16The Classical Radiation Field

16
The Classical Radiation Field

16.1

FIELD COMPONENTS OF THE RADIATION FIELD

Equation (15-22a) is valid for any acceleration of the electron. However, it is convenient to describe (15-22a) in two different regimes, namely, for nonrelativistic
speeds ðv=c ( 1Þ and for relativistic speeds ðv=c ’ 1Þ. The field emitted by an accelerating charge observed in a reference frame where the velocity is much less than the
speed of light, that is, the nonrelativistic regime, is seen from (15-22a) to reduce to


e
EðX, tÞ ¼
½n  ðn  v_ ފ
ð16-1Þ
4"0 c2 R
where EðX, tÞ is the field vector of the radiated field measured from the origin, e is the
charge, c is the speed of light, R is the distance from the charge to the observer,
n ¼ R=R is the unit vector directed from the position of the charge to the observation
point, and v_ is the acceleration (vector) of the charge. The relation between the
vectors X and n is shown in Fig. 16-1.
To apply (16-1), we consider the (radiated) electric field E in spherical
coordinates. Since the field is transverse, we can write
E ¼ E u þ E u

ð16-2Þ

where u and u are unit vectors in the  and  directions, respectively. Because we
are relatively far from the source, we can take n to be directed from the origin and
write n ¼ ur , where ur is the radial unit vector directed from the origin. The triple

vector product in (16-1) can then be expanded and written as
ur  ður  v_ Þ ¼ ur ður Á v_ Þ À v_

ð16-3Þ

For many problems of interest it is preferable to express the acceleration of the
charge v_ in Cartesian coordinates, thus
v_ ¼ x€ ux þ y€uy þ z€uz

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

ð16-4Þ


Figure 16-1

Vector relation for a moving charge and the radiation field.

where the double dot refers to twofold differentiation with respect to time. The unit
vectors u in spherical and Cartesian coordinates are shown later to be related by
ð16-5aÞ
ur ¼ sin  cos ux þ sin  sin uy þ cos uz
u ¼ cos  cos ux þ cos  sin uy À sin uz

ð16-5bÞ

u ¼ À sin ux þ cos uy

ð16-5cÞ

ux ¼ sin  cos ur þ cos  cos u À sin u

ð16-6aÞ

uy ¼ sin  sin ur þ cos  sin u þ cos u

ð16-6bÞ

uz ¼ cos ur À sin u

ð16-6cÞ

or

Using (16-5) and (16-6), we readily find that (16-3) expands to
ur ður Á v_ Þ À v_ ¼ Àu ðx€ cos  cos  þ y€ cos  sin  À z€ sin Þ
þ u ðÀx€ sin  þ y€ cos Þ

ð16-7Þ

We see that ur is not present in (16-7), so the field components are indeed transverse
to the direction of the propagation ur .
An immediate simplification in (16-7) can be made by noting that we shall only
be interested in problems that are symmetric in . Thus, we can conveniently take
 ¼ 0. Then, from (16-1), (16-2), and (16-7) the transverse field components of the
radiation field are found to be
e
E ¼
½x€ cos  À z€ sin Š
ð16-8Þ

4"0 c2 R
e
E ¼
½y€ Š
ð16-9Þ
4"0 c2 R
Equations (16-8) and (16-9) are the desired relations between the transverse radiation
field components, E and E , and the accelerating charge described by x€ , y€ , and z€.
We note that E , E , and  refer to the observer’s coordinate system, and x€ , y€ , and z€
refer to the charge’s coordinate system.
Because we are interested in field quantities that are actually measured, namely,
the Stokes parameters, in spherical coordinates the Stokes parameters are defined by
S0 ¼ E EÃ þ E EÃ
ð16-10aÞ
S1 ¼ E EÃ À E EÃ
S2 ¼

E EÃ

S3 ¼

iðE EÃ

þ

E EÃ

À

E EÃ Þ

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

ð16-10bÞ
ð16-10cÞ
ð16-10dÞ


pffiffiffiffiffiffiffi
where i ¼ À1. While it is certainly possible to substitute (16-8) and (16-9) directly
into (16-10) and find an expression for the Stokes parameters in terms of the acceleration, it is simpler to break the problem into two parts. Namely, we first determine
the acceleration and the field components and then form the Stokes parameters
according to (16-10).
16.2

