Electrostatics: Surface charge density-numerical solution

In summary, the conversation discusses the problem of numerically calculating an electrostatic problem involving a disk at potential V0. The equation for the problem is given and it is mentioned that it can be solved using matrix algebra techniques. To solve it numerically, the equation needs to be discretized and a numerical integration technique can be used. The A_{ii} terms are referred to as self-potential terms and can be assumed to be constant.
  • #1
juvan
14
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Hi, I'm very new here, 10min old, but the problem with my knowledge, or better, lack there of, at this time is very hard, so I need help. I am trying to numerically calculate a certain electrostatic problem (attached an image "prob.jpg" to clarify). I have a disk at potential V0, and with this equation

[itex]V(\rho,\varphi)=\frac{1}{4\pi\epsilon_0}\int\int \frac{\sigma(\rho',\varphi')\rho'd\rho'd\varphi'}{ \sqrt{\rho^2+\rho'^2 - 2\rho\rho'cos(\varphi-\varphi')}} (1)[/itex]
I'm trying to then numerically calculate for σ (attached 2 images "disk.png","charge_on_disk.png" to clarify), this integral equation. So I basically get an equation A*σ=b (A-matrix, σ,b-vectors). As for [itex] A_{ij} [/itex] where [itex]i \neq j[/itex], i think there should be no problem with equation (1), but what about [itex]A_{ii}[/itex], what is.. or how should I set up the equation for these terms, since if I used the same equation (1), I would get [itex]\frac{something}{0}[/itex]. I have no idea where to even start the thought process, to go about setting up the equation.
For the sake of being brief, I will stop, and ask that if I have made anything unclear just say so and I will try to explain further(better).

Thank you.

EDIT: I tried setting all the terms to 1 [itex]A_{ii}=1[/itex] and the resulting graph seems correct, but that's just guess work, I have no basis for setting it to a constant 1.
 

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  • #2
To answer your question, the equation you are trying to solve is called the integral equation of the electrostatic potential. You can find more information about this equation and how to solve it in many textbooks or online resources. Generally speaking, the equation you have written down is a linear equation and can be solved using matrix algebra techniques. In order to solve this equation numerically, you need to discretize the integral equation into a finite number of points. This means that you need to divide the disk into a certain number of equal elements, and at each element calculate the potential. To do this, you can use a numerical integration technique, such as the trapezoidal rule or Simpson's rule. This will give you an approximation of the integral equation, and you can then solve it with a matrix equation. The A_{ii} terms are sometimes referred to as the self-potential terms, and they represent the contribution of the point itself to the total potential. In most cases, these terms can be assumed to be constant, with the value being equal to the given potential at that point. Hope this helps!
 

1. What is surface charge density in electrostatics?

Surface charge density is a measure of the amount of electric charge per unit area on the surface of a conductor or dielectric material. It is denoted by the symbol σ and is expressed in units of coulombs per square meter (C/m^2).

2. How is surface charge density calculated?

The surface charge density can be calculated by dividing the total amount of charge on the surface by the surface area. It can also be calculated by taking the limit of the charge density as the surface area approaches zero.

3. What is the significance of surface charge density in electrostatics?

Surface charge density plays a critical role in understanding the behavior of electric fields and the interaction between charged objects. It helps determine the strength and direction of electric fields on the surface of a material, as well as the forces between charged objects.

4. What factors affect the surface charge density?

The surface charge density is affected by the amount of charge present, the surface area of the material, and the dielectric constant of the material. It can also be influenced by the presence of other nearby charged objects.

5. How is the numerical solution of surface charge density used in practical applications?

The numerical solution of surface charge density is used in various practical applications, such as designing electronic circuits, analyzing the behavior of capacitors and other electrical devices, and studying the effects of lightning strikes on structures. It is also used in the development of anti-static materials and coatings for electronic equipment.

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