Problem 1:
A fixed dipole moment is pointing in the x-direction while moving in the z-direction with a constant velocity v << c. What are the instantaneous electric and magnetic fields at a point (x, y = 0, z) away from the dipole?
Problem 2:
Cs55137 is a common laboratory radioactive source of electrons and gamma rays. Eight percent of the time Cs55137 beta decays to the ground state of Ba56137. The net atomic mass difference between the two isotopes is 1.18 MeV/c2. A 180 degree spectrometer is used to measure the beta decay spectrum. The spectrometer has a radius R = 3.8 cm.
(a)
Write down the reaction for the beta decay.
(b) Calculate the maximum momentum of the beta decay electron/positron.
Express the result in MeV/c.
(c) What is the vector direction of the spectrometer magnetic field relative
to the drawing?
(d) What is the magnetic field setting of the spectrometer for the maximum
energy of the electron/positron to arrive at the detector?
Provide a numerical answer with units.
Problem 3:
A point magnetic moment m is at rest in frame K' and in that frame produces a vector potential A' = m'´r'/r'3 (Gaussian units) and no scalar potential (f = 0). Frame K' moves with constant velocity v << c along the x-axis of frame K, so that an observer in K sees the moment moving with velocity v = bcx/x. Show that to first order in b the observer in K detects an electric dipole moment p = b´m as well as an undiminished (to first order) magnetic moment m.
Problem 4:
An excited nucleus of 57Fe formed by the radioactive decay of 57Co emits a gamma ray of 1.44 x 104 eV. In the process, there is conservation of energy and m0c2 = gma0c2 + hn, where m0c2 is the initial mass of the nucleus and ma0c2 is its mass after the emission of the gamma ray. There is also conservation of momentum, hn/c = gma0u, where u is the recoil velocity of the iron nucleus. The energy released by the reaction is Er = (m0 - ma0)c2.
(a) Show that
hn
= Er(m0
+ ma0)/(2m0)
= (1 - Er/(2m0c2))Er.
Thus hn
< Er:
part of Er
goes to the photon, and the other part supplies kinetic energy to the recoiling
nucleus.
(b) Set m0 = 57*1.7*10-27 kg, and show that
Er/(2m0c2))
~ 1.3*10-7.
Thus the fraction of the available energy
Er
that appears as recoil is small.
(c) Mössbauer discovered in 1958 that, with solid iron, a significant fraction
of the atoms recoil as if they were locked rigidly to the rest of the solid.
This is the Mössbauer
effect. If the sample has a mass of 1 gram, by what fraction is the gamma
ray energy shifted in the recoil process?
(d) A sample of normal 57Fe
absorbs gamma rays of 14.4 keV by the inverse recoilless process much more
strongly than it absorbs gamma rays of any nearby energy. The excited
nuclei thus formed reemit 14.4 keV radiation in random directions some time
later. This is resonant scattering. If a sample of activated
57Fe
moves in the direction of a sample of normal
57Fe,
what must be the value of the velocity v that will shift the frequency of the
gamma rays, as seen by the normal nuclei, by 3 parts in 1013?
This is one line width.
(e) A Doppler shift in the gamma ray results in a much lower absorption by a
nucleus if the shift is of the order of one line width or more. What
happens to the counting rate of a gamma-ray detector placed behind the sample of
normal 57Fe
when the source of activated "Fe moves
(i) toward the normal 57Fe,
(ii) away from it?
(f) If a 14.4 keV gamma ray travels 22.5 meters vertically upward, by what
fraction will its energy decrease?
[Gravitation redshift, a thought experiment:
Suppose
particle of rest mass m is dropped from the top of a tower and falls freely with
acceleration g. It reaches the ground with a velocity v = (2gh)1/2,
so its total energy E, as measured by an observer at the foot of the tower is E
= mc2 + (1/2)mv2 + O(v4) = mc2 + mgh
+ O(v4).
Suppose an observer has some magical method of converting all this energy into a
photon of the same energy. Upon its arrival at the top of the tower with
energy E the photon is again
magically changed into a particle of rest mass
m =
E/c. Energy conservation requires
that m = m. Therefore
E/E = mc2/(mc2
+ mgh + O(v4)) = 1 - gh/c2 + O(v4).]
(g) A normal 57Fe
absorber located at this height must move in
what direction and at what speed in order for resonant scattering to occur?
Problem 5:
A 30 GeV proton passes 10-7cm away from a
hydrogen atom.
(a) Estimate the peak magnitude of the electric field and the duration of the
electric field pulse to which the atom is subjected.
(b) Do the same for a 30 GeV electron passing at the same distance.
You may use mpc2 = 1GeV and
mec2 = 0.5
MeV.
Problem 6:
A line of charge with charge density
l C/m is fixed at rest along the x’
axis of a reference frame S’. A test charge q is at rest in S’ at (0,0,z’ =
d). S’ is in constant motion with velocity v = vi with respect to
a reference frame S.
(a) Calculate the electric field of the line of charge in the rest frame S’ and
the force on q.
(b) Calculate the electric and magnetic fields of the line of charge measured
by an observer at rest in S.
(c) Calculate the force measured by the observer in S on the test charge q.
Problem 7:
A long superconducting solenoid with radius R is at rest in frame K. It
has its axis along the z-axis and a current flowing on the surface produces a
uniform field inside.
(B = Bk, r < R; B = 0, r > R.)
An observer moves with uniform velocity v = vi (v << c)
along the x-axis. Write down the electric and magnetic fields in the rest
frame K' of the observer for r < R and r > R.