RELATION BETWEEN THE UNIT VECTOR IN SPHERICAL
COORDINATES AND CARTESIAN COORDINATES

We derive the relation between the vector in a spherical coordinate system and a
Cartesian coordinate system.
The rectangular coordinates x, y, z are expressed in terms of spherical coordinates r, ,  by the equations:
x ¼ xðr, , Þ

y ¼ yðr, , Þ

z ¼ zðr, , Þ

ð16-11Þ

Conversely, these equations can be expressed so that r, ,  can be written in terms

of x, y, z. Then, any point with coordinates (x, y, z) has corresponding coordinates
(r, , ). We assume that the correspondence is unique. If a particle moves from a
point P in such a way that  and  are held constant and only r varies, a curve in
space is generated. We speak of this curve as the r curve. Similarly, two other
coordinate curves, the  curves and the  curves, are determined at each point as
shown in Fig. 16-2. If only one coordinate is held constant, we determine successively
three surfaces passing through a point in space, these surfaces intersecting in the
coordinate curves. It is generally convenient to choose the new coordinates in such a
way that the coordinate curves are mutually perpendicular to each other at each
point in space. Such coordinates are called orthogonal curvilinear coordinates.
Let r represent the position vector of a point P in space. Then
r ¼ xi þ yj þ zk

ð16-12Þ

From Fig. 16-2 we see that a vector vr tangent to the r curve at P is given by
   
@r
@r
dsr
v¼ ¼
Á
ð16-13Þ
@r
@sr
dr

Figure 16-2 Determination of the r, , and  curves in space.

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

where sr is the arc length along the r curve. Since @r=@sr is a unit vector (this ratio is
the vector chord length Ár, to the arc length Ásr such that in the limit as Ásr ! 0 the
ratio is 1), we can write (16-13) as
v r ¼ hr ur

ð16-14Þ

where ur is the unit vector tangent to the r curves in the direction of increasing arc
length. From (16-14) we see then that hr ¼ dsr =dr is the length of vr .
Considering now the other coordinates, we write
v r ¼ hr ur

v  ¼ h u 

v  ¼ h u

ð16-15Þ

so (16-14) can be simply written as
v k ¼ hk uk

k ¼ r, , 

ð16-16Þ

where uk ðk ¼ r, , Þ is the unit vector tangent to the uk curve. Furthermore, we see
from (16-13) that
 

dsr @r
ð16-17aÞ
hr ¼
¼
dr @r
 
ds  @r 
h ¼
¼
d @
 
ds  @r 
h ¼
¼ 
@
d

ð16-17bÞ

ð16-17cÞ

Equation (16-17) can be written in differential form as
dsr ¼ hr dr

ds ¼ h d

ds ¼ h d

ð16-18Þ

We thus see that hr , h , h are scale factors, giving the ratios of differential distances
to the differentials of the coordinate parameters. The calculations
  of vk from (16-15)
leads to the determination of the scale factors from hk ¼ vk  and the unit vector
from uk ¼ vk =hk .
We now apply these results to determining the unit vectors for a spherical
coordinate system. In Fig. 16-3 we show a spherical coordinate system with unit
vectors ur , u , and u . The angles  and  are called the polar and azimuthal angles,

Figure 16-3

Unit vectors for a spherical coordinate system.

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


respectively. We see from the figure that x, y, and z can be expressed in terms of r, 
and  by
x ¼ r sin  cos 

y ¼ r sin  sin 

z ¼ r cos 

ð16-19Þ

Substituting (16-19) into (16-12) the position vector r becomes
r ¼ ðr sin  cos Þi þ ðr sin  sin Þj þ ðr cos Þk

ð16-20Þ

From (16-13) we find that
vr ¼

@r
¼ sin  cos i þ sin  sin j þ cosk
@r

ð16-21aÞ

v ¼

@r
¼ r cos  cos i þ r cos  sin j À r sin k
@

ð16-21bÞ

v ¼

@r
¼ Àr sin  sin i þ r sin  cos j
@

ð16-21cÞ

The scale factors are, from (16-17),
 
@r
hr ¼   ¼ 1

@r
 
 @r 
h ¼   ¼ r
@
 
 @r 
h ¼   ¼ r sin 
@
Finally, from (16-21) and (16-22) the unit vectors are
v
ur ¼ r ¼ sin  cos i þ sin  sin j þ cos k
hr

