11. Radiation
What is radiation?
Radiation sources
Antenna: radio waves, microwaves
Atoms, molecules: light, infrared light, UV, X-rays
Nuclei: Gamma rays
We shall consider:
Point-like source at rest with time-dependent charge distribution
Accelerated point charge
Point-like source
  r
d  r
d  
radiation zone
long wavelength
limit
d
Multipole expansion
V (r , t ) 
1
40

 (r ' , t r )
r  r'
d '
 0 J (r ' , t r )
A (r , t ) 
d '

4 r  r'
r  r'
retarded time : tr  t 
c
1  Q rˆ  p(t0 ) rˆ  p (t0 ) 
V (r , t ) 


2

40  r
r
rc 
r
retarded time : t0  t 
c
0 p (t0 )
A (r , t ) 
4 r
1  rˆ  p (t0 ) 
Vrad (r, t ) 
40  rc 
0 p (t0 )
A rad (r, t ) 
4 r
Radiation fields
Dipole fields
B rad  
Erad 
B rad
Erad
No em monopole fields
0
(t0 )]
[rˆ  p
4rc
0
(t0 ))]
[rˆ  (rˆ  p
4r
ˆ
0 p(t0 )

sin(  )
4rc
ˆ
0 p(t0 )

sin(  )
4r
Acoustic monopole
fields do exist.
Radiated power
d P
0 4 p02 2

sin 
2
d
32 c
d  sin dd
0 4 p02
P 
12 2c
Why is the sky blue?
Oscillating dipole
p  qz (t )zˆ  qd cos(t )zˆ
Larmor formula
0 q 2 a 2
P
6c
a - accelerati on
Holds in general if v  c.
Radiation from a moving point charge
r
2
2
E(r, t ) 
[(
c

v
)u  r  (u  a)]
3
40 ( r  u)
q
 (tr ) u  crˆ  v a  w
 (tr )
r  r  w(tr ) v  w
q rˆ  (u  a)
Erad (r, t ) 
40 r ( rˆ  u)3
S rad 
1
0
2
(E rad  B rad )   0 Erad
cr̂
B(r, t ) rad
1
 rˆ  E(r, t )
c
instantano usly : v  0
0 q a

16 2c
2
S rad
2
2
 sin  

 rˆ
 r 


 S  da
Total radiated power P 
sphere
Larmor formula
0 q 2 a 2
P
6c
Holds also for velocities
small as compared to c.
Arbitrary velocity
Power radiated by the charge
dP  rˆ  u 
2
2

c 0 Erad rˆ
d  c 
rˆ  (u  a)
dP
q

d 16 2 0 ( rˆ  u)5
2
2
2

0q   2 v  a 
1
P
a 
,  
6c 
c 
1  (v / c)2
2
6
Lienard’s formula
Bremsstrahlung
v and a are parallel
dP 0q2a 2
sin 2 

d 16 2c (1   cos )5
0q2a 2 6
1
v
P
, 
, 
2
6c
c
1 
First commercial X-ray tube
1896
Wilhelm Conrad Roentgen
discovered X-rays 1895
Nobelprice 1901
1896
1960
Synchrotron radiation
dP 0q2a 2 [(1   cos )2  (1   2 ) sin 2  cos2  ]

d 16 2c
(1   cos )5
0q2a 2 4
1
v
P
, 
, 
2
6c
c
1 
BESSY synchrotron X-ray source in Berlin