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 = -\partialB/\partialt [3]

\nabla x H = \partial D/\partialt [4]

D = \epsilonE + P [5]

B = \muH + \muM [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[\muH + \muM].H/\partialt - \partial[\epsilonE + P].E /\partialt

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 \epsilonE^2 + 1/2\mu H^2) + E.\partialP/\partialt + \muH.\partialM/\partialtbrowsing 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! :)