16The Classical Radiation Field

However, it is con-venient to describe 15-22a in two different regimes, namely, for nonrelativistic erating charge observed in a reference frame where the velocity is much less than the s

The Classical Radiation Field

Equation (15-22a) is valid for any acceleration of the electron However, it is con-venient to describe (15-22a) in two different regimes, namely, for nonrelativistic erating 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

4"0c2R

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 _vv is the acceleration (vector) of the charge The relation between the vectors X and n is shown inFig 16-1

To apply (16-1), we consider the (radiated) electric field E in spherical coordinates Since the field is transverse, we can write

where u
and uare 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

For many problems of interest it is preferable to express the acceleration of the charge _vv in Cartesian coordinates, thus

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

or

Using (16-5) and (16-6), we readily find that (16-3) expands to

urður_vvÞ  _vv ¼ u
ðx€cos
cos  þ €ycos
sin   €zz sin
Þ

We see that uris 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

 ¼0 Then, from (16-1), (16-2), and (16-7) the transverse field components of the radiation field are found to be

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 €zz

We note that E
, E, and
refer to the observer’s coordinate system, and €x, €y, and €zz 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

Figure 16-1 Vector relation for a moving charge and the radiation field

where i ¼ ffiffiffiffiffiffiffi

1

p 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 accel-eration, 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)

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 coordi-nates r,
,  by the equations:

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

From Fig 16-2 we see that a vector vr tangent to the r curve at P is given by

v ¼@r

@r¼

@r

@sr



 dsr dr

ð16-13Þ

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

where sris the arc length along the r curve Since @r=@sris a unit vector (this ratio is the vector chord length
r, to the arc length
srsuch that in the limit as
sr!0 the ratio is 1), we can write (16-13) as

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 hr¼dsr=dr is the length of vr

Considering now the other coordinates, we write

so (16-14) can be simply written as

where ukðk ¼ r,
, Þ is the unit vector tangent to the ukcurve Furthermore, we see from (16-13) that

hr¼dsr

dr ¼

@r

@r



h
¼ds
d
¼

@r

@



h¼ds d ¼

@r

@



Equation (16-17) can be written in differential form as

We thus see that hr, h
, hare scale factors, giving the ratios of differential distances

to the differentials of the coordinate parameters The calculations of vkfrom (16-15) leads to the determination of the scale factors from hk¼vk

 and the unit vector from uk¼vk=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

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

Substituting (16-19) into (16-12) the position vector r becomes

From (16-13) we find that

vr¼@r

v
¼@r

v¼@r

The scale factors are, from (16-17),

hr¼ @r

@r











h
¼ @r

@



h¼ @r

@











Finally, from (16-21) and (16-22) the unit vectors are

ur¼vr

u
¼v

u¼v

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

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 hisChapter 6

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 formula-tion 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 quad-ratic 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)],

=  E ¼ @H

=  H ¼ "@E

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  =  E ¼ H @H

Next, we take the scalar product of (16-25b) and E so that we have

E  =  H ¼ "E @E

We now subtract (16-26b) from (16-26a):

H  =  E  E  =  H ¼ H @H

@t "E 

@E

The left-hand side of (16-27) is recognized as the identity:

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

H @H

@t ¼

1 2

@

and

E @E

@t ¼

1 2

@

Then, using (16-28) and (16-29), (16-27) can be written as

=  ðE  HÞ þ@

@t

ðH  HÞ þ "ðE  EÞ

2

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

=  j þ@

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

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

w ¼H

2þ"E2

where

Thus, (16-30) can be written as

=  S þ@w

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

=  S ¼ @w

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:

In free space the solution of Maxwell’s equations yields plane-wave solutions:

We can use (16-25a) to relate E to H:

=  E ¼ @H

Thus, for the left-hand side of (16-25a) we have, using (16-36a),

=  E ¼ =  ½E0eiðkr!tÞ

where we have used the vector identity

Similarly, for the right-hand side we have

Thus (16-25a) becomes

n  E ¼ H

where

n ¼k

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

From the vector identity:

we see that (16-41) reduces to

In Cartesian coordinates the quadratic term in (16-43) is written out as

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 deriva-tions 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 under-taken 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

hSi ¼1

2ðE  H



From (16-40) we have

and substituting (16-46) into (16-45) leads immediately to

hSi ¼1

2c"0ðE  E



Thus, Maxwell’s theory justifies the use of writing the intensity I as

for the time-averaged intensity of the optical field

In spherical coordinates the field is written as

so the Poynting vector (16-47) becomes

hSi ¼c"0

2 ðEE



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

d¼

c"0

2 ðE  E



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

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

We denote

where  is the angle between n and _vv and   j jdenotes that the absolute magnitude is

to be taken Using (16-52) and (16-53), we then find (16-51) becomes

dP d¼e

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

4"0c2Rðx€cos
cos  þ €ycos
sin   €zz sin
Þ ð16-55aÞ

The total radiated power over the sphere is given by integrating (16-51) over the solid angle:

P ¼c"0 2

Z2

0

Z 0

We easily find that

Z 2

0

Z  0

ðEEÞR2sin
d
d ¼ 4e

2 162"2c4ðj jx€2þjyyj€2Þ ð16-57aÞ and

Z 2

0

Z  0

ðE
E
ÞR2sin
d
d ¼ 4e

2 3ð162"2

0c4Þðjxxj€2þ jyyj€2þ4 €zzj j2Þ ð16-57bÞ wherej j 2 ðÞðÞ Thus, adding (16-57a) and (16-57b) yields

Z 2

0

Z  0

ðEEþE
E
ÞR2sin
d
d ¼4

3

e2 4"0c4ðj€rrj2Þ ð16-58aÞ where

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

P ¼2 3

e2

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

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 accelera-tion of an electron, that is, Larmor’s radiaaccelera-tion 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