Solving a Charge at the Center of a Sphere Problem

[BOAS]

Homework Statement

A positive charge q is placed at the center of a hollow electrically neutral conducting sphere (inner radius R₁ 9cm, outer radius R₂ 10 cm.

Using Gauss' law determine the electric field of every point in space, as a function of r (the distance from the center of the sphere). Only the algebraic expression is required.

Homework Equations

The Attempt at a Solution

I'm unsure of how to go about this problem, all the examples I have seen discuss thin spheres.

A positive charge at the center of the sphere would induce a positive charge on the surface of the sphere, whilst a negative one would be induced on the inner surface.

I think it is safe to assume the charge at the center is a point and therefore the charge will be evenly distributed creating a symmetrical electric field.

$\Phi _{0} = ( \Sigma \cos \phi) \Delta A$

I know I need to 'construct' a gaussian surface with radius r (r>R₂) concentric with the shell, but I don't know how to use the information about the two radii I was given - Are they important here?

Since the electric field is everywhere perpendicular to the gaussian surface, $\phi = 0^{o}$ and $\cos \phi = 1$.

The electric charge has the same value all over the surface, so we can say that;

$\Phi _{0} = E( \Sigma \Delta A) = E(4 \pi r^{2})$

Setting $\Phi _{0} = \frac{q}{\epsilon _{0}}$

We can say that $E = \frac{q}{4 \pi \epsilon _{0} r^{2}}$

For r > R₂ since Gauss' law also shows us that there is no net charge inside the sphere.

The above makes sense to me, but at no point did I use the radii given to me in the question...

What am I doing wrong?

Thanks!

BOAS

 

[Doc Al]

I know I need to 'construct' a gaussian surface with radius r (r>R₂) concentric with the shell, but I don't know how to use the information about the two radii I was given - Are they important here?

They are important because they mark off three distinct regions: r < R1; R1 < r < R2; r > R2.

You only dealt with the last region. What about the others?

 

[BOAS]

Ok, that was surprisingly obvious.

I'll have three distinct expressions, does that satisfy the question of finding the electric field of every point in space?

I suppose that it does, but on first reading of the question I was expecting a single expression.

Thanks for your help, I'm confident I can do this now.