zd
~
= 3 - 5 .
3
|r+d|
r
r
(1.15)
Use eqn.(1.15) to calculate the total electric field at the point of
observation, P, correct to terms of order (d/r). The terms
proportional to (1/r2) cancel leaving the field
Ed = E+ + E- =
1  (3zqd)r
qd
- 3  .

5
4π ε0 
r
r 
(1.16)
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By definition the dipole moment of the pair of charges is given
by p=qd. Moreover, zqd= r·p, ie. the scalar product of the dipole
moment and the position vector r. Finally, the expression for the
electric field generated by a stationary point dipole can be written
Ed =
1  3(p·r)r
p
- 3  .

5
4π ε0 
r
r 
(1.17)
Although this expression has been obtained for the particular case in
which p is oriented along z, the result eqn.(1.17) is perfectly
general and is valid for any orientation of p.
Expression (1.17) is so fundamental that, like Coulomb's law, it
should be committed to memory. The field distribution around a point
dipole is shown in fig.(1.7). The magnetic field generated by a
stationary point dipole is, of course, zero.
It is usefull to write the dipole field in terms of its
components in a spherical polar co-ordinate system in which the
dipole is oriented along the z-axis, see fig.(1.8):
Er =
1 2pCosθ
,
4π ε0
r3
(1.18a)
Eθ =
1 pSinθ
.
4π ε0 r3
(1.18b)
22
2E0
R
P
E0
E0
2E
2E00
E0 =
1 |P|
4π ε0 R3
Fig.(1.7) The electric field distribution around a point dipole.
The electric field distribution has rotational symmetry
around the dipole axis
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z
Er
r
Eθ
θ
Dipole P
Er =
1 2pCosθ
4π ε0
r3
Eθ =
1 pSinθ
.
4π ε0 r3
xy Plane
These components are independent of the angle φ.
Fig.(1.8) The electric field of a dipole oriented along the z-axis,
written in spherical polar co-ordinates.
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The Point Magnetic Dipole.
Consider a spinless charge q revolving in a circular orbit of
radius a with a speed v such that (v/c)<<1. One can use eqn.(1.6) to
calculate the electric and magnetic fields at a distance from the
center of the current loop which is very large compared with the
orbit radius a, see fig.(1.9). Using the binomial expansion and
keeping only
the lowest order terms, it can be shown that the time
averaged electric field is given by Coulomb's law:
E =
q r
.
4π ε0  r3
The lowest order correction term upon taking the time average is
proportional to (a/r)2, see problem (1.8). The magnetic field can be
calculated using eqn.(1.6b). The velocity of the particle is
proportional to the orbit radius, and therefore when the time
averages are worked out the lowest order non-vanishing term is
proportional to (a/r)2; see problem (1.8). The time-averaged magnetic
field turns out to be given by
µ
3(m·r)r
m
Bd = 4π0 
- 3  ,
5
r
r 

(1.19)
Notice that this expression has exactly the same form as eqn.(1.17)
for the electric field distribution around an electric dipole moment
p. Here the vector m is called the orbital magnetic dipole moment
associated with the current loop, and
25
z
P
r
y
q
x
a
a = aCosφ ûx + aSinφ ûy
 dφ 
v = a  dt
 
(-Sinφ
Therefore,
and m =
ûx + Cosφ ûy)
 dφ 
axv = a2  dt ûz
 
q(axv)
qa2 dφ 
=
2
2  dt ûz
Fig.(1.9) A charged particle following a circular orbit of radius a,
and travelling with the speed v, generates a magnetic
qav
dipole moment |m|= 2 Ampère-meters2.
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m =
qa2 dφ
2
2 (dt) uz Coulomb-m /sec.
(1.20)
Note that |m|= IA where I= qv/2πa is the current in the loop, and
A= πa2 is the area of the loop. Since the speed of the particle is
given by v=a(dφ/dt), the magnitude of the magnetic moment can also be
written in terms of the angular momentum of the circulating charge:
|m| =
qav
q
2 = 2mp mpav ,
where the mass of the charged particle is mp and its angular momentum
is L= mpav. Thus the magnetic moment m is related to the particle
angular momentum L by the relation
q
m = (2m ) L .
p
(1.21)
For an electron q= -1.60x10-19 C= -|e| so that the magnetic moment
and the angular momentum are oppositely directed. The angular
momentum is quantized in units of /h, therefore the magnetic moment of
a particle is also quantized. The quantum of magnetization for an
orbiting electron is called the Bohr magneton, µB. It has the value
e/h
µB = 2m
= 9.27x10-24 Coulomb-m2/sec.
e
(The units of µB can also be expressed as Ampères-m2 or as
Joules/Tesla).
