W9_parallel_resonance.eps

Week 10: Maxwell’s Equations and Light 357

If we substitute the second equation in for the last term, we get:

d^2 Ex

dz^2

=−

d

dt

μ 0 ǫ 0

dEx

dt

=μ 0 ǫ 0

d^2 Ex

dt^2

(853)

or

d^2 Ex

dz^2 −μ^0 ǫ^0

d^2 Ex

dt^2 = 0 (854)

We stare at this for a moment, our brains dulled by too much algebra.Then, through

the fog, alightbegins to shine through, dim at first, then ever brighter until it rivals the

sun! Holy Smoke, Batman, haven’t we seen that equation, or one sort of like it, before?

Wehave! In the first part of the course we went to considerable (although much less)

pains to derive the one-dimensionalwave equation for a string:

d^2 y(x, t)

dx^2 −

1

v^2

d^2 y(x, t)

dt^2 = 0 (855)

for ay-displaced string, where the wave propagated at speedvin the±xdirection! Well,

it seems that Maxwell’s Equations tell us that thex-component of the electric field in

a region of space far from any sources satisfies a wave equation too! I wonder (you ask

yourself) what thespeedof this wave is?

Well, comparing the two equations, we see that:

v^2 =

1

μ 0 ǫ 0

=

4 π

μ 04 πǫ 0

=

4 π

μ 0

1

4 πǫ 0

=

ke

km

(856)

and if we do only atinybit of arithmetic with the only two constants I really required

you to memorize/learn for this part of the class we get:

v^2 =

9 × 109

10 −^7

= 9× 1016

meters^2

second^2

(857)

or:

v=c= 3× 108

meters

second. (858)

This particular speed was first estimated during the very first daysof systematic scientific

exploration based on observations of variations in the period of oneof Jupiter’s moons.

It was known within a few percent by the mid-1800s, and experiments were being done

that were rapidly adding significant digits to the quantity (it is currently one of the most

accurately known physical constants). This quantity is thespeed of light.

The electric field wave propagates at thespeed of light!

Andthat, boys and girls, is why Maxwell got his name on thewhole setof Maxwell’s

Equations for his one measely term. He proposed (correctly) thatlight is an electro-

magnetic waveand in so doing, transformed the still partially disparate electric and

magnetic fields into a single unified field theory and revolutionized our understanding of,

well,everything. You. Me. Stuff. Whatisn’tmade up of electric charges anddoesn’t

interact via the electromagnetic interaction^102?

Well, we haven’t quite shownallof that yet. But now you can see how it goes well

enough to complete most of what we sill need to do even without my help. If we take the

secondof the two equations (Ampere’s Law) and differentiate both sides with respect to

zand substitute in the first (Faraday’s Law) for the right hand side we get:

d^2 By

dz^2

−μ 0 ǫ 0

d^2 By

dt^2

=

d^2 By

dz^2

−

1

c^2

d^2 By

dt^2

= 0 (859)

(^102) The correct answer: not much...