Does this answer look right? (Infinitely long conductor pipe)

[Jamin2112]

Homework Statement

Fig. 1 shows the cross-section of an infinitely long conducting pipe. The inner radius of the pipe is r = R₁, and the outer radius is r = R₂. Suppose the inner surface has a constant voltage V = V_c > 0, and the outer surface has a constant voltage V = 0. The voltage distribution between the inner and outer surface (blue region) is governed by the Laplace's equation ∇²V=0 expressed in polar coordinates. Solve for V with the given boundary conditions.

[Note: Fig. 1 just looks like a blue annulus centered at the origin in the plane.]

Homework Equations

Don't worry about them for now.

The Attempt at a Solution

Just tell me whether my solution V = (V_c log(r) - log(R₂))/(log(R₁) - log(R₂)) intuitively seems like it's correct. I don't want you to carry out the process of deriving this ... unless you want to.

 

[Jamin2112]

I.e. the voltage at a distance r from the center of the piper is proportional to log(r)

 

[Jamin2112]

EDIT:

Actually I got

V(r,θ) = V_c * [log(r) - log(R₂)] / [log(R₁) - log(R₂)].​

Does that seem right?