18The Radiation of an AcceleratingCharge in the Electromagnetic Field

18
The Radiation of an Accelerating
Charge in the Electromagnetic Field

18.1

MOTION OF A CHARGE IN AN ELECTROMAGNETIC FIELD

In previous chapters the Stokes vectors were determined for charges moving in a
linear, circular, or elliptical path. At first sight the examples chosen appear to have
been made on the basis of simplicity. However, the examples were chosen because
charged particles actually move in these paths in an electromagnetic field; that is, the
examples are based on physical reality. In this section we show from Lorentz’s force
equation that in an electromagnetic field charged particles follow linear and circular
paths. In the following section we determine the Stokes vectors corresponding to
these physical configurations.
The reason for treating the motion of a charge in this chapter as well as in the
previous chapter is that the material is necessary to understand and describe the
Lorentz–Zeeman effect. Another reason for discussing the motion of charged particles in the electromagnetic field is that it has many important applications. Many
physical devices of importance to science, technology, and medicine are based on our
understanding of the fundamental motion of charged particles. In particle physics
these include the cyclotron, betatron, and synchrotron, and in microwave physics the
magnetron and traveling-wave tubes. While these devices, per se, will not be discussed here, the mathematical analysis presented is the basis for describing all of
them. Our primary interest is to describe the motion of charges as they apply to
atomic and molecular systems and to determine the intensity and polarization of the
emitted radiation.
In this chapter we treat the motion of a charged particle in three specific
configurations of the electromagnetic field: (1) the acceleration of a charge in an
electric field, (2) the acceleration of a charge in a magnetic field, and (3) the acceleration of a charge in perpendicular electric and magnetic fields. In particular, the
motion of a charged particle in perpendicular electric and magnetic fields is extremely interesting not only from the standpoint of its practical importance but because
the paths taken by the charged particle are quite beautiful and remarkable.

Copyright © 2003 by Marcel Dekker, Inc. All Rights Reserved.


In an electromagnetic field the motion of a charged particle is governed by the
Lorentz force equation:
F ¼ q½E þ ðv  Bފ

ð18-1Þ

where q is the magnitude of the charge, E is the applied electric field, B is the applied
magnetic field, and v is the velocity of the charge. The background to the Lorentz
force equation can be found in the texts given in the references. The text by G. P.
Harnwell on electricity and magnetism is especially clear and illuminating. Quite
understandably, because of the importance of the phenomenon of the radiation of
accelerating charges in the design and fabrication of instruments and devices, many
articles and textbooks are devoted to the subject. Several are listed in the references.
18.1.1

Motion of an Electron in a Constant Electric Field

The first and simplest example of the motion of an electron in an electromagnetic
field is for a charge moving in a constant electric field. The field is directed along
the z axis and is of strength E0. The vector representation for the general electric
field E is
E ¼ Ex ux þ Ey uy þ Ez uz

ð18-2Þ

Since the electric field is directed only in the z direction, Ex ¼ Ey ¼ 0, so

E ¼ Ez uz ¼ E0 uz

ð18-3Þ

For simplicity the motion of the electron is restricted to the xz plane and is initially
moving with a velocity v0 at an angle  from the z axis. This is shown in Fig. 18-1.
Because there is no magnetic field, the Lorentz force equation (18-1) reduces to
m€r ¼ ÀeE

Figure 18-1

ð18-4Þ

Motion of an electron in the xz plane in a constant electric field directed along

the z axis.

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where m is the mass of the electron. In component form (18-4) is
mx€ ¼ 0

ð18-5aÞ

my€ ¼ 0

ð18-5bÞ

mz€ ¼ ÀeEz ¼ ÀeE0

ð18-5cÞ

At the initial time t ¼ 0 the electron is assumed to be at the origin of the coordinate
system, so
xð0Þ ¼ zð0Þ ¼ 0

ð18-6Þ

Similarly, the velocity at the initial time is assumed to be
x_ ð0Þ ¼ vx ¼ v0 sin 

ð18-7aÞ

z_ð0Þ ¼ vz ¼ v0 cos 

ð18-7bÞ

There is no force in the y direction, so (18-5b) can be ignored. We integrate (18-5a)
and (18-5c) and find
x_ ðtÞ ¼ C1
z_ðtÞ ¼ À

