# EM Problem

1. Feb 10, 2005

### cepheid

Staff Emeritus
Consider the electromagnetic field in a linear medium with material properties $\epsilon \ \ \text{and} \ \ \mu$. Calculate $\nabla \cdot \mathbf{S}$ for the energy flux $\mathbf{S = E \times H}$.

My work:

$$\nabla \cdot \mathbf{S} = \nabla \cdot \frac{1}{\mu}(\mathbf{E \times B })$$

$$= \frac{1}{\mu}[ \nabla \cdot(\mathbf{E \times B })]$$

$$= \frac{1}{\mu}[\mathbf{B} \cdot(\nabla \times \mathbf{E} }) - \mathbf{E} \cdot(\nabla \times \mathbf{B} }) ]$$

$$= \frac{1}{\mu}[\mathbf{B} \cdot(-\frac{\partial \mathbf{B}}{ \partial t}) - \mathbf{E} \cdot(\mu\epsilon\frac{\partial \mathbf{E}}{ \partial t})]$$

I guess my question is: is this the result? I have no idea what this problem wants. Are there at least any more immediately obvious simplifications?

2. Feb 10, 2005

### s_a

Almost there. Remember E . dE/dt = 1/2 d/dt |E|^2 and B . dB/dt = 1/2 d/dt |B|^2 (sorry I don't know Latex). Use these results in your last equation, and not surprisingly you should get the expression for the power per unit volume of the electric and magnetic fields.

3. Feb 10, 2005

### xanthym

That's the classic result for Del*S. A next step (in another assignment) would be derivation of the total flux of S through a closed surface enclosing charges and currents. That would utilize Gauss's Divergence theorem with your present result and other Maxwell's Equations.

$$\int\limits_{Surf}^{} \vec{S} * \vec{n} dA = \int\limits_{Vol}^{} \nabla*\vec{S} dV$$

~

Last edited: Feb 10, 2005
4. Feb 11, 2005

### cepheid

Staff Emeritus