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## Homework Statement

Charge is uniformly distributed with charge density ρ inside a very long cylinder of radius R.

Find the potential difference between the surface and the axis of the cylinder.

Express your answer in terms of the variables ρ, R, and appropriate constants.

## Homework Equations

[tex]\int \overrightarrow{E}.d\overrightarrow{A}=\frac{Q }{\epsilon _{0}}[/tex]

[tex]\Delta V = -\int_{i}^{f}\overrightarrow{E}.d\overrightarrow{s}[/tex]

## The Attempt at a Solution

I am struggling with determining which Gaussian surface to use. If I use a cylinder, then the cylinder would have an infinite area, right? How can I deal with that? If I use a sphere (since I am trying to find the potential difference between only two points, one on the surface and one on the axis), what will be the charge inside the sphere?

If I use a sphere as my Gaussian surface, I get:

[tex]E = \frac{\rho }{4\pi R^{2}\epsilon _{0}}[/tex]

[tex]\Delta V = \frac{\rho }{4\pi R^{2}\epsilon _{0}} \int_{0}^{R}dR=\frac{\rho }{4\pi R\epsilon _{0}}[/tex]

But this is wrong.

If I use a cylinder as my Gaussian surface instead, I get the following, but it doesn't look right:

[tex] \Delta V = \frac{\rho }{2\pi \epsilon_{0}}\int_{0}^{R}\frac{1}{R} dR [/tex]

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