(1.21)
27
In addition to their orbital angular momentum, charged particles
possess intrinsic or spin angular momentum, S. There is also a
magnetic moment associated with the spin. The magnetic moment due to
spin is usually written
q
ms = (g 2m ) S.
p
(1.22)
For an electron q=-|e|, and g=2.00. The spin of an electron has the
magnitude |S|= /h/2; consequently, the intrinsic magnetic moment
carried by an electron due to its spin is just 1 Bohr magneton, µB.
The total magnetic moment generated by an orbiting particle which
carries a spin moment is given by the vector sum of its orbital and
spin magnetic moments. The total magnetic moment associated with an
atom is the vector sum of the orbital and spin moments carried by all
of its constituent particles, including the nucleus. The magnetic
field generated by a stationary atom at distances large compared with
the atomic radius is given by eqn.(1.19) with m equal to the total
atomic magnetic dipole moment.
(1.4) The Electric and Magnetic Dipole Densities.
Let us now turn to the definitions of the electric dipole moment
density, P, and the magnetic dipole density, M, which occur in
Maxwell's equations (1.9).
The Definition of the Electric Dipole Density, P.
Think of an idealized model of matter in which all of the atoms
are fixed in position. In the presence of an electric field each
28
atom will develop an electric dipole moment; the dipole moment
induced on each atom will depend upon the atomic species. Some
atomic configurations also carry a permanent electric dipole
moment by virtue of their geometric arrangement: the water
molecule, for example carries a permanent dipole moment of
6.17x10-30 Coulomb-meters (see problem (1.12). Let the dipole
moment on atom i be pi Coulomb-meters. Select a volume element ∆V
located at some position r in the matter. At some instant of
time, t, measure the dipole moment on each atom contained in ∆V
and calculate their vector sum, Σpi. This moment will fluctuate
with time, so it is necessary to perform a time average over an
interval which is long compared with atomic fluctuations but
short compared with times of experimental interest; let this time
average be <Σpi>. Then the electric dipole density is given by
P(r,t) =
<Σpi>
∆V
Coulombs/m2 .
(1.23)
The shape and size of ∆V are unimportant: the volume of ∆V should
be large compared with an atom, but small compared with the
distance over which P varies in space.
In a real material the atoms are not generally fixed in
position. In a solid they jiggle about more or less fixed sites.
In liquids and gasses they may, in addition, take part in mass
flow as matter flows from one place to another. This atomic
motion considerably complicates the calculation of the electric
dipole density because the effective electric dipole moment on an
atom moving with respect to the observer includes a small
29
contribution from its magnetic dipole moment. The correction
terms are of order (v/c) times the magnetic dipole moment and may
be ignored for our purposes.
The Definition of The Magnetic Dipole Density, M.
This vector quantity is defined by
M(r,t) =
<Σmi>
∆V
Amps/m,
(1.24)
where <Σmi> is a suitable time average over the atomic magnetic
moments contained in a small volume element, ∆V, at time t and
centered at the position specified by r. It is assumed that the
atoms are stationary. If they are not, the magnetization density
contains contributions which are proportional to (vi/c) times the
various atomic electric dipole moments; the velocity vi is the
velocity of atom species i with respect to the observer. We shall
not be concerned with this correction which is, in most cases,
very small. As above, the volume element ∆V is supposed to be
large compared with an atomic dimension but small compared with
the length scale over which M varies in space.
30
(1.5) Return to Maxwell's Equations.
Now let us return to Maxwell's equations (1.9). A vector field is
completely specified, apart from a constant field, by its curl and
its divergence. Therefore, the two vector fields E and B are
specified (ie. can be calculated) given the four source terms
ρf(r,t), Jf(r,t), P(r,t), and M(r,t). Maxwell's equations should be
committed to memory, along with the prescriptions necessary for
calculating the divergence and curl of a vector field in the three
major systems of co-ordinates: (1) cartesian co-ordinates (x,y,z);
(2) cylindrical polar co-ordinates (r,θ,z); and (3) spherical polar
co-ordinates, (r,θ,φ); see fig.(1.10). Vector derivatives are
reviewed by M.R.Spiegel, Mathematical Handbook of Formulas and
Tables, Schaum's Outline Series, McGraw-Hill, N.Y., 1968: Chapter 22.
