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

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• pherytic
In summary: If you just want to understand the basic ideas, Jackson is a much better choice. 6.93 is covered in most textbooks.
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.

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.

vanhees71
Meir Achuz said:
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.

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

## 1. What is the difference between the energy variation for dielectrics and that for conductors?

The energy variation for dielectrics, as described by Zangwill's Electrodynamics, takes into account the polarization of the material, while the energy variation for conductors only considers the free charges on the surface of the material. This means that the energy variation for dielectrics is dependent on the electric field within the material, while the energy variation for conductors is not.

## 2. How does the energy variation for dielectrics affect the behavior of electric fields?

The energy variation for dielectrics introduces a term known as the electric displacement field, which is related to the polarization of the material. This displacement field can alter the behavior of electric fields, causing them to be weaker or stronger depending on the material's polarization.

## 3. Can the energy variation for dielectrics be used to explain the behavior of insulating materials?

Yes, the energy variation for dielectrics can be used to explain the behavior of insulating materials. Insulators are materials that have a high resistance to the flow of electric current and are often used as dielectrics. By considering the polarization and electric displacement field, the energy variation for dielectrics can accurately describe the behavior of insulating materials.

## 4. How does the energy variation for dielectrics relate to the concept of capacitance?

The energy variation for dielectrics is directly related to capacitance, which is a measure of a material's ability to store electric charge. The electric displacement field, which is a key component of the energy variation for dielectrics, is directly proportional to the capacitance of a material. This means that materials with a higher capacitance will have a larger electric displacement field and vice versa.

## 5. Are there any practical applications of the energy variation for dielectrics?

Yes, the energy variation for dielectrics has many practical applications, particularly in the field of electrical engineering. It is used to design and analyze capacitors, insulators, and other electronic components. It is also essential in understanding the behavior of materials in the presence of electric fields, which is crucial in many technological applications such as semiconductors and solar cells.

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