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- Homework Statement
- A sphere with radius R and the dielectric constant χ_e has been charged with a uniform free

space charge density with the value ρ_f (e.g by being bombarded with α-particles which,

after being braked to rest, can no longer move).

b)Now suppose that the free charge distribution is not completely immobile, but that we can

have currents of free charge described by Ohm's law with the conductivity ## \sigma_{\alpha}## At t = 0

we imagine that, as in a), the free space charge density is ρ_f everywhere in the globe.

Determine how the free charge density in the globe varies in space and time for t> 0.

- Relevant Equations
- ## \vec \nabla \cdot \vec J = - \frac{\partial \rho }{\partial t} ##

## \vec J = \sigma_{\alpha} \vec E ##

Hello, I wonder if you could give me some advice to how solve this question. What I was thinking to solve it was to determine J by using Ohms law, ## \vec J = \sigma_{\alpha} \vec E ## I already determined the E field for for the sphere, I got it from a) ("a)" was to determined all the bound charges and show that the sum of them is zero, I attached the solution) If you see there I solve the electric field by Using ## \vec D = \epsilon E ## where i I got ## \vec D## from gauss law for a gaussiuan sphere with radius r< R (inside the sphere) and got that the eletric field of this gaussian sphere is $$ \vec E = \frac{r \rho_f}{4\pi R^3}\frac{1}{\epsilon_0 \left ( 1+ X_e \right ) } \hat r$$. But I wonder if in this case( b) will r = R? or how can I proceed?

Edit: I realized that the field was incorrect, I re-wrote it and got $$ \vec E = \frac{1}{3}\frac{\rho_f r}{\epsilon_0 (1+X_e)} $$

Edit: I realized that the field was incorrect, I re-wrote it and got $$ \vec E = \frac{1}{3}\frac{\rho_f r}{\epsilon_0 (1+X_e)} $$

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