Calculating Electromagnetic Field Energy Flux in Linear Medium

[cepheid]

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?

 

[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.

 

[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.

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

~

 

[cepheid]

Thanks for all your help!

 

[dextercioby]

Are epsilon and mu constants??In the general case,they aren't,but i assume the problem asked for the simplest of them all.

Daniel.