ð16-22aÞ
ð16-22bÞ
ð16-22cÞ

ð16-23aÞ

u ¼

v
¼ cos  cos i þ cos  sin j À sin k
h

ð16-23bÞ

u ¼

v
¼ À sin i þ cos j
h

ð16-23cÞ

which corresponds to the result given by (16-6) (it is customary to express ux , uy , uz
as i, j, k).
We can easily check the direction of the unit vectors shown in Fig. 16-3 by
considering (16-23) at, say,  ¼ 0 and  ¼ 90 . For this condition (16-23) reduces to
ur ¼ k

ð16-24aÞ

u ¼ j

ð16-24bÞ

u ¼ Ài

ð16-24cÞ

which is exactly what we would expect according to Fig. 16-3.
An excellent discussion of the fundamentals of vector analysis can be found in
the text of Hilderbrand given in the references at the end of this chapter. The
material presented here was adapted from his Chapter 6.

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

16.3

RELATION BETWEEN THE POYNTING VECTOR AND THE
STOKES PARAMETERS

Before we proceed to use the Stokes parameters to describe the field radiated by
accelerating charges, it is useful to see how the Stokes parameters are related to the
Poynting vector and Larmor’s radiation formula in classical electrodynamics.
In Chapter 13, in the discussion of Young’s interference experiment the fact
was pointed out that two ideas were borrowed from mechanics. The first was the
wave equation. Its solution alone, however, was found to be insufficient to arrive at a
mathematical description of the observed interference fringes. In order to describe
these fringes, another concept was borrowed from mechanics, namely, energy.
Describing the optical field in terms of energy or, as it is called in optics, intensity,
did lead to results in complete agreement with the observed fringes with respect to
their intensity and spacing. However, the wave equation and the intensity formulation were accepted as hypotheses. In particular, it was not at all clear why the
quadratic averaging of the amplitudes of the optical field led to the correct results.
In short, neither aspect of the optical field had a theoretical basis.
With the introduction of Maxwell’s equations, which were a mathematical
formulation of the fundamental laws of the electromagnetic field, it was possible
to show that these two hypotheses were a direct consequence of his theory. The first
success was provided by Maxwell himself, who showed that the wave equation of
optics arose directly from his field equations. In addition, he was surprised that his
wave equation showed that the waves were propagating with the speed of light. The
other hypothesis, namely, the intensity formed by taking time averages of the quadratic field components was also shown around 1885 by Poynting to be a direct
consequence of Maxwell’s equations. We now show this by returning to Maxwell’s
equations in free space [see Eqs.(15-1)],
= Â E ¼ À
=ÂH¼"

@H
@t

ð16-25aÞ

@E
@t

ð16-25bÞ

=ÁE¼0

ð16-25cÞ

=ÁB¼0

ð16-25dÞ

and where we have also used the constitutive equations, (15-6). First, we take the
scalar product of (16-25a) and H so that we have
H Á = Â E ¼ ÀH Á

@H
@t

ð16-26aÞ

Next, we take the scalar product of (16-25b) and E so that we have
E Á = Â H ¼ "E Á

@E
@t

ð16-26bÞ

We now subtract (16-26b) from (16-26a):
H Á = Â E À E Á = Â H ¼ ÀH Á

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

@H
@E
À "E Á
@t
@t

ð16-27Þ


The left-hand side of (16-27) is recognized as the identity:
= Á ðE Â HÞ ¼ H Á ð= Â EÞ À E Á ð= Â HÞ

ð16-28Þ

The terms on the right-hand side of (16-27) can be written as
HÁ

@H 1 @
¼
ðH Á HÞ

@t
2 @t

ð16-29aÞ

EÁ

@E 1 @
¼
ðE Á EÞ
@t 2 @t

ð16-29bÞ

and

Then, using (16-28) and (16-29), (16-27) can be written as
!
@ ðH Á HÞ þ "ðE Á EÞ
= Á ðE Â HÞ þ
¼0
@t
2

