Solving Charge Distribution for Spheres with Different Material Properties

• IxRxPhysicist
In summary, the conversation is about a question in Jackson 1.4 regarding the properties of three spheres with a total charge Q on them. One of the spheres has a charge distribution that varies as rn, causing trouble with the units. Suggestions are made to use ρ(r) = ρ0 * r^n, but the units of ρ0 will not be \rm \frac{Coulombs}{cm^3}, they will be \rm \frac{Coulombs}{cm^{3+n}}. This can be resolved by integrating the total charge over the sphere to determine the constant ρ0. The conversation also discusses the use of LaTex in responses.
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) ∝ ρo*(rn+1/rn)

buuuuut that still leaves me some messed up units.

Also, ρ0 = 3Q/(4πr3)

Any ideas?

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.

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.

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.

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

1. What is the charge distribution for spheres with different material properties?

The charge distribution for spheres with different material properties refers to the way in which electric charge is distributed on the surface of a sphere made of different materials. This is important because it affects the electric field and potential around the sphere.

2. How is the charge distribution calculated?

The charge distribution for spheres with different material properties can be calculated using Gauss's law, which relates the electric flux through a closed surface to the enclosed charge.

3. What factors affect the charge distribution on a sphere?

The charge distribution on a sphere is affected by the material properties of the sphere, the amount of charge present, and the distance from the center of the sphere.

4. Why is it important to solve for charge distribution on spheres with different material properties?

Solving for charge distribution on spheres with different material properties is important because it allows us to understand and predict the behavior of electric fields and potential in various situations, and can also have practical applications in industries such as electronics and energy.

5. Are there any limitations to the model used for solving charge distribution on spheres with different material properties?

Yes, there are limitations to the model used for solving charge distribution on spheres with different material properties. This model assumes that the sphere is a perfect conductor, which may not be accurate for real-world materials. Additionally, it does not take into account the effects of external electric fields or non-uniform charge distributions.

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