W9_parallel_resonance.eps

360 Week 10: Maxwell’s Equations and Light

In order for this to be true,φ= 0 – the electric and magnetic fieldsdohave to have the

same phase (and frequency and wavelength) and we have nowproventhis, and:

E 0 x=cB 0 y (875)

The electric and magnetic fields are not independent! The magnitude, phase, and fre-

quency of one is determined completely by the other.

This is a wave propagating to theright, as noted. Let’s try the exact same solution for

the independent solution:

Ey(z, t) = E 0 ysin(kz−ωt) (876)

Bx(z, t) = B 0 xsin(kz−ωt+φ) (877)

Note that we have assumed nothing other thanEyis coupled toBx(because that’s what

Ampere/Faraday tell us). Again we substitute – using the form of Faraday’s Law we

derived forEy– and get:

d

dzE^0 ysin(kz−ωt) =

d

dtB^0 xsin(kz−ωt+φ) (878)

kE 0 ysin(kz−ωt) = −ωB 0 xsin(kz−ωt+φ) (879)

E 0 ysin(kz−ωt) = −

ω

k

B 0 xsin(kz−ωt+φ) (880)

E 0 ysin(kz−ωt) = −cB 0 xsin(kz−ωt+φ) (881)

This time we see that the two fields must be in phase and that:

E 0 y=−cB 0 x (882)

For a wave propagating to the right,bothof the independent components ofE~are related

to the coupled components ofB~such that:

|vE|=c|B~| (883)

and so that the E-fieldcrossed into the B-field points in the direction of the wave’s

propagation. That is, if we let the fingers of our right hand line up withE~and curl them

intoB~, our thumb points in the direction of propagation. This also works for waves

propagating in the−xdirection, e.g.E 0 xsin(kz+ωt) (try it!).

10.3: The Poynting Vector

OK, so now we have the harmonic electric and magnetic field, and bothare in phase

and have amplitudes related byc. We know that there is someenergyin these fields

described by theenergy densityof the electric and magnetic fields respectively:

ηe =

1

2

ǫ 0 E^2 (884)

ηm =

1

2 μ 0

B^2 (885)

Now, however, that energy isn’t sitting still. It ismoving, being carried by the wave

from one point to another. We can easily see that energy must be carried by the wave

by imagining a source that is turned on (the dipole moment is pulled out and released to

oscillate, if you like) at timet= 0. Some distance away from the source at first there is

no field – our “Lorentz model” atom was spherically symmetric and produced no field –

and then the fieldreachesit some time after the dipole is excited and starts to oscillate.