Phys 743 - EM II
1.
A rectangular opening with sides of length a and 
defined by 
exists in a flat sheet
filling the xy plane. A plane wave is normally incident with its
polarization vector in the y direction. Calculate the angular
distribution, at large distances, of the power coming through the opening,
in the Kirchhoff approximation (or the Smythe-Kirchhoff approximation, as
you prefer.) Actually I do not care about polarization effects or the
``obliquity prefactor''; I am asking for the dominant diffraction effect.
2.
Starting from the Lorentz transformation, obtain the formula for the addition of velocities, and show that it is consistent with the result of Fizeau's experiment. It is enough to consider a non-dispersive medium.
3.
Light is incident on a small dielectric sphere. What is the angular distribution of the scattered radiation if the incident light is (a) linearly polarized? (b) unpolarized?
4.
Which, if any, of these quantities are Lorentz invariant?
(a) 
(b) 
(c) 
(d) 
Briefly explain your answers. You should have seen some of these quantities before in a context that makes obvious their Lorentz invariance, or the lack of it. Lorentz-transforming each one in long hand is not advisable (but will work for people who can do perfect algebra).
5.
In the decay 
what are (a) the
maximum energy and (b) the minimum energy of the 
? Assume
that the K is at rest. I am asking for a formula, but if time permits you
can get an extra point for plugging in the numbers correctly. The masses
involved, in MeV, are 494, 106, 135, and zero.
1.
The diffraction integral is
where 
is the area of the opening (I have thrown in the factor 
to make F dimensionless). In our case
and we find
The transmitted power is proportional to 
See Comment 1 for
all the other factors (not required).
2.
From the Lorentz transformation
and also 
get
and similarly for 
. In Fizeau's experiment, 
is the speed of light in a medium, seen by a comoving observer. The
observer in the lab sees light moving with speed
3.
(a) 
where 
is the angle from the
incident polarization vector 
If 
(b) 
, where 
is the angle from
the incident direction (as in (a)).
4.
(a) 
is a Lorentz scalar, equal to 
(b) 
is not a Lorentz scalar. If it
were, 
and 
would each be Lorentz scalars, due
to (a). One can note that 
is
the energy density, which is the 
component of the 4-d stress tensor.
(c) 
is a Lorentz scalar, equal to 
(d) 
is not a Lorentz scalar, because
it is a 3-d vector. One can note that 
is the Poynting vector.
5.
(a) 
is maximum when 
and 
go off in the
opposite directions, and 
(see Comment 2).
Squaring the four-vector relation 
we obtain
(b) 
is minimum when 
and 
go off in opposite
directions and 
is created at rest. Then 
See Estevez and Suen, page 255)
Kirchhoff-Smythe is given by Jackson's Eq. (9.156):
with
Plain Kirchoff is given by Eqs. (9.125) and (9.7):
In either case we need to compute the diffraction factor
We find
For Kirchhoff-Smythe we compute the transmitted power per unit solid angle 
and divide by the incident power 
Using
we find
Estevez and Suen write the ``obliquity factor'' in square brackets as 
For Kirchhoff we have similarly
Comment 2. End points of decay spectra
In the answer to question 5a, the condition 
may need
some comment. In general, when a particle of mass M decays into several
particles of masses 
the maximum of 
occurs
when all the other particles come out with the same velocity v in the
direction opposite to particle 1. If 
is finite and 
(for
instance) there is an apparent contradiction because the speed of particle 3
must be c, but the speed of particle 2 must be less than 
To figure
out what happens we take the limit 
Because 
, we see that 
In principle the maximum 
in 
can
be used to decide if the neutrino has a finite mass. In practice it is hard
to attain the necessary accuracy (even in other, more favorable cases). By
the way, the neutrino involved here is the 
neutrino or antineutrino,
according to
