lightarrow, I am aware of the mathematical formulation of the two laws that I referred to. They mean exactly what I said, except that I didn't mention that the fields that these changes in E or B create are in a direction perpindicular to the direction of the wave's propagation (if the left was nabla•**E** or nabla•**B** then the fields generated by the changing fluxes would be in the same direction as c). I took it as a given that the fields were perpindicular to the direction of propagation as light is a transverse wave. I know that your calculations are correct (they appear many different places and I have taken the time to check the math, assuming that rot(rot**V**))=grad(div**V**)-nabla^{2}**V** is true, they work out), my problem is that this solution seems to be just as infallible as yours, yet only one can be right.

Does anybody see what's wrong here?

Let's take Faraday's Law:

rot

**E** = -∂

**B**/∂t

integrating on a surface σ:

∫rot

**E**•d

**σ** = -(∂/∂t)∫

**B**•d

**σ**using Stokes theorem:

∫

**E**•d

**l** = -(∂/∂t)Φ

_{B}where: the first integral is computed on the edge of the surface σ, d

**σ** =

**N**dσ where

**N** is the versor of the element of area dσ and d

**l** is the line element of that edge; Φ

_{B} is the total flux of

**B** through σ.

Now let's take as σ a disk with radius r, so the edge is the circle of radius r;

if **B** is uniform through the disk and perpendicular to it, for cylindrical symmetry the field

**E** in the circle is tangent to it and with uniform intensity E and we can write:

2πrE = -πr

^{2}∂B/∂t -->

--> E = -(r/2)∂B/∂t

For a sinusoidal time-varying B, that is, B = B

_{0}sin(ωt) we have:

E = -(r/2)ωB

_{0}cos(ωt)

and so E is 90° out of phase in time, with respect to B, as you say.

Note however the key hypotesis:

if **B** is uniform through the disk; in our case instead,

**B** is not uniform (it varies sinusoidally also with space and not only with time), so we have to use Faraday's Law in the differential form:

rot

**E** = - ∂

**B**/∂t

and so we have to compute also the spatial derivatives of

**E**; for this reason the "i" factor comes in both members of that equation and so the phase is the same (I remind you that a factor "i" equals a 90° phase difference: i = e

^{iπ/2}).

However, in the post where I computed

**E** and

**B** for an EM wave, I should have said that I took a

*plane polarized wave* (because that's more simple); I'm not sure of what could come out if the wave wouldn't be that way, actually.

About:

rot(rot

**V**))=grad(div

**V**)-nabla

^{2}**V**it's not very simple to prove it, but it's a well known mathematical rule; you can find it in (good) electrodynamics books.