Electric field lines produced by two eccentric charged cylinders

[Razvan]

Homework Statement

This is the theory for a laboratory session. If we have two hollow cylinders, one with radius r1 and the other with radius r7, the center of the first one being placed at a distance d from the center of the second one, what are the surfaces having the same electric potential?

Homework Equations

I would like to know the meaning and the derivation of those three equations in the picture. The book says that they replace the cylinders with their electrical axis placed at the points A and B, whose positions are determined using the electrical images method (this is my translation, it might be incorrect).

The Attempt at a Solution

I thought that the electric field lines must be perpendicular both to the "inside" and the "outside" cylinders, but, unfortunately, I can't go any further.

Thank you!

 

[Razvan]

Let me put it in another way. The cylinder with radius r7 is conneted to the ground, while the cylinder with radius r1 is charged. I should find a way to replace the cylinders with "lines", or if we look at a section, replace the circles with two points. The book says they use the electrical images method, but I've never studied it. Can you please explain how they arrived at those three relations? Thank you.

 

[Mark44]

Moved by mentor.

 

[BvU]

Hello Razvan, and welcome to PF.
I take it you are learning about electrostatics at a college or university level ?
And you know how to evaluate electric and potential fields of point charges, dipoles, etc ?
And you know that electric field lines are perpendicular to the surface of a conductor ?

In that case you want to read about the method of image charges: if you can somehow generate a field with aconcentric cylindrical equipotential surfaces (in the areas in your experiment where there are no charges present), you have THE solution of the Poisson equation for those areas (there it is the Laplace eqn), no matter how different the charge configuration OUTSIDE those areas is.

 

[Razvan]

Thank you for your response. The answer to all your questions is yes. I started studying Electrotechnics (translated name) this semester only. I don't think we are supposed to know how to derive those 3 relations, it is more a curiosity of mine.

As for the things I have already studied, I know that the potential V(x,y) (in a plane) is the mechanical work needed to move a unit charge from "infinity" to the point (x,y), a function which is proportional to the inverse of the distance from the point (x,y) to the position of the charge.

I also know that the Laplacian is the sum of the second order partial derivatives, which I think should be zero at the points where there are no charges.

As for the link you have provided, i tried finding an electrical image for each point on the inner circle, but the result (the curve outside the "grounded" circle) is not a circle anymore.

Any more hints would be greatly appreciated. Thank you.

 

[BvU]

Ah, here we don't have point charges but line charges. That way we get equiptential cylinders, just what we need.
Too late and too lazy to work it out (that's your part anyway). For a start, look here: 4.6.3

 

[Razvan]

I'm sorry for asking for help again, but I don't seem to be able to find the right equations. I think the relations (18) to (25) from the site you suggested might be helpful, but I'm incapable of applying them to my example. I think I forgot to mention that I am given the charge density per unit length for the inner cylinder (but I don't think it changes the location, maybe just the magnitude of the "image" charge). If you could just indicate which equations I should use, I am more than happy to do the rest of the work.

 

[BvU]

The sketch attached to post 1 suggests you place image line charges at A and B.
They draw V = 0 halfway in between, meaning the $\lambda_B = -\lambda_A$ (equal but opposite sign, not equal to charge density in your problem yet).
MIT stuff shows how to derive that equipotential surfaces are cylinders.
In fact that was all that was asked for in the original posting...

If you want V=0 on one of these cylinders, you can subtract a little $\lambda_C$ from both $\lambda_A$ and $\lambda_B$, such that V=0 coincides with the outer cylinder.

This post has some similarity (but it is about spheres and point charges, not cylinders and line charges).

And there is this one (page 3!).