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