ÀeE0 t
þ C2
m

ð18-8aÞ
ð18-8bÞ

where C1 and C2 are constants of integration. From the initial conditions, C1 and C2
are easily found, and the specific solution of (18-8) is
x_ ðtÞ ¼ v0 sin 
z_ðtÞ ¼

ÀeE0 t
þ v0 cos 
m

ð18-9aÞ
ð18-9bÞ

Integrating (18-9) once more yields
xðtÞ ¼ v0 t sin 
zðtÞ ¼

ÀeE0 t2
þ v0 t cos 
2m

ð18-10aÞ

ð18-10bÞ

where the constants of integration are found from (18-6) to be zero. We can eliminate t between (18-10a) and (18-10b) to obtain
!
eE0
zðtÞ ¼ À
ð18-11Þ
x2 þ ðcot Þx

2mv20 sin2 
which is the equation of a parabola. The path is shown in Fig. 18-2.
Inspecting (18-11) we see that if  ¼ 0 then zðtÞ ¼ 1. That is, the electron
moves in a straight line starting from the origin 0 along the z axis and ‘‘intercepts’’
the z axis at infinity (1). However, if  is not zero, then we can determine the

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Figure 18-2

Parabolic path of an electron in a constant electric field.

positions x(t) where the electron intercepts the z axis by setting z(t) ¼ 0 in (18-11).
On doing this the intercepts are found to occur at
xðtÞ ¼ 0

xðtÞ ¼

mv20
sin 2
eE0

ð18-12aÞ

ð18-12bÞ

The first value corresponds to our initial condition x(0) ¼ z(0) ¼ 0. Equation
(18-12b) shows that the maximum value of x is attained by setting  ¼ 45 , so
xmax ¼

mv20
eE0

ð18-13Þ

This result is not at all surprising, since (18-11) is identical in form to the equation
for describing a projectile moving in a constant gravitational field. Finally, the
maximum value of z is found from (18-11) to be
!
1 mv20
zðtÞ ¼
sin 2
2 eE

ð18-14aÞ

or
1
zmax ¼ xmax
2
where we have used (18-12b).

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ð18-14bÞ


18.1.2 Motion of a Charged Particle in a Constant Magnetic Field
We now consider the motion of an electron moving in a constant magnetic field. The

coordinate configuration is shown in Fig. 18-3. In the figure B is the magnetic field
directed in the positive z direction. The Lorentz force equation (18-1) then reduces
to, where the charge on an electron is q ¼ Àe,
F ¼ Àeðv  BÞ

ð18-15Þ

Equation (18-15) can be expressed as a differential equation:
m€r ¼ Àeðv  BÞ

ð18-16Þ

where m and r¨ are the mass and acceleration vector of the charged particle, respectively. In component form (18-16) is
mx€ ¼ Àeðv  BÞx

ð18-17aÞ

my€ ¼ Àeðv  BÞy

ð18-17bÞ

mz€ ¼ Àeðv  BÞz

ð18-17cÞ

where the subscript on (v  B) refers to the appropriate component to be taken. The
vector product v  B can be expressed as a determinant


 ux uy uz 





y_
z_ 
ð18-18Þ
v  B ¼  x_


 Bx By Bz 
where ux, uy, and uz are the unit vectors pointing in the positive x, y, and z directions,
respectively and the velocities have been expressed as x_ , y_ , and z_. The constant
magnetic field is directed only along z, so Bz ¼ B and Bx ¼ By ¼ 0. Then, (18-18)
and (18-17) reduce to
mx€ ¼ Àeðy_ BÞ

ð18-19aÞ

my€ ¼ ÀeðÀx_ BÞ

ð18-19bÞ

mz€ ¼ 0

ð18-19cÞ

Figure 18-3 Motion of an electron in a constant magnetic field.

Copyright © 2003 by Marcel Dekker, Inc. All Rights Reserved.

