Avril’s Radiation Problem

Kirk T. McDonald
Joseph Henry Laboratories, Princeton University, Princeton, NJ 08544
(January 4, 2018; updated January 17, 2018)

1 Problem
Benoit Avril has posed the problem of a positive charge on a circular path of angular velocity
ω spinning in the counterclockwise sense, and a negative charge on a circular path with the
same angular velocity in the clockwise sense.
Deduce the electromagnetic fields, and the radiation, of this configuration supposing that
the radius a of the circular path is small compared to the wavelength λ = 2πc/ω, where c is
the speed of light in vacuum, and that the velocity v = aω is small compared to c.

2 Solution
2.1 vc
In the stated approximation (first studied by Hertz [1]), we consider the electric dipole
moment p of the configuration, taking the circular path to lie in the x-y plane with its
center at the origin,
p = qa(x̂ + iŷ) e−iωt − qa(x̂ − iŷ) e−iωt = 2iqa ŷ e−iωt = ip ŷ e−iωt , (1)
where the magnitude of the dipole moment is p = 2qa, and the particles are at “3 o’clock”
(x = a, y = 0) at time t = 0.
The electric and magnetic fields of an ideal, point Hertzian electric dipole p can be
written (in Gaussian units) as,1
 
ei(kr−ωt) 1 ik
E = ik p (r̂ × ŷ) × r̂
2
+ ip [3(ŷ · r̂)r̂ − ŷ] 3 − 2 ei(kr−ωt), (2)
r r r
 
1 1
B = ik p (r̂ × ŷ)
2
− ei(kr−ωt) , (3)
r ikr2
whose real parts are,
 
sin(kr − ωt) k cos(kr − ωt) sin(kr − ωt)
E = −k p (r̂ × ŷ) × r̂
2
+ p [3(ŷ · r̂)r̂ − p̂] − , (4)
r r2 r3
 
sin(kr − ωt) cos(kr − ωt)
B = −k p (r̂ × ŷ)
2
+ , (5)
r kr2
where r̂ = r/r is the unit vector from the center of the dipole to the observer, p = p sin ωt ŷ
is the electric dipole moment vector, ω is the angular frequency, and k = ω/c = 2π/λ is the
wave number.
1
See, for example, sec. 9.2 of [2].

1
 We say that the radiation part of these fields are the terms that vary as 1/r:
sin(kr − ωt)
Erad = k 2p r̂ × (r̂ × ŷ) , (6)
r
sin(kr − ωt)
Brad = −k 2p (r̂ × ŷ) . (7)
r
In the near zone of the dipole, where kr < ∼ 1, the radiation fields are smaller that the other
components of E and B. The most prominent feature of the fields in the near zone is that
the electric field looks a lot like that of an electrostatic dipole, as shown in the figure below.
Because field patterns that look like radiation are discernable only for r > ∼ λ, there may be
an impression that the radiation is created at some distance from an antenna, rather than
at the antenna itself.

Since the radiated power comes from the antenna (from the power supply that drives
the antenna), there must be a flow of energy out from the surface of the antenna into the

2
surrounding space. The usual electrodynamic measure of energy flow is Poynting’s vector
[3] (in a medium with unit relative permeability),
c
S= E × B. (8)
4π
When we use the fields (4)-(5) to calculate the Poynting vector we find six terms, some
of which do not point along the radial vector r̂:
  2 
c sin (kr − ωt) cos(kr − ωt) sin(kr − ωt)
S = k p [(r̂ × ŷ) × r̂] × (r̂ × ŷ)
4 2
+
4π r2 kr3
 2
cos (kr − ωt) − sin2(kr − ωt)
+k 2p2 [3(ŷ · r̂)r̂ − ŷ] × (r̂ × ŷ)
r4
 
k 1
+ cos(kr − ωt) sin(kr − ωt) 3 − 5
r kr
  2 
c 4 2 2 sin (kr − ωt) cos(kr − ωt) sin(kr − ωt)
= k p sin θ r̂ +
4π r2 kr3
 2
cos (kr − ωt) − sin2(kr − ωt)
+k 2p2 [(3 cos2 θ − 1) r̂ − 2 cos θ ŷ]
r4
 
