How can I use vector identities to prove and explain the Ponyting theorem?

[the_godfather]

Homework Statement

Prove and explain the Ponyting theorem

Homework 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]

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/$\partial$tbrowsing 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

 

[G01]

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

Use the chain rule to expand

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

 

[the_godfather]

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

not sure why you need to chain rule though?

 

[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.

 

[the_godfather]

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.

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

 

[G01]

No problem! :)