Equation (18-19c) is of no interest because the motion along z is not influenced by
the magnetic field. The equations of motion are then
x€ ¼

ÀeB
y_
m

ð18-20aÞ

eB
x_
ð18-20bÞ
m
Equation (18-20a) and (18-20b) can be written as a single equation by introducing
the complex variable (t):
y€ ¼

ðtÞ ¼ xðtÞ þ iyðtÞ

ð18-21Þ

Differentiating (18-21) with respect to time, we have
_ ¼ x_ þ iy_

ð18-22aÞ

€ ¼ x€ þ iy€

ð18-22bÞ

Multiplying (18-20b) by i and adding this result to (18-20a) and using (18-22a)
leads to
ieB
_ ¼ 0
ð18-23Þ
m
The solution of (18-23) is readily found by assuming a solution of the form:
€ À

ðtÞ ¼ e!t

ð18-24Þ

Substituting (18-24) into (18-23) we find that
!ð! À i!c Þ ¼ 0

ð18-25Þ

where !c ¼ eB/m is the frequency of rotation, known as the cyclotron frequency.
Equation (18-25) is called the auxiliary or characteristic equation of (18-23),
and from (18-25) the roots are ! ¼ 0, i!c. The general solution of (18-23) can be
written immediately as
ðtÞ ¼ c1 þ c2 ei!c t

ð18-26Þ

where c1 and c2 are constants of integration. To provide a specific solution for

(18-23), we assume that, initially, the charge is at the origin and moving along the
x axis with a velocity v0. Thus, we have
xð0Þ ¼ 0
x_ ð0Þ ¼ v0

yð0Þ ¼ 0
y_ ð0Þ ¼ 0

ð18-27aÞ
ð18-27bÞ

which can be expressed in terms of (18-21) and (18-22a) as
ð0Þ ¼ xð0Þ þ iyð0Þ ¼ 0

ð18-28aÞ

_ð0Þ ¼ x_ ð0Þ þ iy_ ð0Þ ¼ v0

ð18-28bÞ

This leads immediately to
c1 ¼ Àc2
c2 ¼

iv0
!c

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ð18-29aÞ

ð18-29bÞ


so the specific solution of (18-26) is
ðtÞ ¼ À

iv0
ð1 À ei!c t Þ
!c

ð18-30Þ

Taking the real and imaginary part of (18-30) then yields
v0
sin !c t
!c
v
yðtÞ ¼ À 0 ð1 À cos !c tÞ
!c
xðtÞ ¼

ð18-31aÞ
ð18-31bÞ

or
v0
sin !c t
!c
v
v

y þ 0 ¼ 0 cos !c t
!c !c
xðtÞ ¼

Squaring and adding (18-32a) and (18-32b) give

  2
v 2
v
x2 þ y þ 0 ¼ 0
!c
!c

ð18-32aÞ
ð18-32bÞ

ð18-33Þ

which is an equation of a circle with radius v0/!c and center at x ¼ 0 and y ¼ Àv0/!c.
Equations (18-32) and (18-33) show that in a constant magnetic field a charged
particle does indeed move in a circle. Also, (18-32) describes a charged particle
moving in a clockwise direction as viewed along the positive axis toward the
origin. Equation (18-33) is of great historical and scientific interest, because it is
the basis of one of the first methods and instruments used to measure the ratio
e/m, namely, the mass spectrometer. To see how this measurement is made, we
note that since the electron moves in a circle, (18-33) can be solved for the condition
where it crosses the y axis, which is x ¼ 0. We see from (18-33) that this occurs at
y¼0
y¼À

ð18-34aÞ
2v0
!c

ð18-34bÞ

We note that (18-34b) is twice the radius  ( ¼ v0/!c). This is to be expected because
the charged particle moves in a circle. Since !c ¼ eB/m, we can solve (18-34b) for e/m
to find that
 
e
2v0
¼À
ð18-35Þ
m
By
The initial velocity 0 is known from equating the kinetic energy of the electron
with the voltage applied to the charged particle as it enters the chamber of the mass
spectrometer. The magnitude of y where the charged particle is intercepted (x ¼ 0) is
measured. Finally, the strength of the magnetic field B is measured with a magnetic
flux meter. Consequently, all the quantities on the right side of (18-35) are known, so
the ratio e/m can then be found. The value of this ratio found in this manner agrees
with those of other methods.

