# Jackson 1.12 & 1.13: Prove Thomson's & Insulated Conductor Theorem

• nctweg
In summary, the theorem is saying that if a number of conducting surfaces are fixed in position and a given total charge is placed on each surface, then the electrostatic energy in the region bounded by the surfaces is a minimum. The theorem also says that if an uncharged, insulated conductor is introduced into the region bounded by the surfaces, the electrostatic energy is lowered.
nctweg

## Homework Statement

1.12 -
Prove Thomson's theorem : If a number of conducting surfaces are fixed in
position and a given total charge is placed on each surface, then the electrostatic
energy in the region bounded by the surfaces is a minimum when the
charges are placed so that every surface is an equipotential.

1.13 -
Prove the following theorem: If a number of conducting surfaces are
fixed in position with a given total charge on each, the introduction of an
uncharged, insulated conductor into the region bounded by the surfaces
lowers the electrostatic energy.

## The Attempt at a Solution

So I haven't actually started them yet because I don't quite understand the geometry I'm being asked about. Note - I am not really looking for help on how to do the problems (at least not yet, I want to give it at least a week before giving in).

My questions are simple and perhaps dumb; are these conducting surfaces connected? It says that there is a region bounded by the surfaces but if that were the case and they were conducting, wouldn't they just end up forming some kind of closed shape with the charge spread throughout (rather than, as the problem indicates, each surface having Q total charge). My other thought is that Jackson's just referring to a general region that's between the surfaces but not necessarily closed but I really can't tell which is what the question is asking about.

Also in problem 1.13, is the insulated conductor a surface or does it have volume? (Or does it not make a difference?). Are we just approximating the insulation as being thin enough that it makes no difference?

Thanks for the help!

The surfaces are not connected.

nctweg
Okay, thanks for confirming my stupid question. I figured as much but I did want to check first.

## 1. How did Jackson prove Thomson's theorem?

Jackson used mathematical techniques to derive the equations for electric and magnetic fields in an insulating material, and showed that the ratio of the electric and magnetic fields is equal to the speed of light squared. This is known as Thomson's theorem.

## 2. What is the significance of Thomson's theorem?

Thomson's theorem is significant because it provides a mathematical relationship between the electric and magnetic fields in an insulating material. It also helps to understand the behavior of electromagnetic waves in insulating materials, such as how they propagate and interact with each other.

## 3. How does the Insulated Conductor Theorem relate to Thomson's theorem?

The Insulated Conductor Theorem is an extension of Thomson's theorem and applies to the case of a conductor surrounded by an insulating material. It states that the electric field inside a conductor is zero, and the ratio of the electric and magnetic fields is equal to the speed of light squared.

## 4. What is the practical application of Thomson's and Insulated Conductor Theorem?

Thomson's and Insulated Conductor Theorem have many practical applications in the study of electromagnetism and the design of electrical systems. They are used to analyze the behavior of electromagnetic waves in insulating materials and to calculate the electric and magnetic fields in various situations.

## 5. Are Thomson's and Insulated Conductor Theorem still valid today?

Yes, Thomson's and Insulated Conductor Theorem are still valid today and have been extensively tested and validated through experiments and observations. They are fundamental principles in the study of electromagnetism and are used in various fields such as physics, engineering, and telecommunications.

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