W9_parallel_resonance.eps

342 Week 10: Maxwell’s Equations and Light

• Physicists usually rearrange them to make the equations connecting fields tosources
  stand out from the equations that have no source terms (because we have yet to see
  a magnetic monopole):
  ∮

S

E~·nˆdA =^1

ǫ 0

∫

V/S

ρedV (779)

∮

C

B~·d~ℓ−d

dt

μ 0 ǫ 0

∫

S/C

E~·nˆdA = μ 0

∫

S/C

J~·nˆdA (780)

∮

S

B~·nˆdA = 0 (781)

∮

C

E~·d~ℓ+d

dt

∫

S/C

B~·nˆdA = 0 (782)

This way, the symmetryis compelling!Two inhomogeneous equations have source

terms connected to electric charge, two homogeneous equationshave thesame form

but lack the source terms, at least until monopoles are discovered.

• If one applies these equations to asource-free volume of spacewhere electric and
  magnetic fields are varying, one can show that they lead to the followingwave
  equationsfor theelectromagnetic fieldpropagating in (say) thez-direction:

∂^2 E~

∂z^2 −

1

c^2

∂^2 E~

∂t^2 = 0 (783)

∂^2 B~

∂z^2

−

1

c^2

∂^2 B~

∂t^2

= 0 (784)

The ∂

2

∂z^2 symbol in this expression, let me remind you, just means to take the

derivative of the functionsE~(~x, t) andB~(~x, t) with respect to thez-coordinate

only, pretending that the other coordinates are constants. In this equation,

c=

√

ke

km=

1

√ǫ

0 μ 0

= 3× 108 meters per second (785)

is thespeed of light in a vacuum, which we can see iscompletely determinedfrom

Maxwell’s equations.

Since Maxwell’s equations are laws of nature and expected to hold in allinertial

reference frames, it is entirelyreasonableto expect the speed of light to be constant

in all reference frames! This postulate, together with some very simple assumptions

about coordinate transformations, suffices to derive the theoryof relativity!

• We will study the details of at least certain simple solutions to these wave equations
  over the next few weeks. For the moment, the most important solution for you to
  learn is:

Ex(z, t) = E 0 xsin(kz−ωt) (786)

By(z, t) = B 0 ysin(kz−ωt) (787)

known as aharmonic plane wavetravelling in thez-direction. Note thatExand

Byarein phaseand do not have independent amplitudes – their amplitudes are

connected by Maxwell’s equations (Faraday or Ampere’s law) andEx=cBy. There

is an identical pair of solutions with a differentpolarization:

Ey(z, t) = E 0 ysin(kz−ωt) (788)

Bx(z, t) = −B 0 xsin(kz−ωt) (789)

that also propagate in thez-direction, as determined from the derivation of the

wave equations above.