# PHY 3221 Classical Mechanics, Homework #11, due 8:30 am, 04/23/12 1 v

```PHY 3221 Classical Mechanics, Homework #11, due 8:30 am, 04/23/12
1 Two equal mass particles are moving in opposite directions. In the frame of an inertial observer,
the right moving particle has speed v1 and the left moving particle has speed v2 . At t = 0 in the
observer’s inertial frame, the right moving particle is at the origin and the left moving particle is
a distance d to the right of the origin.
a) Write down equations for the positions of the two particles at arbitrary times after t = 0 (until
they collide).
b) Find the time and position at which they collide.
c) Find the velocity of the center of momentum of the system of two particles.
d) Find x0 , the position of the center of momentum at t = 0.
e) Find the speed of each particle, relative to the center of momentum, before they collide.
f) If the two particles stick together when they collide, what is the final mass of the product in terms
of the initial mass of each particle.
1
2 This question concerns photons interacting with matter.
a) Two photons of the same frequency collide head-on and annihilate. Do the conservation laws
allow them to produce a particle at rest?
b) Show that the conservation laws do not allow a single photon to decay into an electron-positron
pair.
c) Show that by “adding” two identical photons travelling in the same direction, it is possible to
produce a photon (and not some massive particle) of double the initial single-photon energy.
d) A photon is massless and travels at the speed of light. What can you say about the four-vector
describing the photon when it is traveling in a medium with a refractive index greater than 1?
2
3 Suppose that the electromagnetic antisymmetric four-tensor (named after Faraday) is given by:

F &micro;ν
0
Bz
−B
0

z
=
By
−Bx
Ex /c Ey /c
−By
Bx
0
Ez /c

−Ex /c
−Ey /c 
.
−Ez /c
0
a) Show that K &micro; = q(F &middot; V )&micro; is compatible with the Lorentz force law F = q(E + v &times; B).
b) Explain why a γ occurred in relation to your answer in part (a).
c) Show that, under a Lorentz transformation arising from a boost in the x-direction, the electromagnetic fields transform as:
Ex′ = Ex
Bx′ = Bx
Ey′ = γ(Ey − vx Bz )
By′ = γ(By + vx Ez /c2 )
Ez′ = γ(Ez + vx By )
Bz′ = γ(Bz − vx Ey /c2 ).
d) Show that E′ &middot; B′ = E &middot; B, so that this quantity must be a scalar under Lorentz transformations.
If B (but not E) is zero in some frame, what can you say about the angle between B and E in
any other frame?
e) Show that E ′ 2 − c2 B ′ 2 = E 2 − c2 B 2 , so that this quantity must also be a scalar under Lorentz
transformations. If B (but not E) is zero in some frame, is there any frame in which E vanishes?
3
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