by equating 
to the energy of a
charged sphere of radius 
(up to factors of order 1). What is 
and
how is it related to the Thomson cross section?
Phys 743 - EM II
Final exam - 18 Dec. 1995
1.
Light is incident on a free electron at rest. Classically, what is the angular distribution of the scattered radiation if the incident light is (a) linearly polarized? (b) circularly polarized ( c) unpolarized? Are they the same as for the scattering from a small dielectric sphere? Explain.
2.
Light is incident on a free electron at rest. Classically, what is
the total scattering cross section (Thomson cross section)? One can define a
classical electron radius by equating to the energy of a
charged sphere of radius (up to factors of order 1). What is and
how is it related to the Thomson cross section?
3.
(a) Consider a uniform medium (a plasma or a metal, if you wish)
containing N free electrons per unit volume. What is the effective
dielectric constant 
of this medium?
(b) Now consider a sphere of radius 
containing just one
electron and assume that the electron is somehow uniformly distributed over
the volume of the sphere. Compute the photon cross section for this (small)
sphere and compare it with the Thomson cross-section. Are they the same?
Explain.
4.
A photon of frequency 
is incident on a free electron at rest. If
the photon's frequency after collision is 
and its
scattering angle in the laboratory is 
, what is the relation
between 
and 
(Compton's formula)? What are the
energy and the momentum of the recoiling electron?
5.
Write down the Lagrangian for an electron in an electromagnetic field and show that the resulting Euler equations are the same as the Einstein-Lorentz equations of motion. You can use either the 3-dimensional formulation or the manifestly covariant 4-dimensional formulation.
6.
(a) An electron of energy E is moving in the xy plane in the
presence of a constant magnetic field 
What are the
angular frequency 
and the radius R of the resulting orbit, as
measured in the lab? Give numbers for 
and
when 
.
(b) What are the corresponding frequencies and radii in the electron's rest frame?
(c) Suppose the electron's velocity makes an angle 
with 
. What are then the angular frequency of the xy motion and the
radius of the spiral in the lab? In the electron's rest frame? (Formulas
only).
7.
Consider an electron moving perpendicularly to a constant magnetic field and
describe the emitted radiation when its energy is (a) 
(b) 
.
Solutions
1.
The angular distribution for Thomson scattering is the same as for the scattering from a small dielectric sphere, or from any object with an isotropic polarizability in the dipole approximation.
(a) 
where 
is the angle between the
incident polarization 
and the scattering direction 
. See Comment 1.
(b) and (c) 
, where 
is the scattering angle. The polar plot is a ``peanut''.
2.
The Thomson cross section is 
By setting 
we find 
Thus 
Not required: if the electron were a uniformly charged sphere, its
energy would be 
3.
From 
and 
we obtain, using 
Then the cross section for the sphere is
This reduces to the Thomson result for 
where 
However, 
corresponds to an energy much greater than 
The Thomson result is not valid at these
energies and the comparison fails.
Here are the numbers: with 
, we
find that 
and 
More directly, the ratio of 
to 
is just 
4.
The final electron 4-momentum is 
or more explicitly
The freshman way to obtain the Compton formula is to compute 
from this and to note that it equals 
The
equivalent 4-vector way is to write out 
and to use 
and so on.
Either way we obtain
(compare Jackson, page 682). Using 
this gives the
more standard Compton formula
One can use (2) to eliminate 
or 
from the
right hand side of the equations (1) and get other fine formulas. For
instance
5.
The 4-d treatment is in Jackson, page 576. The 3-d Lagrangian, with 
is
Then, using Cartesian components
The Euler equations 
give then
These are the components of the Einstein-Lorentz equations 
6.
(a) In the lab, 
and 
where v is the speed and p the magnitude of the momentum. For 
and 
For 
and 
(b) In the electron's rest frame, 
at any energy and R is the same as in the lab.
(c) In the lab, 
independent of 
and 
In the electron's rest frame, 
at any energy and R is the same as in the lab.
7.
Anything relating to circular motion from Jackson's pages 658-679 is acceptable here. Non-relativistic formulae apply at 1eV, highly relativistic formulae apply at 1GeV. A common error was to draw the radiation patterns as in Jackson Fig. 14.4, which is for a different case. In particular, the relativistic pattern for synchrotron radiation peaks in the direction of the velocity v and looks like a "club" in a polar plot (a cross section of the plot does not display the "rabbit ears" seen in Fig. 14.4). Compare also Jackson's Fig. 14.10 (correct) with Fig. 14.5 (not applicable).
-------------------
Comment 1: Dipole scattering patterns
For linear incident polarization (problem 1a) 
gives 
With 
and 
, we have
For circular incident polarization (problem 1b), one can use 
with 
to obtain 
which
is the same as 
For unpolarized incidence
(problem 1c), one can average eq. (3) over 
This is a formal way of doing things and is spelled out here because the
method used in class seems to have left some people insecure. That method
was to start from 
and sum over two orthogonal orientations of 
-------------------
Comment 2 : Numbers for orbits in magnetic fields (problem 6a)
In cgs units, the angular frequency in the lab is 
In MKS
units, it is 
A common error was to plug MKS units into the
cgs formula.
For 
we can put 
Then the angular
frequency is
The speed is given by
and then 
These values are
typical for cyclotron resonance in solids.
For 
the electron is highly relativistic, because 
and
Then 
The speed is practically c, hence the radius is 
These values are typical for electron synchrotrons.
A more direct way to get R for E = 1 GeV
(when 
also) is provided by
Jackson's eq. (12.24), which comes from R = pc/eB:
