Induced charg by chagr distribution

[gabrielbhl]

Hi everybody
It's easy (and well known) to calculate the charge distribution of a condutor in the presence of a point chage. BUT if we change this point charge and put an charge distribution... like a condutor disc, how can I calculate the induced chage of the other conductor??

THANKS

 

[gabrielbhl]

Even using the method of image charges is is too dificul to calculate... I have no idea

 

[Born2bwire]

If you have a geometry where you can use image charges then nothing is changed by replacing the point charge with a distribution of charges. Electromagnetics follows linear superposition and since this is a static problem your total solution will be the solution for each individual charge in your distribution. So you can just find the image of the charge distribution and calculate the field from the image.

Of course using images is only limited to a small subset of problems, but the point still stands. If you can solve for a single charge then you can just extend the method for a distribution of charges. The primary difference will lie in that you will have to integrate over the charge distribution (much like you would sum up over a finite set of charges) to get the total field.

This is assuming that your charge distribution will not change in the presence of an applied field. But if you were to charge a conducting disc then the charge distribution on your disc will change due to the interaction of the original charge distribution with the induced charge distribution on your conductor. In this case it is a much more difficult problem. However, you can solve this problem more easily by first noting that a conductor is an equipotential surface. So instead of trying to work from the standpoint of:
I have charge distribution A on a conducting object. I bring in conducting object B. What is the new charge distributions on A and B?

You work from the viewpoint of:
I have an object A with a potential of X. I bring in B. What is the charge distribution on A and B where A has a potential of X and B has a potential of 0 when they are infinitely separated?

You can represent this problem as a differential equation, Laplace's equation, and using the appropriate boundary conditions you can then solve for the potential on the surfaces of the objects. There are some problems where you can solve this analytically (method of images is just an example of this) but sometimes you need to use numerical techniques like method of moments or relaxation methods.

 

[gabrielbhl]

man.. thanks a lot, really, i was freaking out here... I needed to hear (read) it from someone. Thanks!

 

[PJC01]

for your question! This is a great topic to discuss and explore. To answer your question, the induced charge on a conductor due to a charge distribution can be calculated using the principle of superposition. This principle states that the total charge on a conductor is the sum of the charges induced by each individual charge in the distribution.

To calculate the induced charge, we first need to determine the electric field at each point on the conductor's surface due to the charge distribution. This can be done using Coulomb's law or Gauss's law, depending on the distribution's symmetry. Once we have the electric field, we can use the fact that the induced charge is proportional to the electric field at that point. This means that the greater the electric field, the greater the induced charge will be.

In the case of a conductor disc, we can use the same approach. We would first determine the electric field at each point on the disc's surface due to the charge distribution. Then, we would use the proportionality between the electric field and induced charge to calculate the total induced charge on the disc.

It's important to note that the induced charge on a conductor will always be on its surface and will redistribute itself in a way that minimizes the total energy of the system. This is known as the principle of minimum energy. So, even if the charge distribution is changed, the induced charge on the conductor will always adjust to minimize the total energy.

I hope this helps answer your question and sparks further interest in the fascinating world of induced charges and charge distributions. Keep exploring and asking questions!