W9_parallel_resonance.eps

Week 10: Maxwell’s Equations and Light 365

and

∂Ey

∂z

= ∂Bx

∂t

(901)

∂Bx

∂z

= μ 0 ǫ 0

∂Ey

∂t

(902)

and from this show that (Ex, By) and (Ey, Bx) both satisfy the wave equation for a

z-directed wave.

Problem 3.

Show thatf(z±vt) satisfies the wave equation:

∂^2 f

∂x^2

−

1

v^2

∂^2 f

∂t^2

(903)

Show (by drawing appropriate pictures that convinceyouthat it is true so that you

understandit) that these are left and right propagating waves respectively.

Finally show thatF 0 cos(kz±ωt) is a function that has this form, so that harmonic

travelling waves manifestly satisfy the wave equation!

Problem 4.

A

R

Sun

Payload Light Sail

Some science fiction stories, notably ones by Larry Niven, portrayspace travel around the

solar system occurring with no expenditure of reaction fuel using alight sail. A light sail

is an enormous, extremely thin, perfectly reflecting mirror arranged like a parachute so

that it can ”lift” a payload/space capsule attached to the sail by shroud lines. Radiation

pressure from sunlight exerts a force on the sail sufficient to lift the mass directly out

from the sun, and by altering the angle of the sail one can ”tack” in arbitrary directions.

This problem analyzes the plausibility of this proposal. Start by computing the force

exerted by sunlight on a perfectly reflecting sail at normal incidence a distanceRaway

from the center of the sun. Note well that a reflecting sail will exerttwicethe force that

an absorptive sail would (why?). Next, make a reasonable assumption for the density of

the sail material and compute the maximum thickness of a sheet of itthat is capable of

lifting its own weight against the gravitational pull of the sun. Using this information,

youdecide if the idea of sailing directly away from the sun (with or without apayload)

is plausible. Does your answer depend on how far away from the sun you are?

Of course, this simple no-orbit radial model is naive. In reality, the starting and ending

point of any journey areorbitsaround the sun; a payload won’t fall into the sun even if

it has no light sail at all as long as it is in a solar orbit, and one has to do a lot of work

on a mass to take itoutof a solar orbit if it starts in one.

In general, to go from one orbit to another, it suffices toadd energy(and angular momen-

tum in the proper measure) to the orbiting object (or take them away, of course) in the

correct direction using an angled light sail. Making any assumptions that you like, make

an argument for or against light sails as a means of moving a significantpayload mass