Description
Problems
1. Collision with a particle at rest, Jackson, 11.23: In a collision process a particle
of mass m2, at rest in the laboratory, is struck by a particle of mass m1, momentum
pLAB and total energy ELAB. In the collision the two initial particles are transformed
into two others of mass m3 and m4. The configurations of the momentum vectors in
the center of momentum (cm) frame (traditionally called the center-of-mass frame)
and the laboratory frame are shown in the figure.
(a) (10 pts) Use invariant scalar products to show that the total energy W in the cm
frame has its square given by
W2 = m2
1 + m2
2 + 2m2ELAB (1)
and that the cms 3-momentum p
0
is
p
0 =
m2pLAB
W
(2)
(b) (8 pts) Show that the Lorentz transformation parameters βcm and γcm describing
the velocity of the cm frame in the laboratory are
βcm =
pLAB
m2 + ELAB
, γcm =
m2 + ELAB
W
(3)
(c) (8 pts) Show that the results of parts (a) and (b) reduce in the nonrelativistic
limit to the familiar expressions,
W ‘ m1 + m2 +
m2
m1 + m2
p
2
LAB
2m1
(4)
p
0 ‘
m2
m1 + m2
pLAB, βcm ‘
pLAB
m1 + m2
(5)
2. Converting photons to electron and positron: Consider two photons with different energies that annihilate (in the vacuum) and produce an electron-positron pair.
(I.e. a reaction with two photons – in, and electron and positron – out.)
(a) (8 pts) For what ranges of initial photon energies and angles between their directions of propagation can this reaction take place? (In other words, give a relation,
perhaps an inequality, that may contain photon energies, the angle, electron mass,
speed of light, etc.)
(b) (6 pts) Consider now a head-on collision (the angle is π radians) and the photons
of the same energy. Calculate the numerical value of the minimal photon energy
required for the reaction to take place. Express the answer in SI units (Joules).
3. Field tensor: Consider the electromagnetic field tensor F
µν = ∂
µAν − ∂
νAµ
in the
conventions of our course (SI units, (+, −, −, −) metric tensor).
(a) (15 pts) Starting from the definition in terms of the potential Aµ
, derive the
matrix (F
µν) in terms of the components of the electric and magnetic fields, Ex,
Ey, Ez, Bx, By, Bz.
(b) (15 pts) Consider a Lorentz boost with relativistic velocity β in the positive
x direction. Derive the components of the electric and magnetic fields in the
moving frame in terms of the corresponding quantities in the original frame by
transforming F
µν
.
(c) (10 pts) Derive the transformation properties of the electric and magnetic fields
under parity (space inversion) and time reversal.
4. To B~ E~ or not to B~ E~ : In a reference frame K there are a constant electric E~ and a
magnetic B~ fields such that E~⊥B~ .
(a) (10 pts) With what velocity a reference frame K0
should be moving with respect
to K so that in K0
there is only electric or only magnetic field ? Derive the value
of the corresponding field in the K0
frame as function of the original fields.
(b) (10 pts) Does the solution always exist ? Is it unique ?
