# Gauss’ law

**1. The problem statement, all variables and given/known data**

A positive charge q is placed at the center of a hollow electrically neutral conducting sphere (inner radius R_{1} 9cm, outer radius R_{2} 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.

**2. Relevant equations**

**3. 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.

[itex]\Phi _{0} = ( \Sigma \cos \phi) \Delta A[/itex]

I know I need to ‘construct’ a gaussian surface with radius r (r>R_{2}) 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, [itex]\phi = 0^{o}[/itex] and [itex] \cos \phi = 1[/itex].

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

[itex]\Phi _{0} = E( \Sigma \Delta A) = E(4 \pi r^{2}) [/itex]

Setting [itex]\Phi _{0} = \frac{q}{\epsilon _{0}} [/itex]

We can say that [itex] E = \frac{q}{4 \pi \epsilon _{0} r^{2}}[/itex]

For r > R_{2} 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

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