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

 

[paranormal]

I can understand your confusion about energy density in electromagnetism. I took a look at the pdf you provided and I can provide some clarification.

Firstly, the surface integral is 0 because the surface being integrated over is a closed surface, enclosing a volume. This means that any energy entering the volume must also exit the volume, resulting in a net energy of 0. This is a fundamental principle in electromagnetism known as the divergence theorem.

Regarding the (deltaD). E term, this is a mathematical manipulation used to simplify the equation and make it easier to solve. The author is essentially taking the derivative of the electric displacement field (D) with respect to the electric field (E), which is a common technique in electromagnetism.

As for the "half" that pops out, this is simply a mathematical constant that arises from the integration process. It is a result of the integration limits and the shape of the surface being integrated over.

I hope this helps to clarify some of your questions about energy density in electromagnetism. Keep studying and don't be afraid to ask for further clarification if needed. Understanding these concepts is crucial for a deeper understanding of electromagnetism. Good luck with your studies!