# Potential above a charged shell

#### Melawrghk

1. The problem statement, all variables and given/known data
A very long insulating cylindrical shell of radius 6.90 cm carries charge of linear density 8.40 microC/m spread uniformly over its outer surface. Find the potential of a point located 4.4cm above the surface of the shell

2. Relevant equations

3. The attempt at a solution

I used Gaussian surface to find the electric field created by the shell outside (Q = linear charge density):
QL/$$\epsilon_{0}$$ = E*2*pi*d*L,
where L is length, d is distance to point from the axis of the cylinder.
The Ls cancel out, rearranging, I got:
E=$$\frac{Q}{2pi*d*\epsilon_{0}}$$

I then integrated that to get the potential:
V=$$\frac{Q}{2pi*\epsilon_{0}} \int {(1/d)dd}$$
The integral's upper limit is d-r, and lower limit is r.
I got the equation below as a result:
V=$$\frac{Q}{2pi*d*\epsilon_{0}} ln(\frac{d-r}{r})$$

Substituting in the numbers I got V=-6.8*10^4, but that's wrong. Is it my method? Should I slice the cylinder into rings and do V from each one, then integrate the whole thing?

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#### gabbagabbahey

Homework Helper
Gold Member
I then integrated that to get the potential:
V=$$\frac{Q}{2pi*d*\epsilon_{0}} \int {(1/d)dd}$$
What's that extra 1/d doing out front of the integral?

The integral's upper limit is d-r, and lower limit is r.
Why is that?

#### Melawrghk

What's that extra 1/d doing out front of the integral?

Why is that?
First one: typo, sorry :)

Second one: now that I think about, I'm not sure. I was thinking of using distance from the cylinder's axis and then r would be the smallest so that you're still on the outside (well, surface) and d-r would be anything beyond that. Should those limits be d and 0 instead?..

EDIT: I got it! Instead of using d as distance to center of the cylinder, I used d as distance to surface and for gaussian surface I had:
E=Q/(epsilon*2*pi*(d+r))
And integrated taht with upper limit being 'd' and lower being 0. And it worked! :D Thanks so much.

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"Potential above a charged shell"

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