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18.1.3

Motion of an Electron in a Crossed Electric and

Magnetic Field

The final configuration of interest is to determine the path of an electron which moves
in a constant magnetic field directed along the z axis and in a constant electric field
directed along the y axis, a so-called crossed, or perpendicular, electric and magnetic
field. This configuration is shown in Fig. 18-4.
For this case Lorentz’s force equation (18-1) reduces to
mx€ ¼ Àeðy_ BÞ

ð18-36aÞ

my€ ¼ ÀeE þ eðx_ BÞ

ð18-36bÞ

mz€ ¼ 0

ð18-36cÞ

From (18-21) and (18-22), (18-36) can be written as a single equation:
€ À i!c _ ¼ À

ieE
m

ð18-37Þ

where !c ¼ eB/m. Equation (18-37) is easily solved by noting that if we multiply by
eÀi!c t then (18-37) can be rewritten as



d Ài!c t
ÀieE Ài!c t
_
ðe
Þ ¼
ð18-38Þ
e
dt
m
Straightforward integration of (18-38) yields


 
eE
ic1 i!c t
¼
tÀ
e þ c2
m!c
!c

ð18-39Þ

where c1 and c2 are constants of integration. We choose the initial conditions to be
xð0Þ ¼ 0

yð0Þ ¼ 0

x_ ð0Þ ¼ v0

y_ ð0Þ ¼ 0

ð18-40aÞ
ð18-40bÞ

The specific solution of (18-39) is
 ¼ a þ ibð1 À cos Þ þ b sin 

Figure 18-4

Motion of an electron in a crossed electric and magnetic field.

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ð18-41aÞ


where
 ¼ !c t

ð18-41bÞ

a¼

eE
m!2c

ð18-41cÞ

b¼

v0 À eE=m!c
!c

ð18-41dÞ

Equating the real and imaginary parts of (18-41a) and (18-21), we then find that
xðÞ ¼ a þ b sin 

ð18-42aÞ

yðÞ ¼ bð1 À cos Þ

ð18-42bÞ

Equation (18-42) is well known from analytical geometry and describes a general cycloid or trochoid. Specifically, the trochoidal path is a prolate cycloid, cycloid,
or curtate cycloid, depending on whether a < b, a ¼ b, or a > b, respectively. We
can easily understand the meaning of this result. First, we note that if the applied
electric field E were not present then (18-42) would reduce to the equation of a circle
of radius b, so the electron moves along a circular path. However, an electric field in
the y direction forces the electron to move in the same direction continuously as the
electron moves in the circular path. Consequently, the path is stretched, so the circle
becomes a general cycloid or trochoid. This ‘‘stretching’’ factor is represented by the
term a in (18-42a). We note that (18-40) shows  ¼ 0 corresponds to the origin.
Thus,  is measured from the origin and increases in a clockwise motion.
We can easily find the maximum and minimum values of x() and y() over a
single cycle of . The maximum and minimum values of y() are simply 0 and 2b and
occur at  ¼ 0 and , respectively. For x() the situation is more complicated. From
(18-42a) the angles where the minimum and maximum values of x() occur are

pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi!
2
2
À1 Æ b À a
 ¼ tan
ð18-43Þ
a
The negative sign refers to the minimum value of x(), and the positive sign refers to
the maximum value of x(). The corresponding maximum and minimum values of
x() are then found to be
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi!
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
2
2
À1 Æ b À a
ð18-44Þ
xða, bÞ ¼ a tan
Æ b2 À a2
a
In particular, if we set b ¼ 1 in (18-43) and (18-44) we have
pffiffiffiffiffiffiffiffiffiffiffiffiffi!
Æ
1 À a2
 ¼ tanÀ1
a
xðaÞ ¼ a Á tan

À1

pffiffiffiffiffiffiffiffiffiffiffiffiffi!

