# Potential of Concentric Cylindrical Insulator

1. Jan 26, 2013

### InertialRef

1. The problem statement, all variables and given/known data

An infinitely long solid insulating cylinder of radius a = 3.6 cm is positioned with its symmetry axis along the z-axis as shown. The cylinder is uniformly charged with a charge density ρ = 25.0 μC/m3. Concentric with the cylinder is a cylindrical conducting shell of inner radius b = 19.5 cm, and outer radius c = 23.5 cm. The conducting shell has a linear charge density λ = -0.41μC/m.

What is V(P) – V(R), the potential difference between points P and R? Point P is located at (x,y) = (41.0 cm, 41.0 cm).

https://www.smartphysics.com/Content/Media/Images/EM/06/h6_cylinder.png [Broken]

2. Relevant equations

V(r) = ∫Edr
$\Phi$ = $\oint$EdA
$\Phi$ = qenclosed/$\epsilon$o*A

3. The attempt at a solution
I understand that I am to solve for E, and then take the integral in terms of r. But I'm still confused. I solved for E previously to get the following:

E$\oint$dA = qenclosed/$\epsilon$o
E*A = qenclosed/$\epsilon$o

A = 2*pi*r*h
qenclosed = λ(h) + ρ(volume of cylinder)

Therefore:

E(2*pi*r*h) = [λ(h) + ρ(volume of cylinder)]/$\epsilon$o
E = [λ(h) + ρ(volume of cylinder)]/($\epsilon$o)*(2*pi*r*h)

I'm confident about taking the integral of that equation above, but I'm just not sure what interval I'm supposed to take it over. Any help is appreciated. :)

Last edited by a moderator: May 6, 2017
2. Jan 26, 2013

### haruspex

You know what direction that field is in at any given point. At a point x along the the line from R to P, what is the strength and direction of the field? What is its component in the direction RP? (This can also be done vectorially of course.)