How Do You Calculate Average Charge Density in a Cylinder?

[Roodles01]

Homework Statement

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.

Homework 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?

The Attempt at a Solution

ρ = Cr² / l
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Can someone confirm this or point me in the right direction, please.

 

[rolotomassi]

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

 

[Roodles01]

∫ Cr² dr

C ∫ r² dr (0 < l < L)

CL³/3

 

[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 $$dV = rdrd\theta dz$$

 

[Roodles01]

∫_s ρ_(r) dV = Q

∫ Cr² * 4πr² dr = 4Cπ ∫ r⁴ dr

so
Q = 4CπR⁵ / 5

 

[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.