Cylinder with uniform charge distribution - algebra block!

I have a cylinder whose radius, r = a & length = L

charge distribution through cylinder is

ρ = C/2 * r^{3}

where r = radial distance in cylindrical coordinates

C = constant

show that the average charge density ρbar = Ca^{3} / 5

* * * * * * * * * * * * *

I have average charge = tot. charge / tot. volume

aAGH! Amendment

vol = ∏r^{2}L

i.e. ρ / ∏r2L

= ((C/2)*r^{3}) / ∏r^{2}L

= Cr^{3} / 2∏r^{2}L

= Cr / 2∏L

= Cr^{2} / 6.28L

Now this should make more sense to me but doesn't.

Re: Cylinder with uniform charge distribution - algebra block!

$\displaystyle \rho$ is not the "tot. charge", it is the charge at a single point. It's been a few years since I took calculus (so maybe someone else can correct me if I'm mistaken), but I believe you need to calculate:

$\displaystyle \int_{r=0}^{r=a}{\int_{\theta=0}^{\theta=2\pi}{ \frac{C}{2} r^3 \,d\theta\,dr\ }} = 2\pi \int_0^a{ \frac{C}{2} r^3 \,dr }$

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Re: Cylinder with uniform charge distribution - algebra block!

Of course.

All makes perfect sense.

Attachment 27867