1. Larmor Formula: radiation from non-relativistic particles
Feb. 28, 2011
Larmor Formula: radiation from non-relativistic particles
Dipole Approximation
Thomson Scattering

2. The E, B field at point r and time t depends on
the retarded position r(ret) and retarded time t(ret) of the charge.
Let
Field of particle w/ constant velocity
Transverse field due to acceleration

3. Qualitative Picture:
transverse “radiation”
field propagates at velocity c

4. Radiation from Non-Relativistic Particles
For now, we consider non-relativistic particles, so
Then E the RADIATION FIELD is
and

6. Magnitudes of E(rad) and B(rad):
Poynting vector is in n direction with magnitude

7. ASIDE: If
Show that
Need two identities:
So…
Now

9. Energy flows out along direction
with energy dω emitted per time per solid angle dΩ
cm2
ergs/s/cm2 so

10. Integrate over all dΩ to get total power
LARMOR’S FORMULA
emission from a single
accelerated charge q

11. NOTES 1. Power ~ q2 2. Power ~ acceleration 2 3. Dipole pattern:
No radiation emitted along
the direction of acceleration.
Maximum radiation is emitted
perpendicular to acceleration.
4. The direction of is determined by
If the particle is accelerated along a line, then the radiation is
100% linearly polarized in the plane of

12. The Dipole Approximation
Generally, we will want to derive for a collection of particles
with
You could just add the ‘s given by the formulae derived
previously, but then you would have to keep track of all the
tretard(i) and Rretard(i)

13. One can treat, however, a system of size L with “time scale for changes”
tau where
so differences between tret(i) within the system are negligible
Note: since frequency of radiation
then or If
This will be true whenever the size of the system is small
compared to the wavelength of the radiation.

14. Can we use our non-relativistic expressions for
? yes.
Let l = characteristic scale of particle orbit
u = typical velocity
tau ~ l/u
tau >> L/c  u/c << l/L
since l<L, u<c  non-relativistic

15. Using the non-relativistic expression for E(rad):
If Ro = distance from field to system, then we can write L
where
Dipole Moment

16. Emitted Power
power per solid angle
power
DIPOLE APPROXIMATION
FOR NON-RELATIVISTIC
PARTICLES

17. What is the spectrum for this Erad(t)?
Simplify by assuming the dipole moment is always in same direction,
let
then

18. Let fourier transform of
then

19. Then
Recall from the discussion of the Poynting vector:
Integrate over time:
(1)

20. Parseval’s Theorem for Fourier Transforms 
(2)
Since E(t) is real
FT of E 
(See Lecture notes for Feb. 16) so
Thus (2) 

21. Substituting into (1) 
Thus, the energy per area per frequency 
substituting
and
integrate over solid angle

22. NOTE:
1. Spectrum ~ frequencies of oscillation of d (dipole moment)
2. This is for non-relativistic particles only.

23. Thomson Scattering
Rybicki & Lightman, Section 3.4

24. Thomson Scattering
EM wave scatters off a free charge. Assume non-relativistic: v<<c.
E field
e = charge
Incoming E field
in direction
electron
Incoming wave: assume linearly polarized. Makes charge oscillate.
Wave exerts force:
r = position of charge

25. Dipole moment: so
Integrate twice
wrt time, t
So the wave induces an oscillating dipole
moment with amplitude

26. What is the power? Recall time averaged power / solid angle
(see next slide) So

27. Aside:
Time Averages
The time average of the signal is denoted by angle brackets , i.e.,
If x(t) is periodic with period To, then

28. The total power is obtained by integrating over all solid angle:or

29. What is the Thomson Cross-section?
Incident flux is given by the time-averaged Poynting Vector
Define differential cross-section: dσ
scattering into solid angle dΩ
cross-section per
solid angle
cm2 /ster
Power per solid angle
erg /sec /ster
Time averaged
Poynting Vector
erg/sec /cm2

30. Thus so
since
(polarized incident light)
we get
Thomson cross section
classical electron
radius

31. Integrate over dΩ to get TOTAL cross-section for Thomson scattering
Thomson Cross-section
NOTES:
1. Thomson cross-section is independent of frequency.
Breaks down when hν >> mc2, can no longer ignore relativistic effects.
2. Scattered wave is linearly polarized in ε-n plane

32. Electron Scattering for un-polarized radiation
Unpolarized beam = superposition of 2 linearly polarized beams with
perpendicular axes

33. Differential Cross-section
Average for
2 components
Thomson
cross-section
for unpolarized
light