Getting Magnetic Field from Electric Field

[schrodingerscat11]

Homework Statement

Hi! There is a formula in Griffiths that relates the electric field and the magnetic field:
\mathbf{\tilde{B}}(\mathbf{r},t)=\frac{1}{c}\mathbf{\hat{k}}\times\mathbf{\tilde{E}}

Question: Is this formula applicable for plane waves only? Can I use this for spherical waves? Thank you very much.

Homework Equations

See above.

The Attempt at a Solution

I was told this is just for plane waves, but I want to be sure. Thank you! :)

 

[DEvens]

I know what the k vector is for a plane wave. And with a little re-reading notes on this stuff I could reproduce the construction that gets you your relation between B, K, and E for a plane wave.

What would be the equivalent of a k vector for a spherical wave? Can you write down the E and B as functions of position and time for a spherical wave? It's going to have to be some kind of function of radius. How can you write something that looks like the k dot E or k dot B in a plane wave but that gives you something that is a function of radius? Hint: It's going to involve the radius vector. Can you demonstrate your solution obeys Maxwell's equations? Does your solution have a k vector? Hint: plane waves have a single axis of travel and so there is a single k vector for all positions and time. That clearly can't be true for a spherical wave.

And, having done all of that, can you then write k x E and compare it to B?