# Charge density in a cylinder

1. May 28, 2015

### Roodles01

1. The problem statement, all variables and given/known data
A cylinder of radius a and length l has charge distribution

ρ=Cr2
where C is a constant and r is radial distance in cylindrical coordinates.
Derive an expression for the average charge density within the cylinder.

2. Relevant equations
Well, charge density given is within the volume, I think.
So for a point on the axis of the cylinder should be ρ divided by the length shouldn't it? Or is that being too simple?

3. The attempt at a solution

ρ = Cr2 / l

Can someone confirm this or point me in the right direction, please.

2. May 28, 2015

### rolotomassi

First you need to integrate the distribution over the cylinder to find the total charge.

3. May 28, 2015

### Roodles01

∫ Cr2 dr

C ∫ r2 dr (0 < l < L)

CL3/3

4. May 28, 2015

### rolotomassi

The radius 0 < r < a .
The length L.

You need to integrate over the VOLUME of the cylinder to find the total charge in the volume. I advise working in cylindrical co-ordinates. where $$dV = rdrd\theta dz$$

5. May 29, 2015

### Roodles01

s ρ(r) dV = Q

∫ Cr2 * 4πr2 dr = 4Cπ ∫ r4 dr

so
Q = 4CπR5 / 5

6. May 29, 2015

### rolotomassi

You need to integrate over the cylinder.

$$\int dV \ = \iiint rdrd\theta dz\ = \int_{0}^{L} dz \ \int_{0}^{2\pi}d\theta \int_{0}^{a}rdr$$ This is the volume integral for a cylinder and as you can see, doing the integral gives the volume of a cylinder of radius, a, and length, L. But since your integrating a function over this volume, you want
$$\int \rho(r) dV$$ You can split the integral similarly to find the total charge.

Last edited: May 29, 2015
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