Description
Problems
1. Non-relativistic particle in homogeneous magnetic field: Consider the motion
of a non-relativistic point particle in a static homogeneous magnetic field, ignoring
radiation. Assume B~ is in the z direction, B~ (~r, t) = B0zˆ.
(a) (20 pts) Starting from the Lorenz force, derive the trajectory ~r0(t) of the particle
and identify the non-relativistic cyclotron frequency. What is the change of kinetic
energy of the particle with time ?
2. Fields in a hollow cylinder: Using cylindrical coordinates (ρ, ϕ, z), consider electric
and magnetic fields
E~ = ˆρ
c1
ρ
, B~ = ˆϕ
c2
ρ
,
with constant c1 and c2 inside the volume bounded by a ≤ ρ ≤ b, i.e. inside an infinitely
long cylinder with a hole.
(a) (10 pts) Determine the Poynting vector S~ inside the volume.
(b) (10 pts) Determine the total flux of energy in the fields through a cross-sectional
surface with a ≤ ρ ≤ b.
(c) (10 pts) Determine the energy per unit length dE/dz and the momentum per unit
length d~p/dz in the fields (for a ≤ ρ ≤ b).
3. Force due to a plane wave: An incident monochromatic plane wave described by a
vector potential A~ = A~
0 cos(ωt − ~k~r) is completely absorbed by a sphere of radius R.
(a) (10 pts) Find the electric and magnetic fields. (Take into account that for a plane
wave ~kA~ = 0.)
(b) (20 pts) Determine the Maxwell’s stress tensor.
(c) (20 pts) Find the force F~ exerted by the wave on the sphere averaged over the
period T = 2π/ω using the result of part (b).