How Do You Calculate Average Charge Density in a Cylinder?

AI Thread Summary
To calculate the average charge density in a cylinder with a charge distribution of ρ = Cr², one must integrate the charge density over the volume of the cylinder. The total charge Q can be found using the volume integral in cylindrical coordinates, specifically integrating ρ(r) dV. The integration involves the limits for radius (0 to a), angle (0 to 2π), and length (0 to L), resulting in the expression Q = 4CπR⁵ / 5. The average charge density is then derived by dividing the total charge by the volume of the cylinder. This method ensures accurate calculation of the average charge density within the specified parameters.
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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.
 
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First you need to integrate the distribution over the cylinder to find the total charge.
 
∫ Cr2 dr

C ∫ r2 dr (0 < l < L)

CL3/3
 
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
 
s ρ(r) dV = Q

∫ Cr2 * 4πr2 dr = 4Cπ ∫ r4 dr

so
Q = 4CπR5 / 5
 
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.
 
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