Electromagnetism: Questions About Energy Density

[Gavins]

Hey I am studying electromagnetism at the moment and one problem I'm having is the energy density.
Instead of typing everything out, this pdf has everything. http://www.physics.sfsu.edu/~lea/courses/ugrad/360notes11.pdf There are some things I don't understand. Why is the surface integral 0? He just says that it's 0 for the usual reasons. Then I'm not sure what he's doing with the (deltaD). E thing. I don't understand how he puts the whole thing in the delta and then a half pops out.

Thank you.

 

[Zorba]

I think he is just using the usual product rule to get the half:
$$\Delta (E \cdot E)=\Delta E \cdot E + E \cdot \Delta E = 2(\Delta E\cdot E)$$
$$\Rightarrow \Delta E\cdot E = \frac{1}{2} \Delta (E \cdot E)$$

As for the integral being zero, I'm not sure, perhaps it's zero in the limit.

 

[sigmaro]

You must be having so much fun, i love the book of Griffiths;
now, he used D because he is dealing with dielectric metarial, there you should not consider bounded charges since they are not cause but result.
div(D)=Pf but div(E)=P (here since i don't have rho, P is charge density) if you use E you can not undrestand if the thing you got is either free or bounded.
Zorba is right, USUALLY V is zero at the infinity, so first integral vanishes.
As you continue, you don't have to be afraid of confusing E or D because from now on you won't deal with charge any more, you can convert D to k€E then as you have same variable in delta or d, you can tell d(E^2)=2EdE, or the reverse.
After all if you like it this way you may not touch it and have some k€ there or you can convert it back and have ½ED

 

[klawlor419]

"Why is the surface integral 0? He just says that it's 0 for the usual reasons."

the electric displacement D vector as you go to infinity is proportional to 1/r^2 and the potential is a 1/r. the dA has a r^2 term in it. therefore making the whole term in the integral a 1/r, hence as you go out to infinity(integrating over all of space), the surface integral goes to zero