Problem 42.
40 (a) Prove that in an electromagnetic plane wave, half the
energy is in the electric field and half in the magnetic field.
(b) Based on your result from part a, find the proportionality con-
stant in the relation dp∝E×Bdv, where dpis the momentum of
the part of a plane light wave contained in the volume dv. The vec-
torE×B, multiplied by the appropriate constant, is known as the
Poynting vector, and even outside the context of an electromagnetic
plane wave, it can be interpreted as a momentum density or rate of
energy flow. (To do this problem, you need to know the relativistic
relationship between the energy and momentum of a beam of light
from problem 11 on p. 460.)√41 (a) A beam of light has cross-sectional areaAand power
P, i.e.,P is the number of joules per second that enter a window
through which the beam passes. Find the energy densityU/vin
terms ofP,A, and universal constants.
(b) FindE ̃andB ̃, the amplitudes of the electric and magnetic fields,
in terms ofP,A, and universal constants (i.e., your answer should
notincludeUorv). You will need the result of problem 40a. A real
beam of light usually consists of many short wavetrains, not one big
sine wave, but don’t worry about that.√
.Hint, p. 1033
(c) A beam of sunlight has an intensity ofP/A= 1.35× 103 W/m^2 ,
assuming no clouds or atmospheric absorption. This is known as the
solar constant. ComputeE ̃andB ̃, and compare with the strengths
of static fields you experience in everyday life: E∼ 106 V/m in a
thunderstorm, andB∼ 10 −^3 T for the Earth’s magnetic field.√42 The circular parallel-plate capacitor shown in the figure is
being charged up over time, with the voltage difference across the
plates varying asV =st, wheresis a constant. The plates have
radiusb, and the distance between them isd. We assumed b,
so that the electric field between the plates is uniform, and parallel
to the axis. Find the induced magnetic field at a point between the
plates, at a distanceRfrom the axis. .Hint, p. 1033√43 A positively charged particle is released from rest at the origin
att= 0, in a region of vacuum through which an electromagnetic
wave is passing. The particle accelerates in response to the wave.
In this region of space, the wave varies asE = xˆE ̃sinωt,B =
yˆB ̃sinωt, and we assume that the particle has a relatively large
value ofm/q, so that its response to the wave is sluggish, and it
never ends up moving at any speed comparable to the speed of light.
Therefore we don’t have to worry about the spatial variation of the
wave; we can just imagine that these are uniform fields imposed by
some external mechanism on this region of space.
(a) Find the particle’s coordinates as functions of time.√(b) Show that the motion is confined to−zmax≤z≤zmax, where
zmax= 1.101(
q^2 E ̃B/m ̃^2 ω^3)
.
754 Chapter 11 Electromagnetism