ð16-30Þ

Inspection of (16-30) shows that it is identical in form to the continuity equation for
current and charge:
=Ájþ

@
¼0
@t

ð16-31Þ

In (16-31) j is a current, that is, a flow of charge. Thus, we write the corresponding
term for current in (16-30) as
S ¼ ðE Â HÞ

ð16-32Þ

The vector S is known as Poynting’s vector and represents, as we shall show, the flow
of energy.
The second term in (16-30) is interpreted as the time derivative of the sum of
the electrostatic and magnetic energy densities. The assumption is now made that
this sum represents the total electromagnetic energy even for time–varying fields, so
the energy density w is
w¼

H2 þ "E2
2

ð16-33aÞ

where
H Á H ¼ H2

ð16-33bÞ

E Á E ¼ E2

ð16-33cÞ

Thus, (16-30) can be written as
=ÁSþ

@w
¼0
@t

ð16-34Þ

The meaning of S is now clear. It is the flow of energy, analogous to the flow of
charge j (the current). Furthermore, if we write (16-34) as
=ÁS¼À

@w
@t

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

ð16-35Þ


then the physical meaning of (16-35) (and (16-34)) is that the decrease in the time rate
of change of electromagnetic energy within a volume is equal to the flow of energy
out of the volume. Thus, (16-34) is a conservation statement for energy.
We now consider the Poynting vector further:
S ¼ ðE Â HÞ

ð16-32Þ

In free space the solution of Maxwell’s equations yields plane-wave solutions:
Eðr, tÞ ¼ E0 eiðkÁrÀ!tÞ

ð16-36aÞ

Hðr, tÞ ¼ H0 eiðkÁrÀ!tÞ

ð16-36bÞ

We can use (16-25a) to relate E to H:
= Â E ¼ À

@H
@t

ð16-25aÞ

Thus, for the left-hand side of (16-25a) we have, using (16-36a),
=  E ¼ =  ½E0 eiðkÁrÀ!tÞ Š
¼ ik  E

ð16-37aÞ

where we have used the vector identity
=  ðaÞ ¼ r  a þ r  a

ð16-38Þ

Similarly, for the right-hand side we have
À

@H
¼ i!H
@t

ð16-39Þ

Thus (16-25a) becomes
nÂE¼

H
c"0

ð16-40aÞ

where
n¼

k
k

ð16-40bÞ

since k ¼ !=c. The vector n is the direction of propagation of S. Equation (16-40a)
shows that n, E, and H are perpendicular to one another. Thus, if n is in the direction
of propagation, then E and H are perpendicular to n, that is, in the transverse plane.
We now substitute (16-40a) into (16-32) and we have

S ¼ c"0 ½E  ðn  Eފ

ð16-41Þ

From the vector identity:
a  ðb  cÞ ¼ ða Á cÞb À ða Á bÞc

ð16-42Þ

we see that (16-41) reduces to
S ¼ c"0 ðE Á EÞn

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

ð16-43Þ


In Cartesian coordinates the quadratic term in (16-43) is written out as
E Á E ¼ Ex Ex þ Ey Ey

ð16-44Þ

Thus, Maxwell’s theory leads to quadratic terms, which we associate with the flow of
energy.
For more than 20 years after Maxwell’s enunciation of his theory in 1865,
physicists constantly sought to arrive at other well-known results from his theory,
e.g., Snell’s law of refraction, or Fresnel’s equations for reflection and transmission
at an interface. Not only were these fundamental formulas found but their derivations led to new insights into the nature of the optical field. Nevertheless, this did not
give rise to the acceptance of this theory. An experiment would have to be undertaken which only Maxwell’s theory could explain. Only then would his theory be
accepted.