It is also worth reading the discussion contained in The Feynman
Lectures on Physics, by R.P.Feynman, R.B.Leighton, and M.Sands,
Addison-Wesley, Reading, Mass.,1964; Volume II, Chapters 2 and 3.
Four very important vector theorems are listed in Appendix(1A).
These theorems should be memorized because they will be used time
after time.
(1.6) Force Density in Matter.
Having calculated the electric and magnetic field distributions
in a piece of material one would like to use them to obtain the force
density acting at each point. The force density is given by the sum
of three terms.
31
(a) Cartesian System.
z
r = xû x +yû y +zû z
P(x,y,z)
y
x
(b) Cylindrical Polar System.
z
r = rû r + zû z
P(r,θ,z)
x = rCosθ
r
y = rSinθ
y
θ
û θ
r
û r
x
(c) Spherical Polar System.
z
û r
r = rû r
P
P(r,θ,φ)
x = rSinθCosφ
θ
y = rSinθSinφ
û φ
r
û θ
y
z = rCosθ
φ
x
Fig.(1.10) The three major co-ordinate systems in common use.
32
(1) A term which is the direct analogue of the force exerted on a
single charged particle moving with a velocity v, ie.
f =q[E + (vxB)]. If this force is added together for all of the
charges contained in a small volume ∆V, averaged over time, and
divided by the volume element ∆V in order to obtain a force density,
the result is
Fq = ρfE + (Jf x B) Newtons/m3.
(1.25)
(2) If the electric field varies from place to place in space
there is a contribution to the force density which is proportional to
the electric dipole density. The origin of this force can be
understood by examining the forces exerted on a model point dipole by
an electric field, fig.(1.11). In fig.(1.11) the calculation has been
explicitly written out for a dipole oriented along the x-axis. If a
similar calculation is carried through for the dipole components
oriented along y and z, and the resulting force components are added
to obtain the total force on a dipole in an inhomogeneous electric
field, the result is
∂E
∂E
∂E
Fx = px( ∂xx) + py( ∂yx) + pz( ∂zx) = p· Ex
(1.26a)
∂E
∂E
∂E
Fy = px( ∂xy) + py( ∂yy) + pz( ∂zy) = p· Ey
(1.26b)
∂E
∂E
∂E
Fz = px( ∂xz) + py( ∂yz) + pz( ∂zz) = p· Ez .
(1.26c)
33
x0
x0+d
x
-q
+q
At x0: the electric field components are Ex,Ey,Ez
∂E
At (x0+d): the electric field components are Ex + ( ∂xx)d
∂E
Ey + ( ∂xy)d
∂E
Ez + ( ∂xz)d
The force components on the negative charge are Fx = -q Ex
Fy
= -q Ey
Fz
= -q Ez
The force components on the positive charge are
∂E
Fx = qEx + qd( ∂xx)
∂E
Fy = qEy + qd( ∂xy)
∂E
Fz = qEz + qd( ∂xz)
The net force components are:
∂E
∂E
Fx = qd ( ∂xx) = px ( ∂xx)
∂E
∂E
Fy = qd ( ∂xy) = px ( ∂xy)
∂E
∂E
Fz = qd ( ∂xz) = px ( ∂xz)
Fig.(1.11) The electric forces acting on a dipole oriented along the
x-axis.
34
These force components can be summed over a small volume element
and time-averaged in the usual way. When divided by the volume of the
element ∆V to obtain force densities, the result is
Fp = (P· Ex)ux + (P· Ey)uy + (P· Ez)uz N/m3.
(1.27)
Similar arguements can be used to show that the electric field
exerts a torque density, Tp, on polarized material, where
Tp = P x E Newtons/m2.
(1.28)
(3) It is not obvious at this stage of developement but magnetic
forces are exerted on the magnetization density by a magnetic field
gradient. The magnetic force density is given by an expression which
is the exact analogue of that for the electric force density,
eqn.(1.27):
FM = (M· Bx)ux + (M· By)uy + (M· Bz)uz Newtons/m3.
