# Induced charge density -- non-zero potential case

• mertcan
In summary, the conversation discusses the challenges of finding induced charge density over surfaces of conductors with non-zero potential, as opposed to the more commonly seen grounded potential case. It is mentioned that this can be solved analytically in some simple cases, but for more complex shapes, Finite Element Analysis or similar methods must be used. The speaker is asking for examples and solutions related to calculating charge density using Finite Element Analysis.
mertcan
Hi, Let's think 2 arbitrary shape conductors with non zero charged. If these 2 conductors are closed, there will be induced charge density over surfaces of these conductors. I have not seen such an example, instead there are lots of problems which involve zero(grounded) potential case and method of images theorem is applied. So, I am asking how can we mathematically find induced charge density over conductors' surfaces including non zero potential case(not grounded)?
Thanks...

Last edited by a moderator:
Finding charge density distribution involves solving Poisson's equation ## \nabla ^2 U = - \frac {\rho}{\epsilon \epsilon_0} ##.
This equation can be solved analytically only in some simple cases when the geometry is rather regular (e.g. planar, cylindrical, spherical, etc.) and these you find in textbooks. In other cases, you can get a power (or other) series expansion. But if you want to solve the Poisson's equation for a completely arbitrary shapes (conductors, dielectrics) you have to use Finite Element Analysis or similar methods.

Then, please let me ask the following question: I am really looking for that kind of examples, could you tell me how I can find some examples and solutions related to calculating charge density with finite element analysis ?

## What is induced charge density in the non-zero potential case?

Induced charge density refers to the redistribution of electric charge on the surface of a conductor in response to an external electric field. In the non-zero potential case, the conductor is not at equilibrium and the charge density is not uniform.

## How is the induced charge density related to the electric potential?

In the non-zero potential case, the induced charge density is directly proportional to the electric potential. This means that as the electric potential increases, the induced charge density also increases.

## What factors affect the magnitude of the induced charge density?

The magnitude of the induced charge density depends on the electric field strength, the size and shape of the conductor, and the material properties of the conductor. A higher electric field strength, larger conductor, and more conductive material will result in a higher induced charge density.

## What is the relationship between the induced charge density and the electric field inside the conductor?

In the non-zero potential case, the electric field inside the conductor is directly proportional to the induced charge density. This means that as the induced charge density increases, the electric field inside the conductor also increases.

## How is the concept of induced charge density used in practical applications?

Induced charge density is an important concept in understanding the behavior of conductors in electric fields. It is used in various practical applications such as capacitors, electrostatic shielding, and the operation of electronic devices.

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