Solving Charge Distribution for Spheres with Different Material Properties

[IxRxPhysicist]

Hey all,
So the question in Jackson 1.4 is that I have 3 spheres that all have a total charge Q on them, but each sphere has different material properties. For instance, I have a conducting sphere, a sphere with a uniform charge distribution, and one with a charge distribution that has a charge distribution that varies as rⁿ. It's the last one I am having trouble with, how can I get an r dependence in ρ without screwing up the units? I tried something like:

ρ(r) ∝ ρ_o*(rⁿ⁺¹/rⁿ)

buuuuut that still leaves me some messed up units.

Also, ρ₀ = 3Q/(4πr³)

Any ideas?

 

[phyzguy]

You can have \rho(r) = \rho_0 \, r^n, but the units of \rho_0 will not be \rm \frac{Coulombs}{cm^3}, they will be \rm \frac{Coulombs}{cm^{3+n}},. And then you have to determine \rho_0 by integrating the total charge over the sphere to give Q.

 

[IxRxPhysicist]

Whoa whoa ρ can take on arbitrary units? Like Coulombs/cm³⁺ⁿ? The units just have to balance out in ∫ ρ dV = Q?...Also as an aside, are you using LaTex or something because I like the format in your response.

 

[phyzguy]

Yes, \rho_0 is just a constant in your equations that can have whatever units it needs to to make the units come out right. In that case it's not a charge density any more.

This link shows you how to include latex in your posts.

 

[IxRxPhysicist]

Interesting, learned something new. I just learned mathematica so I guess I will put LaTex on my to do list. Thanks!