A Variation of Energy for Dielectrics (Zangwill's Electrodynamics)

[pherytic]

Hello PhysicsForums community,

I have been reading through Zangwill's Modern Electrodynamics all on my own, and I've just joined here hoping I can post some questions that come up for me. To start, I am confused about something in section 6.7.1, concerning the variation of total energy U of a dielectric in the presence of a charged conductor. This is given by (6.87)

$$\delta U = \int d^3 r \, \vec E \cdot \delta \vec D$$

where E is the total electric field, D is the auxiliary/displacement field.

Then, the books says (6.93)

$$ \vec E = 1/V(∂U/∂ \vec D)$$

I understand (ignoring any center of mass dependence) that using the logic of total differentials I can write

$$\delta U = (∂U/∂ \vec D) \cdot \delta \vec D$$

So it follows that

$$\int d^3 r \, \vec E \cdot \delta \vec D = (∂U/∂ \vec D) \cdot \delta \vec D$$

But the given equation for E only seems valid if E and D are constant over the volume, which isn't generally true. What am I misunderstanding? How does the equation for E follow?

Thanks for any guidance.

 

[Meir Achuz]

I have looked at my copy of Zangwill. Section 6.7.1 is confused, confusing, and should not be in a textbook.
I have seen simple straightforward derivations of his equation 6.94 in many textbooks. Just look at any other book. Zangwill is not a book you should read or try to understand by yourself.

 

[pherytic]

I have looked at my copy of Zangwill. Section 6.7.1 is confused, confusing, and should not be in a textbook.
I have seen simple straightforward derivations of his equation 6.94 in many textbooks. Just look at any other book. Zangwill is not a book you should read or try to understand by yourself.

I got the opposite advice before I started - that Zangwill was better/clearer than Jackson, and now I am six chapters in (to be fair I can follow ~90% of it without issues).

Also, I was hoping to understand 6.93 (electric field in terms of partial derivative of U) not 6.94.

 

[vanhees71]

Zangwill is not a bad book, but compared to Jackson...