W9_parallel_resonance.eps

Week 10: Maxwell’s Equations and Light 343

In these equations, note well that:

k=

2 π

λ (790)

is thewave numberof the wave, whereλis thewavelengthof the harmonic wave,

while:

ω=^2 π

T

(791)

is theangular frequencyof the wave. The wavelength is thus the “spatial period” of

the wave, whereTis the “temporal period” of the wave that harmonically oscillates

in space and time. This wave propagates in thepositivez-direction as can be seen

by consideringkz−ωt=k(z−ωkt) =k(z−ct). Note well that this uses the result

that:

c=λ

T

=ω

k

(792)

for a harmonic wave.

• The flow of energy in an electromagnetic wave (and field in general) can be deter-
  mined from thePoynting vector:

S~=^1

μ 0

(E~×B~) (793)

The magnitude of the Poynting vector is called theintensityof the electromag-

netic wave – the energy per unit area per unit time or power per unit area being

transported by the wave in the direction of its motion:

I=dP

dA

= d

dA

dU

dt

=|S| (794)

whereUis the energy in the wave. To speak more mathematically precisely to

communicate the transport ofpower(energy per unit time, in watts) across some

given surfaceA, one evaluates theflux of the Poynting vector through the surface:

PA=

∫

A

S~·ˆndA (795)

As you can see one just cannot get away from flux integrals as a wayof representing

the “flow” of energy, current, fluid, orE~orB~field through a surface! As such, it

is a very important idea to conceptually master.

• The Poynting vector can be understood andalmostderived by adding up the total
  energy in the electric and magnetic fields in a volume of space being transported
  perpendicular to a surfaceA. In a time ∆t, all of the energy in a volume ∆V=
  A c∆tgoes through the surface at the end. This is:

∆U= (

1

2 ǫ^0 E

x^2 +^1

2 μ 0 B

(^2) y)A c∆t (796)
If we use|Ex|=c|By|(see above) for a wave travelling in thez-direction and do a
bit of algebra, we can see that:
∆U
A∆t=

1

μ 0 |

E~x||B~y| (797)

which is just the Poynting vector magnitude in thez-direction for these two field

components.