Homework Help: Dielectric sphere - Find the electric field

1. Sep 3, 2007

Qyzren

Consider a dielectric sphere of radius R and dielectric constant e_r. The sphere contains
free charges that have been uniformly distributed with density rho.

(a) Show that electric field inside the sphere is given by
E = [rho*r /(3*e_0*e_r)]* r hat

where r hat is the unit vector pointing in the radial direction.

(b) Show that the electric field outside the sphere is
E = [rho*R³]/[3*e_0*r²] * r hat

(c) Calculate the potential V at the centre of the sphere compared to that at infinity.

Any tips/hints that can help start me off will be appreciated.

2. Sep 3, 2007

Claude.

3. Sep 3, 2007

Qyzren

well gauss' law is just integral of E.dA = Q/e_0 since we have a sphere the SA is just 4*pi*r^2 so it gives E = Q/[4*pi*e_0*R^2].

How does the density rho and dielectric constant e_r affect/influence this?

Thanks everyone for helping.

4. Sep 5, 2007

Reshma

Use the Guass's law in presence of dielectrics.
Within the dielectric the total charge density can be written as:
$$\rho = \rho_b + \rho_f$$
where $\rho_b[/tex] is the bound charge and [itex]\rho_f$ is the free charge density resp.
The Gauss's law will be modified accordingly. Read up on the electric displacement vector and you can easily find the solution.

5. Sep 5, 2007

olgranpappy

It influences it because you don't know what Q is. As you have written it Q is the source of the electric field E, i.e. the true and total Q (both bound and otherwise). But you don't know Q because you only know the *free* charge density.

Luckily, gauss' law will work for the electric displacement too with
$$\int \vec D \cdot \vec dA = Q_{\textrm{free}}^{\textrm{enclosed}}$$

Last edited: Sep 5, 2007
6. Sep 5, 2007

olgranpappy

...and you also have the definition (for a linear dielectric) that
$$\epsilon_r\epsilon_0 \vec E = \vec D$$

So, use what's in my first post to solve for D and then just divide to get E

...I might not have all the factors right since I am used to using Guassian units... but I think I got the placements of the epsilons correct...