(1.29)
The magnetic field also exerts a torque density on the magnetized
matter, and this is given by the expression (analagous with
eqn.(1.28))
TM = M x B
Newtons/m2.
(1.30)
35
(1.7) The Auxillary Fields D,H.
It is sometimes useful to rewrite Maxwell's equations in terms of
E, B, and two new vector fields D, and H. These new fields are
defined by
and
D = ε0E + P
(1.31)
B = µ0(H + M).
(1.32)
Maxwell's equations when written using the derived field
distributions become
∂B
curlE = - ∂t
(1.33a)
divB = 0
(1.33b)
∂D
curlH = Jf + ∂t
(1.33c)
divD = ρf
(1.33d)
Maxwell's equations have a simpler form when written this way, and
may in consequence be easier to remember. Their physical content is,
of course, unaltered by the introduction of the two auxillary fields.
Maxwell's equations have been written in CGS units in
Appendix(1B); this system of units is still used in much of the
current literature on magnetism. The main virtue of the CGS system is
that the fields E, D, B, and H all have the same units; this is
36
particularly usefull for the discussion of free space radiation
problems. In the MKS system these four vectors all have different
units. The main difficulty with the CGS system is that potentials and
currents are not measured in practical units.
37
Appendix (1A)
Four vector theorems which will be extensively used throughout
this book are the following:
(1) The curl of any gradient is zero.
(2) The divergence of any curl is zero.
(3) Gauss' Theorem;
n̂
Volume V
bounded by a
A
cosed surface S
Element of
area dS
Consider a volume V bounded by a closed surface S. An element
of area on the surface S can be specified by the vector
dS= n̂dS where dS is the magnitude of the element of area and
n̂ is a unit vector directed along the outward normal. Let A
be any vector field. Then
∫S(A·n̂)dS
=
∫VdivA
where dτ is an element of volume.
dτ
38
(4) Stokes' Theorem:
n̂
Closed curve C
A
Element of
dL
area dS
Consider any surface S which spans the closed curve C.
Calculate the line integral of a vector A around the curve C;
let the element of length along C be specified by dL. Then
∫A·dL =
C
∫ScurlA·n̂
dS
where the direction of the unit vector n̂ is related to the
direction of traversal of curve C by the right hand rule.
39
Appendix (1B)
In CGS units Maxwell's equations become
1 ∂B
curlE = - c ∂t
(B1)
divB = 0
(B2)
4π 
∂P 
1 ∂E
curlB = c  Jf + c curlM + ∂t  + c ∂t


(B3)
(ρ f
(B4)
divE = 4π
- divP).
In this system of units c= 2.998x1010 cm/sec. and E, and B have the
same units (stat-Volts/cm). However, for historical reasons, the
units of B are known as Gauss. 104 Gauss are equal to 1 Weber/m2: the
unit 1 Weber/m2 is also called a Tesla. The electric field is
measured in stat-Volts/cm where 1 stat-Volt is equal to 299.8 Volts;
(yes, these are the same significant figures as occur in the speed of
light!). An electric field of 1 stat-Volt/cm (sometimes stated as 1
esu/cm) is approximately equal to 30,000 Volts/m.
If auxillary fields D, and H are introduced through the relations
D = E + 4πP
(B5)
B = H + 4πM
(B6)
then in CGS units Maxwell's equations become
40
4π
1 ∂D
curlH = c Jf + c ∂t
(B7)
divD = 4π ρf
(B8)
where the first two equations ( (B1) and (B2) ) remain the same. The
vectors D, and H have the same units as E, and B although for
historical reasons the units of H are Oersteds.
The relation between charge density and current density in the
MKS and the CGS systems can be deduced from the ratio of the proton
charge as measured in both sets of units. This ratio is
ep
ep
esu
= 2.9979x109 .
(1.5)
MKS
It follows from this ratio that 2998 esu/cm3 is equal to 1
Coulomb/m3. Similarly, a current density of 1 Ampère/m2 is equal to
2.998x105 esu/cm2.
The conversion from MKS to CGS magnetic units is easy to remember
since the earth's magnetic field is approximately 1 Oersted which is
equal to 10-4 Tesla (Webers/m2).