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v2
dh
rad µ0q 2v2 1 - sin2 ¸
Ù
c2
= (8.173)
3
dt 6Àc
v2
1 -
c2
Ù
where ¸ is the angle between v and v.
Ù Ù
In the limit v v, sin¸ = 0, which corresponds to bremsstrahlung. For v ¥" v,
sin¸ = 1, which corresponds to cyclotron radiation or synchrotron radiation.
Virtual photons
According to formula (8.98a) on page 122 and Figure 8.5.4 on the next page
q v2
E¥" = Ez = 1 - (x - x0) · x3
Æ
4˵0s3 c2
(8.174)
q b
=
4Àµ0 ³2 (vt)2 + b2/³2 3/2
which represents a contracted field, approaching the field of a plane wave. The
passage of this field  pulse corresponds to a frequency distribution of the field
energy. Fourier transforming, we obtain
"
1 q bÉ bÉ
EÉ,¥" = E¥"(t)eiÉtdt = K1 (8.175)
2À -"
4À2µ0bv v³ v³
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138 ELECTROMAGNETIC RADIATION
vt
q
v = vx
Æ
b
B
E¥" ‘
FIGURE 8.8: The perpendicular field of a charge q moving with velocity
v = vx is E¥" ‘.
Æ
Here, K1 is the Kelvin function (Bessel function of the second kind with ima-
ginary argument) which behaves in such a way for small and large arguments
that
q
EÉ,¥"
4À2µ0bv
EÉ,¥"
showing that the  pulse length is of the order b/v³.
Due to the equipartition of the field energy into the electric and magnetic
fields, the total field energy can be written
bmax "
2 2
U = µ0 E¥" d3x = µ0 E¥" vdt 2Àbdb (8.177)
V bmin -"
where the volume integration is over the plane perpendicular to v. With the
use of Parseval s identity for Fourier transforms, formula (8.12) on page 101,
we can rewrite this as
" bmax "
U = U dÉ = 4Àµ0v EÉ,¥" 2 dÉ2Àbdb
0 bmin 0
(8.178)
q2 " v³/É db
H" dÉ
2À2µ0v b
0 b
min
from which we conclude that
q2 v³
UÉ H" ln (8.179)
2À2µ0v bminÉ
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8.5 RADIATION FROM A LOCALISED CHARGE IN ARBITRARY MOTION 139
where an explicit value of bmin can be calculated in quantum theory only.
As in the case of bremsstrahlung, it is intriguing to quantise the energy into
photons [cf. Equation (8.150) on page 131]. Then we find that
2± c³ dÉ
NÉ dÉ H" ln (8.180)
À bminÉ É
where ± = e2/(4Àµ0hc) H" 1/137 is the fine structure constant.
¯
Let us consider the interaction of two electrons, 1 and 2. The result of this
interaction is that they change their linear momenta from p1 to p 1 and p2 to
p 2, respectively. Heisenberg s uncertainty principle gives bmin
¯
so that the number of photons exchanged in the process is of the order
2± c³ dÉ
NÉ dÉ H" ln p1 - p 1 (8.181)
À hÉ É
¯
Since this change in momentum corresponds to a change in energy hÉ = E1 -
¯
E1 and E1 = m0³c, we see that
cp1
2± E1 - cp 1
dÉ
NÉ dÉ H" ln (8.182)
À m0c2 E1 - E1 É
a formula which gives a reasonable account of electron- and photon-induced
processes.
8.5.5 Radiation from charges moving in matter
When electromagnetic radiation is propagating through matter, new phenom-
ena may appear which are (at least classically) not present in vacuum. As men-
tioned earlier, one can under certain simplifying assumptions include, to some
extent, the influence from matter on the electromagnetic fields by introducing
new, derived field quantities D and H according to
D = µ(t,x)E = ºµ0E (8.183)
B = µ(t,x)H = ºmµ0H (8.184)
Expressed in terms of these derived field quantities, the Maxwell equations,
often called macroscopic Maxwell equations, take the form
" · D = Á(t,x) (8.185a)
"
" × E + B = 0 (8.185b)
"t
" · B = 0 (8.185c)
"
" × H - D = j(t,x) (8.185d)
"t
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140 ELECTROMAGNETIC RADIATION
Assuming for simplicity that the electric permittivity µ and the magnetic
permeability µ, and hence the relative permittivity º and the relative permeab-
ility ºm all have fixed values, independent on time and space, for each type
of material we consider, we can derive the general telegrapher s equation [cf.
Equation (2.26) on page 27]
"2E "E "2E
- õ - µµ = 0 (8.186)
"¶2 "t "t2
describing (1D) wave propagation in a material medium.
In Chapter 2 we concluded that the existence of a finite conductivity, mani-
festing itself in a collisional interaction between the charge carriers, causes the
waves to decay exponentially with time and space. Let us therefore assume that
in our medium à = 0 so that the wave equation simplifies to
"2E "2E
- µµ = 0 (8.187)
"¶2 "t2
If we introduce the phase velocity in the medium as
1 1 c
vÕ = = = " (8.188)
" "
µµ ºµ0ºmµ0 ººm
"
where, according to Equation (1.9) on page 5, c = 1/ µ0µ0 is the speed of light,
i.e., the phase speed of electromagnetic waves in vacuum, then the general
solution to each component of Equation (8.187)
Ei = f (¶ - vÕt) + g(¶ + vÕt), i = 1,2,3 (8.189)
The ratio of the phase speed in vacuum and in the medium
c " " def
= ººm = c µµ a" n (8.190)
vÕ
is called the refractive index of the medium. In general n is a function of both [ Pobierz całość w formacie PDF ]