Image Charges and a hollow conducting sphere

[raving_lunatic]

Homework Statement

Hi! Bear in mind, before shooting me down, that I'm very new to electrostatics and extremely (i.e. today) new to the method of image charges, and all my learning is dodgy book-learning and not learning from asking the learned questions.

This said, I just want my physical logic checking before I make a terrible mistake.

I've just worked out the image system for a point charge q some distance d from the center of an earthed, hollow, conducting sphere of radius a; it consists of a single point charge at some distance b from the center of magnitude q' and I can express these quantities in terms of the sphere's radius and the distance, so I think that's okay.

The second part of the question asks "what's the image system when the sphere is isolated (i.e. not earthed) and initially uncharged?" and then it asks me to find the force on the point charge q, which I can obviously do once I've worked out the magnitude and position of the image charge.

Homework Equations

Coulomb's Law F = q1q2/4∏ε0(r)^2

Potential at the surface of a charged sphere V = q / 4∏ε0(r)

The Attempt at a Solution

Now, I know that the presence of q is going to induce some charge -q' on the hollow sphere, and that the charge -q' will be such that the electric field within the material of the hollow sphere is zero. My argument, which may be catastrophically wrong, is that we can say this is a point charge of -q' at the centre (because a charged hollow sphere behaves like a point charge at its centre).

If this is correct, how can we calculate the magnitude q' of the image charge? I think that the potential at the boundary should be q'/4∏ε₀a, in which case I'm not sure how we can then calculate the magnitude of q'. Any assistance would be greatly appreciated.

 

[Simon Bridge]

The method of images is usually used for a charge outside the sphere.
http://en.wikipedia.org/wiki/Method_of_image_charges#Reflection_in_a_conducting_sphere
http://physicspages.com/2011/12/12/method-of-images-point-charge-and-sphere/

My argument, which may be catastrophically wrong, is that we can say this is a point charge of -q' at the centre (because a charged hollow sphere behaves like a point charge at its centre).

... you'll notice (see links) that the image charge is not in the center of the sphere.