Electric dipole radiation and simple antennas

The simplest radiating system is an electric dipole whose moment oscillates in time with a
well-de¯ned angular frequency ! along the z direction (for instance). We can produce such a
dipole by considering two opposite charges, one of which is ¯xed at the origin while the other
executes simple harmonic motion with amplitude d; so that its position is given by z = d cos !t:
To represent an atom, the moving charge would be e (considered here as a negative quantity),
representing an electron, and the ¯xed charge would be jej, representing the nuclear charge
screened by the other electrons. We will use this notation for the general case as well, where e
could have any value and either sign. In general

pz = ez = ed cos !t (4.15)

and we note that (with the dot denoting di®erentiation with respect to t)

p_ z = ev = ¡ed ! sin !t

pÄz = ev_ = ¡ ed !2 cos !t

According to the Larmor equation (4.13) the total radiated power at time t is then

2 1 pÄz2
P =
3 4¼"0 c3
1
and varies like cos2 !t: Recalling that the time average of cos2 !t is 2 and denoting the time
average by brackets we ¯nd that the average radiated power is
- 2®
2 1 pÄz 1 e2 d2 !4
hP i = = (4.16)
3 4¼"0 c3 3 4¼"0 c3
We have just rewritten the Larmor formula for a simple oscillator, emphasizing the di®erence
between instantaneous power and average power. We note also that the corresponding formulas
in the Gaussian system are obtained by omitting the factor 1=4¼"0 .
Although the wave emitted by the oscillating dipole is a spherical wave, it does not have
the same intensity in all directions. It can be shown that the average power emitted in the
direction that makes an angle µ with the z axis, within a solid angle d-; is

d hP i 1 e2 d2 !4
= sin2 µ (4.16')
d- 8¼ 4¼"0 c3
This can be understood as follows: the radiation is emitted by the component of the emitter's
motion that is perpendicular to the line of sight { see Fig. 4.3(c). Note also that
Z Z ¼
8¼
sin2 µ d- = 2¼ sin3 µ dµ =
0 3

so that (4.16') agrees with (4.16).

The intensity of the radiation is the energy °ux, i.e. the energy dE crossing the area dA per
unit time. Since energy per unit time is power, and at distance R from the source of radiation
dA = R2 d-; we see that the energy °ux, or radiation intensity, is
dP 1 dP
= 2
dA R d-

1
The intensity arriving at time t depends on the state of motion of the source at time t ¡ R=c;
because the signal travels with speed c: For a single dipole source in simple harmonic motion
it varies like cos2 !(t ¡ R=c) and its time average is found by using (4.16').
Finally, it must be mentioned that the Larmor formula is applicable only when the motion
of the particle is non-relativistic, i.e. when v << c: For simple harmonic motion the maximum
value of v is !d, and thus (4.16) and (4.16') are strictly valid only when !d << c, or 2¼d << ¸;
where ¸ is the wavelength. This is not a serious restriction in the case of atoms, where d is
about 1 ngstrom and ¸ is thousands of times larger, for visible light.

Antennas. Simple antennas are most e±cient when they have dimensions comparable to
the wavelength they emit. The theory of the simple dipole emitter is not strictly applicable
to them; however it still gives qualitatively correct answers. An oscillating particle of charge
e can be thought of as a current I = I0 sin !t = 2e! sin !t: Replacing e by I0 =2! and using
! = 2¼c=¸ as well as 1=c = ("0 ¹0 )1=2 ; we can rewrite eqs. (4.16) and (4.16') in the form
µ ¶1=2 µ ¶2
1 ¹0 2¼d
hP i = I02 (4.17)
48¼ "0 ¸
µ ¶1=2 µ ¶2
d hP i 1 ¹0 2¼d
= I02 sin2 µ (4.18)
d- 128¼2 "0 ¸
µ ¶1=2
¹0
The quantity Z0 = has the dimensions of impedance and is called the impedance of
"0
free space; it has the numerical value Z0 = 377 -:
An example often used is the half-wave center-fed linear antenna { see Fig. 4.3(b). The
current distribution, for jzj · d=2; is approximately of the form

I = I0 cos(¼z=d) sin !t

with ! = ¼c=d: Since ¸ = 2¼c=!; we see that d = ¸=2; which explains the name \half-wave".
The radiation pattern of the half-wave antenna is similar to the dipole pattern (4.18), and the
total power is larger than the simple formula (4.17) by a factor of 1.46.

Dipole radiation pattern sin2 µ Cut-out view of the same