Methods of Images between two parallel cylinder

[flux!]

Homework Statement

Using methods of Images, How do I find the potential of the system consisting of two conducting cylinders that are not grounded and separated at a distance of 2D, one in a potential of $V_0$ and the other is $-V_0$?

Homework Equations

The potential due to an infinite line charge is given by
$$V = \frac{\lambda}{2\pi \varepsilon}ln|\frac{b_{ref}}{r}|$$

The Attempt at a Solution

This does not necessarily require solution since the final potential is just the sum of the equation above. But I could get wrong In analyzing how many potentials are involved.

So by methods of Images,

1. We got to first determine the line charge that produce the present potential on each cylinder, which was fairly as simple as using the equation above, then changing pertinent variables. So that already 2 potentials.

2. Next, we got to account for the potential induced by each conducting cylinder to each other, so that is its image. So that is another two potential.

3. Over all we will have 4 potentials affecting any point outside each cylinder, but well, when I looked at the final answer, its just two potential. Moreover, my own answer does not jive with the correct solution of the same problem I have solved without using Methods of Images. Clearly, something went unusual to my analysis of the problem.

4. In summary

What should be the right way of doing methods of Images for this problem?

 

[mark726]

Hello,

Thank you for your question. The method of images is a powerful technique for solving boundary value problems in electrostatics. In the case of two conducting cylinders, there are a few key steps to follow in order to find the potential of the system.

1. Determine the potential on each cylinder due to the external potential V_0 and -V_0: This can be done by using the equation for the potential due to an infinite line charge, as you have already mentioned in your attempt. This will give you the potential on each cylinder at a distance of 2D from the center.

2. Take into account the images of the cylinders: Since the cylinders are not grounded, they will act as images of each other. This means that the potential on each cylinder will be affected by the presence of the other cylinder. To account for this, you can use the method of images to find the potential at a point outside of one cylinder due to the image of the other cylinder.

3. Sum up the potentials: Once you have determined the potential on each cylinder due to the external potential and the images, you can add them together to get the total potential at any point outside of the cylinders.

It is important to note that the potential at a point inside one cylinder will be affected by both the external potential and the image potential of the other cylinder. This is because the potential inside a conductor must be constant and equal to the potential on its surface.

In summary, the method of images involves finding the potential on each conductor due to the external potential and the images of the other conductor, and then summing them up to get the total potential at any point outside of the cylinders. I hope this helps clarify the steps involved in using the method of images for this problem. Let me know if you have any further questions.