Solving Laplace's Equation Homework Statement

[satchmo05]

Homework Statement

I have a really dumb question, but I want to make sure this is right...
So I have the integral (d²V)/(dф²) = 0. I am solving for the potential function on the bounds, 0 < ф < ф_o. I will also be solving on range of ф_o < ф < 2∏.

Homework Equations

The Laplace equation is above.

The Attempt at a Solution

So, I am just having my doubts... When solving for 0 < ф < ф_o, I take the first integral with respect to d/dф, and after this I get dV/dф = arbitrary constant. If I take another integral on this (after multiplying both sides by dф), I get V = ф_o*arbitrary constant, right?!?

When solving for the other bounds, ф_o < ф < 2∏, I get V = arbitrary constant*(2∏ - ф_o), yes?!

Thank you for all help, I apologize for the silly question. I just wanted to make sure...

 

[Dick]

If you mean in one dimension is a solution to Laplace's equation a linear function, a*phi+b. Yes, it is. On the other hand, if you are working in two dimensions and phi is an angular coordinate, then you've got the Laplace equation wrong. It has to involve the other coordinate as well.

 

[dino20290]

Q: Consider a system consisting of a pair of earthed conducting plates at right angles to each other, one at the z=0 and the other in the y=0 plane and a positive point charge Q at (0,1,2). Show that Laplace's equation (with appropriate boundary conditions) can be solved for this system by replacing the conducting plates with a family of image charges.

I really don't know where to start, it'd be great if someone could help please.

 

[Dick]

Start by reviewing how to do it with one plate and why it works. Two plates isn't that much harder.