E=cB and electromagnetic energy density

1. Feb 14, 2015

davidbenari

I have to questions that I cant seem to understand.

I've seen in my textbook the relation E=cB being deduced from Faradays law applied to some type of rectangular loop. The problem is this is done with regards to a planar wave. So is this relationship general for all EM waves? If so, how? And could someone show me a simple way to prove this to myself? If the velocity of a wave is not c but some other (because of refraction or whatever) then is the relation E=vB?

Another question I have is about electromagnetic energy density. Is the formula for EM energy density $U=\frac{\epsilon E^2}{2} + \frac{B^2}{2\mu}$ applied using the values for the magnetic and electric fields at a specific point? That is: similar to a problem with mass density, we assume small infinitesimal volumes in the neighborhood of a plane having the same energy density? (since the E and B values all have the same value at those points)

2. Feb 14, 2015

Staff: Mentor

Planar waves satisfy the maxwell equations only if that relation is true (with magnitudes, not with vectors).

Energy density in "physical" settings is continuous, so it does not vary wildly between points nearby, yes.

3. Feb 14, 2015

davidbenari

I know how to derive E=cB for planar waves, but I was wondering why this relationship holds true for sinusoidal waves too?

Also, if the energy density is $u=\frac{\epsilon E^2}{2} + \frac{B^2}{2\mu}$ could I say that the average energy density is $u_{average}=\frac{\epsilon E^2}{4} + \frac{B^2}{4\mu}$ taking averages of the cosine^2 functions?

Thanks.

4. Feb 15, 2015

Staff: Mentor

I would be surprised if it is true for every point in space for every vacuum solution.

At least for planar waves this is possible if you replace E and B by their peak values.

5. Feb 15, 2015

davidbenari

Can I apply the derivation I've seen of E=cB for planar waves to sinusoidal waves too?

Namely: Imagine a rectangular loop with its vertical side $a$ parallel to the polarisation of the electric field (the B field is therefore creating flux across this loop). I place this rectangular loop in such a way that half of it is immersed in the field, and the other half in pure vacuum since the wave hasn't reach there yet. Both halves have length $cdt$

Applying faradays law we say the line integral yields $-Ea$ and $\frac{d\Phi}{dt}$ is $frac{(B+dB)acdt}{dt}$ which is just $Bacdt$ ignoring the infinitesimal element. Therefore we obtain again E=cB.

Can I do this? Ignore the infinitesimal element?

6. Feb 15, 2015

davidbenari

I have the feeling I don't really understand what a plane wave is. Wikipedia includes sinusoidal waves too. I thought plane waves were this gigantic constant phase planes that travel in some direction…

7. Feb 25, 2015

tech99

8. Feb 25, 2015

tech99

If you want to work out energy density over the duration of a sine wave you should use the first formula using the RMS values of the fields, as in normal engineering practice.

9. Feb 26, 2015

Khashishi

E = cB is only true for some special cases. It certainly isn't true in general. (Consider two vertically polarized beams crossing each other at right angles.)
$U=\frac{\epsilon E^2}{2} + \frac{B^2}{2\mu}$ is true much more generally, for arbitrary fields which contain both propagating waves and static fields which surround sources.