pffiffiffiffiffiffiffiffiffiffiffiffiffi
Æ 1 À a2
Æ 1 À a2
a

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ð18-45Þ

ð18-46Þ


Equation (18-46) shows that x(a) is imaginary for a > 1; that is, a maximum and a
minimum do not exist. This behavior is confirmed in Fig. 18-13 and 18-14 for
a ¼ 1.25 and a ¼ 1.5.
Equation (18-45) ranges from a ¼ 0 to 1; for a ¼ 0 (no applied electric field)
 ¼ =2 and 3/2 (or À/2), respectively. This is exactly what we would expect for a
circular path. Following the conventional notation the path of the electron moves
counterclockwise, so /2 is the angle at the maximum point and 3/2 (À/2) corresponds to the angle at the minimum point. Figure 18-5 shows the change in ðaÞ as
the electric field (a) increases. The upper curve corresponds to the positive sign of the
argument in (18-45), and the lower curve corresponds to the negative sign, respectively. We see that at a ¼ 1 the maximum and minimum values converge. The point
of convergence corresponds to a cycloid. This behavior is confirmed by the curve for
x(a) in the figure for a ¼ 1, as we shall soon see.
The maximum and minimum points of the (prolate) cycloid are given by
(18-46). We see immediately that for a ¼ 0 we have x(0) ¼ Æ1. This, of course,
applies to a circle. For 0 < a < 1 we have a prolate cycloid. For a cycloid a ¼ 1, and
(18-46) gives x(1) ¼ 0 and ; that is, the maximum and minimum points coincide.
This behavior is also confirmed for the plot of x(a) versus a at the value where a ¼ 1.
In Fig. 18-6 we have plotted the change in the maximum and minimum values of x(a)
as a increases from 0 to 1. The upper curve corresponds to the positive sign in

(18-46), and the lower curve corresponds to the negative sign.
It is of interest to determine the points on the x axis where the electron path
intersects or is tangent to the x axis. This is found by setting y ¼ 0 in (18-42b). We see
that this is satisfied by  ¼ 0 or  ¼ 2. Setting b ¼ 1 in (18-42a), the points of
intersection on the x axis are given by x ¼ 0 and x ¼ 2a; the point x ¼ 0 and y ¼ 0,

Figure 18-5 Plot of the angle ðaÞ, Eq. (18-45), for the maximum and minimum points as
the electric field (a) increases.

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Figure 18-6 Plot of the maximum and minimum values of xðÞ written as x(a), Eq. (18-46)
as the electric field (a) increases from 0 to 1.

we recall, is the position of the electron at the initial time t ¼ 0. Thus, setting b ¼ 1 in
(18-42a), the initial and final positions of the electron for a ¼ 0 are at x(i) ¼ 0 and
x(f ) ¼ 0, which is the case for a circle. For the other extreme, obtained by setting a ¼
1, the initial and final intersections are 0 and 2, respectively. Thus, as the magnitude
of the electric field increases, the final point of intersection on the x axis increases. In
addition, as a increases, the prolate cycloid advances so that for a ¼ 0 (a circle) the
midpoint of the path is at x ¼ 0 and for a ¼ 1 the midpoint is at x ¼ .
We now plot the evolution of the trochoid as the electric field E(a) increases.
The equations used are, from (18-42) with b ¼ 1,
xðÞ ¼ a þ sin 

ð18-47aÞ

yðÞ ¼ 1 À cos 

ð18-47bÞ

It is of interest to plot (18-47a) from  ¼ 0 to 2 for a ¼ 0, 0.25, 0.50, 0.75, and 1.0.
Figure 18-7 is a plot of the evolution of x() from a pure sinusoid for a ¼ 0 to a
cycloid for a ¼ 1.
The most significant feature of Fig. 18-7 is that the maxima shift to the right as
a increases. This behavior continues until a ¼ 1, whereupon the maximum point
virtually disappears. Similarly, the minima shift to the left, so that at a ¼ 1 the
minimum point virtually disappears. This behavior is later confirmed for a ¼ 1, a
cycloid.
The paths of the electrons are specifically shown in Figs. 18-8 to 18-15. The
curves are plotted over a single cycle of  (0 to 2). For these values (18-45) shows

Copyright © 2003 by Marcel Dekker, Inc. All Rights Reserved.


Figure 18-7

Plot of xðÞ, Eq. (18-47a), for a ¼ 0 to 1.

Figure 18-8

The trochoidal path of an electron, a ¼ 0 (a circle).

that the path intersects the x axis at 0 and 2a, respectively. We select a to be 0, 0.25,
0.5, . . . , 1.5. The corresponding intersections of the path on the x axis are then (0, 0),
(0, /2), (0, ), . . . , (0, 3). With these values of a, Figs. 18-8 to 18-15 show the
evolutionary change in the path. Figure 18-15 shows the path of the electron as it
moves over four cycles.

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Figure 18-9 The trochoidal path of an electron a ¼ 0.25.

Figure 18-10

The trochoidal path of an electron, a ¼ 0.5.

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Figure 18-11

The trochoidal path of an electron, a ¼ 0.75.

