# Proof of Ponytings theorem

1. Jan 15, 2013

### the_godfather

1. The problem statement, all variables and given/known data

Prove and explain the Ponyting theorem

2. Relevant equations

S = E x H [1]
$\nabla$.S = $\nabla$.(E x H) [2]

$\nabla$ x E = -$\partial$B/$\partial$t [3]

$\nabla$ x H = $\partial$ D/$\partial$t [4]

D = $\epsilon$E + P [5]

B = $\mu$H + $\mu$M [6]

3. The attempt at a solution

I understand I am to use the vector identity to obtain [2]

$\nabla$ . ( E x H ) = ($\nabla$ E).H - ($\nabla$ x H ). E

I then substitute [3] and [4] into [2]

I then use the definition from [5] and [6] and sub them into my equation

I have:

$\nabla$ . S = -$\partial$[$\mu$H + $\mu$M].H/$\partial$t - $\partial$[$\epsilon$E + P].E /$\partial$t

i can then expand out the bracket but I'm not sure what to do next

The result i'm aiming for is

$\nabla$.S = -$\partial$/dt ( 1/2 $\epsilon$E^2 + 1/2$\mu$ H^2) + E.$\partial$P/$\partial$t + $\mu$H.$\partial$M/$\partial$t

browsing google i have found this. 8.87 and 8.88 will give me the answer I want but I'm unfamiliar with them. could anyone shed some light on this

Last edited: Jan 15, 2013
2. Jan 15, 2013

### G01

Your almost there. Here' a hint to shed light on those equations:

Use the chain rule to expand

$$\frac{1}{2}\frac{\partial H^2}{\partial t} = ?$$

3. Jan 15, 2013

### the_godfather

$\frac{2}{2} H . \frac{\partial H} {\partial t}$

not sure why you need to chain rule though?

4. Jan 15, 2013

### G01

The chain rule is where the justification for EQ 8.87 and 8.88 comes from. You should be able to use the chain rule to get the E^2 and H^2 terms in Poynting's theorem from terms like E*dE/dt.

5. Jan 15, 2013

### the_godfather

apologies. Was being dumb for a moment there. I believe I have it now. thanks

6. Jan 16, 2013

### G01

No problem! :)