Charge distribution of a hollow conductor

[Caulibeam]

Homework Statement

We are given that there is a hollow conducting sphere with a charge 1C (to simplify things) inside it and are asked to find the charge per unit area on the inner and outer surface of the sphere. Inner radius of the sphere is 1m and outer radius is 2m

Homework Equations

In the answers given by another student he did this:

surface charge density inner = q/Ainner = 1/ (4pi*1^2) where A is the area
and
surface charge density outer = q/Aouter = 1/ (4pi*2^2)

The Attempt at a Solution

However in my opinion, since it is a conductor. the charge will be equally distributed within the outer and inner surface since its spherical in shape and so

surface charge density = q/(Aouter + Ainner)

 

[anathema]

= 1/ (4pi*1^2 + 4pi*2^2) = 1/ (4pi*5)

First of all, it is important to clarify that the charge per unit area on the inner and outer surfaces of a conducting sphere will not be the same. This is because the charge on the inner surface will contribute to the electric field inside the sphere, while the charge on the outer surface will contribute to the electric field outside the sphere. Therefore, the distribution of charge will be different on the two surfaces.

Now, let's look at the solution given by the other student. They correctly used the formula for surface charge density, which is q/A. However, they made a mistake in determining the area of the inner and outer surfaces. The area of a sphere is given by 4πr^2, where r is the radius. Therefore, the correct formulas for surface charge density on the inner and outer surfaces would be:

surface charge density inner = q/Ainner = 1/ (4pi*1^2) = 1/4π

surface charge density outer = q/Aouter = 1/ (4pi*2^2) = 1/16π

Your attempt at a solution is also incorrect. The total charge on the sphere is still 1C, regardless of the distribution on the inner and outer surfaces. Therefore, the correct formula for surface charge density would be:

surface charge density = q/(Aouter + Ainner) = 1/ (4pi*1^2 + 4pi*2^2) = 1/ (4pi*5) = 1/20π

In conclusion, the charge per unit area on the inner and outer surfaces of the conducting sphere will be different, and the correct formulas for surface charge density are:

surface charge density inner = q/Ainner = 1/4π

surface charge density outer = q/Aouter = 1/16π