If we express E and H in complex terms, then the time-averaged flux of energy
is given by the real part of the complex Poynting vector, so
1
hSi ¼ ðE Â HÃ Þ
2

ð16-45Þ

From (16-40) we have
n  Eà ¼ HÃ

ð16-46Þ

and substituting (16-46) into (16-45) leads immediately to
1
hSi ¼ c"0 ðE Á EÃ Þn
2

ð16-47Þ

Thus, Maxwell’s theory justifies the use of writing the intensity I as
I ¼ Ex Exà þ Ey EyÃ

ð16-48Þ

for the time-averaged intensity of the optical field.
In spherical coordinates the field is written as
E ¼ E u þ E u

ð16-49Þ

so the Poynting vector (16-47) becomes
hSi ¼

c"0
ðE EÃ þ E EÃ Þn
2

ð16-50Þ

The quantity within parentheses is the total intensity of the radiation field, i.e., the
Stokes parameter S0 . Thus, the Poynting vector is directly proportional to the first
Stokes parameter.
Another quantity of interest is the power radiated per unit solid angle, written
as
dP c"0
¼
ðE Á EÃ ÞR2
d
2

ð16-51Þ

We saw that the field radiated by accelerating charges is given by
E¼

e
½n  ðn  v_ ފ
4"0 c2 R

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

ð16-1Þ


Expanding (16-1) by the vector triple product:
E¼

e
½nðn Á v_ Þ À v_ Š
4"0 c2 R

ð16-52Þ

We denote
n Á v_ ¼ jnjjv_ j cos Â

ð16-53Þ

where  is the angle between n and v_ and jÁ Á Áj denotes that the absolute magnitude is
to be taken. Using (16-52) and (16-53), we then find (16-51) becomes
dP
¼ e2 jv_ j sin2 Â
d

ð16-54Þ

We saw that the field radiated by accelerating charges is given by
E ¼

e
ðx€ cos  cos  þ y€ cos  sin  À z€ sin Þ
4"0 c2 R

ð16-55aÞ

E ¼

e
ðÀx€ sin  þ y€ cos Þ
4"0 c2 R

ð16-55bÞ

The total radiated power over the sphere is given by integrating (16-51) over the solid
angle:
Z Z
c"0 2 
P¼
ðE EÃ þ E EÃ ÞR2 sin  d d
ð16-56Þ
2 0
0
We easily find that
Z 2 Z 
ðE EÃ ÞR2 sin  d d ¼
0

0

4e2
ðjx€ j2 þjy€ j2 Þ
162 "20 c4

ð16-57aÞ

4e2
ðjx€ j2 þ jy€ j2 þ 4jz€j2 Þ
3ð162 "20 c4 Þ

ð16-57bÞ

and
Z

2
0

Z



ðE EÃ ÞR2 sin  d d ¼

0

where j ÁÁ j2  ð ÁÁ Þð ÁÁ ÞÃ . Thus, adding (16-57a) and (16-57b) yields
Z 2 Z 
4 e2
ðE EÃ þ E EÃ ÞR2 sin  d d ¼

ðj€rj2 Þ
3 4"0 c4
0
0

ð16-58aÞ

where
r€ ¼ x€ ux þ y€ uy þ z€uz

ð16-58bÞ

Substituting (16-58a) into (16-56) yields the power radiated by an accelerating
charge:
P¼

2 e2
j€rj2
3 4"0 c3

ð16-59Þ

Equation (16-59) was first derived by J. J. Larmor in 1900 and, consequently, is
known as Larmor’s radiation formula.

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


The material presented in this chapter shows how Maxwell’s equations led to
the Poynting vector and then to the relation for the power radiated by the acceleration of an electron, that is, Larmor’s radiation formula. We now apply these results

to obtain the polarization of the radiation emitted by accelerating electrons. Finally,
very detailed discussions of Maxwell’s equations and the radiation by accelerating
electrons are given in the texts by Jackson and Stratton.
REFERENCES
Books
1. Jackson, J. D., Classical Electrodynamics, Wilcy, New York, 1962.
2. Sommerfeld, A., Lectures on Theoretical Physics, Vols. I–V, Academic Press, New York,
1952.
3. Hildebrand, F. B., Advanced Calculus for Engineers, Prentice-Hall, Englewood Cliffs, NJ,
1949.
4. Stratton, J. A., Electromagnetic Theory, McGraw-Hill, New York, 1941.
5. Schott, G. A., Electromagnetic Radiation, Cambridge University Press, Cambridge, UK,
1912.
6. Jeans, J. H., Mathematical Theory of Electricity and Magnetism, 5th ed., Cambridge
University Press, 1948.

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