Figure 18-12

The trochoidal path of an electron, a ¼ 1.0.

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Figure 18-13

The trochoidal path of an electron, a ¼ 1.25.

Figure 18-14

The trochoidal path of an electron, a ¼ 1.5.

18.2

STOKES VECTORS FOR RADIATION EMITTED BY
ACCELERATING CHARGES

We now determine the Stokes vectors for the radiation emitted by the accelerating
charges undergoing the motions described in the previous section, namely, (1) the

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Figure 18-15

The trochoidal path of an electron over four cycles, a ¼ 0.25.

motion of an electron in a constant electric field, (2) the motion of an electron in a
constant magnetic field, and (3) the motion of the electron in a crossed electric and
magnetic field.
The components of the radiation field in spherical coordinates were shown in
Chapter 16 to be
E ¼

e
½x€ cos  À z€ sin Š
4"0 c2 R

ð16-8Þ

E ¼

e
½y€ Š
4"0 c2 R

ð16-9Þ

These equations refer to the observation being made in the xz plane; that is, at  ¼ 0.
The angle  is the polar angle in the observer’s reference frame.
Recall that the Stokes parameters of the radiation field are defined by
S0 ¼ E EÃ þ E EÃ

ð16-10aÞ

S1 ¼ E EÃ À E EÃ

ð16-10bÞ

S2 ¼ E EÃ þ E EÃ

ð16-10cÞ

S3 ¼ iðE EÃ À E EÃ Þ

ð16-10dÞ

In the following problems we represent the emitted radiation and its polarization in
the form of Stokes vectors.

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18.2.1 Stokes Vector for a Charge Moving in an Electric Field
The path of the charge moving in a constant electric field in the xz plane was found
to be
xðtÞ ¼ v0 t sin 
zðtÞ ¼

ÀeE0 t2
þ v0 t cos 
2m

ð18-10aÞ
ð18-10bÞ

We see that the accelerations of the charge in the x and z directions are then
x€ ðtÞ ¼ 0
z€ðtÞ ¼ À

ð18-48aÞ
eE0
m

ð18-48bÞ

Substituting (18-48) into (16-8) and (16-9) yields
E ¼

e2 E0
sin 

m4"0 c2 R

E ¼ 0

ð18-49aÞ
ð18-49bÞ

and we immediately find from (18-49) that the Stokes vector is
0
1
1
!2
B À1 C
e2 E 0
C
sin2 B
S¼
@0 A
2
m4"0 c R
0

ð18-50Þ

Equation (18-50) shows that the emitted radiation is linearly vertically polarized. It
also shows the accelerating electron emits the familiar dipole radiation pattern
described by sin2 , so the intensity observed along the z axis is zero ( ¼ 0) and
is a maximum when viewed along the x axis ( ¼ /2).
Before we finish the discussion of (18-50) there is another point of interest that
should be noted. We observe that in (18-50) there is a factor of e2 =4"0 mc2 . We now

ask the question, what, if any, is the meaning of this quantity? The answer can be
obtained by recalling that the electric field E ‘‘outside’’ of an electron is given by
e
ur
ð18-51Þ
E¼
4"0 r2
where r is the distance from the center of the electron and ur is the unit radius vector.
We now imagine the electron has a radius a and compute the work that must be done
to move another (positive) charge of the same magnitude from the surface of this
electron to infinity. The total work, or energy, required to do this is
Z1
W ¼ Àe
E Á dr
ð18-52Þ
a

where dr is drur. Substituting (18-51) into (18-52) gives
Z1
e2
dr
e2
¼
W¼
4"0 a r2 4"0 a

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ð18-53Þ

We now equate (18-53) to the rest mass of the electron mc2 and find that
a¼

e2
4"0 mc2

ð18-54Þ

Thus, the factor e2/4"0mc2 is the classical radius of the electron. The value of
a is readily calculated from the values e ¼ 1.60 Â 10À19 C, m ¼ 9.11 Â 10À31 kg,
and c ¼ 2.997 Â 108 m/sec, which yields
a ¼ 2:82 Â 10À15 m

ð18-55Þ

We see that the radius of the electron is extremely small. The factor e2/4"0mc2
appears repeatedly in radiation problems. Later, it will appear again when we consider the problem where radiation is incident on an electron and is then re-emitted,
that is, the scattering of radiation by an electron.
18.2.2

