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UNIVERSITY OF LEIPZIG
INSTITUTE FOR THEORETICAL PHYSICS
Department: Theory of Elementary Particles
Electrodynamics
Lecturer: PD Dr. A. Schiller
List of problems 9:
24. A localized electric charge distribution produces an electric field, E =
−∇Φ. Into this field is placed a small localized time-independent current
density J(x), which generates a magnetic field H.
(a) Show that the momentum of these electromagnetic fields can be transformed to
1 Z
Pfield = 2 Φ(x) J(x) d3 x
c
provided that the product Φ H falls off rapidly enough at large distances. How rapidly is “rapidly enough”?
(b) Assuming that the current distribution is localized to a region small
compared to the scale of variation of the electric field, expand the
electrostatic potential in a Taylor series and show that
Pfield =
1
E(0) × m
c2
where E(0) is the electric field at the current distribution and m is
the magnetic moment caused by the current.
25. Starting with the retarded solution to the three-dimensional wave equation
Ψ(x, t) =
Z
f x0 , t0 = t −
|x − x0 |
|x−x0 |
c
d 3 x0 ,
show that the source f (x0 , t0 ) = δ(x0 ) δ(y 0 ) δ(t0 ), equivalent to a t = 0 point
source at the origin in two spatial dimensions, produces a two-dimensional
wave,
2c Θ(ct − ρ)
Ψ(x, y, t) = √ 2 2
c t − ρ2
where ρ2 = x2 + y 2 and Θ(ξ) is the unit step function [Θ(ξ) = 0(1) if
ξ < (>)0].
26. A transverse electromagnetic wave (E and H components are perpendicular
to the propagation direction and perpendicular to each other) is incident
normally in vacuum on a perfectly absorbing flat screen.
(a) From the law of conservation of linear momentum, show that the pressure (called radiation pressure) exerted on the screen is equal to the
field energy per unit volume in the wave.
(b) In the neighborhood of the earth the flux of electromagnetic energy
from the sun is approximately 1.4 kW/m2 . If an interplanetary “sailplane” had a sail of 1 g/m2 of area and negligible other weight, what
would be its maximum acceleration in meters per second squared due
to the solar radiation pressure?