k 1
+ cos(kr − ωt) sin(kr − ωt) 3 − 5 , (9)
r kr

where θ is the angle between vectors r and p. As well as the expected radial flow of energy,
there is a flow in the direction of the dipole moment p = ip ŷ. Since the product
cos(kr − ωt) sin(kr − ωt) can be both positive and negative, part of the energy flow is inwards
at times, rather than outwards as expected for pure radiation.
However, we obtain a simple result
 if we consider
only the time-averaged Poynting vector,
S. Noting that cos2 (kr − ωt) = sin2 (kr − ωt) = 1/2 and cos(kr − ωt) sin(kr − ωt) =
(1/2) sin 2(kr − ωt) = 0, eq (9) leads to,

ck 4 p2 sin2 θ
S = r̂. (10)
8πr2
The time-average Poynting vector is purely radially outwards, and falls off as 1/r2 at all
radii, as expected for a flow of energy that originates in the oscillating point dipole. The
time-average angular distribution d P  /dΩ of the radiated power is related to the Poynting
vector by,
d P  ck 4 p2 sin2 θ p2 ω 4 sin2 θ
= r2 r̂ · S = = , (11)
dΩ 8π 8πc3
which is the expression usually derived for dipole radiation in the far zone. Here we see that
this expression holds in the near zone as well.
We conclude that radiation, as measured by the time-averaged Poynting vector, exists in
the near zone of the present example (and of all antennas) as well as in the far zone.

3
2.2 v≈c
When the charges move with velocity close to that of light the fields are considerably different
from those in sec. 2.1, as first studied by Heaviside in sec. 534, pp. 432-498 of [4]. A
graphical method of determining the “radiation” field (which falls off as 1/r) of a single
charge according to a distant observer was given on p. 445, which was also discussed by
Feynman, chap. 34 of [5].

While the electric field would be sinusoidal with time for v  c, if v ≈ c the electric field
is very large when the charge is heading directly towards the observer (at the retarded time
t = t − r/c). This large field occurs once a revolution of the charge in its orbit.

This large
field can be thought of as a “searchlight” beam of angular extent 1/γ = 1 − v 2/c2 that
rotates with angular velocity ω.2
For the case of two opposite charges moving oppositely around a circle as in sec. 2.1, the
fields of the two charges add. For a distant observer on the positive x-axis, the two charges
head directly towards him/her at the same time, so only a single large pulse of radiation
would be detected each cycle, which pulse would have four times the intensity of the pulse
in case of only a single charge. For an observer in the x-y plane at angle θ to the x-axis, two
large pulses would be detected each cycle, separated by time θ/ω;3 these pulses would have
the same intensity as that in case of a single charge.

References
[1] H. Hertz, Die Kräfte electrischer Schwimgungen, behandelt nach der Maxwell’schen
Theorie, Ann. d. Phys. 36, 1 (1889),
http://kirkmcd.princeton.edu/examples/EM/hertz_ap_36_1_89.pdf
The Forces of Electrical Oscillations Treated According to Maxwell’s Theory, Nature
39, 402, 450, 547 (1889), http://kirkmcd.princeton.edu/examples/EM/hertz_nature_39_402_89.pdf
reprinted in chap. 9 of Electric Waves (Macmillan, 1900),
http://kirkmcd.princeton.edu/examples/EM/hertz_electric_oscillations.pdf

[2] J.D. Jackson, Classical Electrodynamics, 2nd ed. (Wiley, 1975),

4
 http://kirkmcd.princeton.edu/examples/EM/jackson_ce2_75.pdf

[3] J.H. Poynting, On the Transfer of Energy in the Electromagnetic Field, Phil. Trans.
Roy. Soc. London 175, 343 (1884),
http://kirkmcd.princeton.edu/examples/EM/poynting_ptrsl_175_343_84.pdf

[4] O. Heaviside, Electromagnetic Theory, Vol. III (Electrician Printing, London, 1912),
http://kirkmcd.princeton.edu/examples/EM/heaviside_electromagnetic_theory_3.pdf

[5] R.P. Feynman, R.B. Leighton and M. Sands, The Feynman Lectures on Physics, Vol. 1
(Addison Wesley, 1964), http://www.feynmanlectures.caltech.edu/I_34.html