Download Midterm Exam No. 03 (Spring 2015) PHYS 520B: Electromagnetic Theory

Midterm Exam No. 03 (Spring 2015)
PHYS 520B: Electromagnetic Theory
Date: 2014 Apr 15
1. (40 points.) A particle of mass m and charge q moving in a uniform magnetic field B and
uniform electric field E experiences a velocity dependent force F given by the expression
m
dv
= q [E + v × B] ,
dt
(1)
where v(t) = dx/dt is the velocity of the particle in terms of its position x(t). Choose
the magnetic field to be along the positive z direction, B = Bẑ, and the electric field to
be along the positive y direction, E = E ŷ.
(a) For the case when the particle starts at rest at the origin at time t = 0, use the
initial conditions
v(0) = 0 x̂ + 0 ŷ + 0 ẑ,
x(0) = 0 x̂ + 0 ŷ + 0 ẑ,
(2)
(3)
to solve the differential equation in Eq. (1) to find the position x(t) and velocity v(t)
as a function of time. Use ω = qB/m.
(b) In particular, prove that the particle takes a path along a cycloid. That is, the
particle moves as though it were a spot on the rim of a wheel rolling along the xaxis. It satisfies the equation of a circle of radius R whose center (vt, R, 0) travels
along the x-direction at constant speed v,
(x − vt)2 + (y − R)2 = R2 .
(4)
Determine the radius R and speed v in terms of E, B, q, and m. Show that v = ωR.
2. (30 points.) The path of a relativistic particle moving along a straight line with constant
(proper) acceleration α is described by the equation of a hyperbola
x2 − c2 t2 = x20 ,
x0 =
c2
.
α
(5)
This is the motion of a particle ‘dropped’ from x = x0 at t = 0 in a region of constant
(proper) acceleration.
(a) Will a photon dispatched to ‘chase’ this particle at t = 0 from x = 0 ever catch up
with it? If yes, when and where does it catch up?
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(b) Will a photon dispatched to ‘chase’ this particle at t = 0 from 0 < x < x0 ever catch
up with it? If yes, when and where does it catch up?
(c) Will a photon dispatched to ‘chase’ this particle, at t = 0 from x < x0 ever catch up
with it? If yes, when and where does it catch up?
3. (30 points.) The Poincaré formula for the addition of (parallel) velocities is
v=
va + vb
va vb ,
1+ 2
c
(6)
where va and vb are velocities and c is speed of light in vacuum. (For the following
assume that the Poincaré formula holds for all speeds, subluminal (vi < c), superluminal
(vi > c), and speed of light.) Analyse what is obtained if you add a subluminal speed to
a superluminal speed? That is, is the resultant speed subluminal or superluminal.
Hint: Analyse the case
va
c
(7)
= − ± δ,
c
vb
for infinitely small δ > 0.
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