Stokes Vector for a Charge Accelerating in a Constant
Magnetic Field

In the previous section we saw that the path described by an electron moving in a
constant magnetic field is given by the equations:
xðtÞ ¼

v0
sin !c t

!c

yðtÞ ¼ À

v0
ð1 À cos !c tÞ
!c

ð18-31aÞ

ð18-31bÞ

where 0 is the initial velocity and !c ¼ eB/m is the cyclotron frequency. Using the
exponential representation:
Refei!c t g ¼ cos !c t

ð18-56aÞ

RefÀiei!c t g ¼ sin !c t

ð18-56bÞ

we can then write
x ¼ c ðÀiei!c t Þ

ð18-57aÞ

y þ c ¼ c ðei!c t Þ

ð18-57bÞ

where
c ¼

v0
!c

ð18-57cÞ

The accelerations x€ ðtÞ and y€ ðtÞ are then
x€ ðtÞ ¼ ic !2c ei!c t

ð18-58aÞ

y€ ðtÞ ¼ Àc !2c ei!c t

ð18-58bÞ

Copyright © 2003 by Marcel Dekker, Inc. All Rights Reserved.


and the radiation field components become
E ¼

Àiec !2c
cos ei!c t
4"0 c2 R

ð18-59aÞ

E ¼

ec !2c i!c t
e
4"0 c2 R

ð18-59bÞ

From the definition of the Stokes parameters in (16-10) the Stokes vector is
0
1
1 þ cos2 

2
B 1 À cos2  C
ec
C
ð18-60Þ
S¼
!4c B
@
A
2
0
4"0 c R
2 cos 
which is the Stokes vector for elliptically polarized light radiating at the same frequency as the cyclotron frequency !c. Thus, the Stokes vector found earlier for a
charge moving in a circle is based on physical reality. We see that (18-60) reduces to
right circularly polarized light, linearly horizontally polarized light, and left circularly polarized light for  ¼ 0, /2, and , respectively.
18.2.3 Stokes Vector for a Charge Moving in a Crossed Electric and

Magnetic Field
The path of the electron was seen to be a trochoid described by
xðÞ ¼ a þ b sin 

ð18-42aÞ

yðÞ ¼ bð1 À cos Þ

ð18-42bÞ

where
 ¼ !c t
eE
a¼
m!2c
v À eE=m!c
b¼ 0
!c

ð18-41bÞ
ð18-41cÞ
ð18-41dÞ

Differentiating (18-42a) and (18-42b) twice with respect to time and using (18-56)
then gives
x€ ðtÞ ¼ ib!2c ei!c t

ð18-61aÞ

b!2c ei!c t

ð18-61bÞ

y€ðtÞ ¼

and we immediately find that the Stokes vector is
0
1
1 þ cos2 
B 1 À cos2  C
C
S ¼ b2 !4c B
@
A
0
2 cos 
which, again, is the Stokes vector for elliptically polarized light.

Copyright © 2003 by Marcel Dekker, Inc. All Rights Reserved.

ð18-62Þ


With this material behind us we now turn our attention to the Lorentz–Zeeman
effect and see how the role of polarized light led to the acceptance of Maxwell’s
electrodynamical theory in optics.

REFERENCES
Books
1. Jackson, J. D., Classical Electrodynamics, John Wiley, New York, 1962.

2. Sommerfeld, A., Lectures on Theoretical Physics, Vols. I–V, Academic Press, New York,
1952.
3. Harnwell, G. P., Principles of Electricity and Electromagnetism, McGraw-Hill, New York,
1949.
4. Humphries, S., Jr., Charged Particle Beams, John Wiley, New York, 1990.
5. Hutter, R. C. E. and Harrison, S. W., Beam and Wave Electronics in Microwave Tubes,
D. Van Nostrand Princeton, 1960.
6. Panofsky, W. K. H. and Phillips, M., Classical Electricity and Magnetism, AddisonWesley, Reading, MA, 1955.
7. Goldstein, H., Classical Mechanics, Addison-Wesley Reading, MA, 1950.
8. Corben, H. C. and Stehle, P., Classical Mechanics, John Wiley, New York, 1957.

Copyright © 2003 by Marcel Dekker, Inc. All Rights Reserved.