Solving Sphere Image Problem: Find Potential & 2nd Image Charge

[CalcYouLater]

Homework Statement

A point charge is located outside of a grounded conducting sphere. Find the potential outside the sphere.

This problem was a solved example. It was solved by placing an image charge inside the sphere so that the potential from the charge outside the sphere would cancel the potential from the charge inside the sphere at the surface of the sphere.

The follow up question is:
By adding a second image charge the same method used above will handle the case of a sphere at any potential V_0. What charge should you use, and where should you put it?

Homework Equations

The Attempt at a Solution

It seems that by exploiting the principle of superposition, an easy solution is to add another image charge at the center. Then that charge would have to be q= 4*Pi*epsilon*R*V_0. Where "R" is the radius of the sphere.

But is that the only possible choice? Since the conductor is an equipotential, doesn't that mean that I could place the added charge anywhere inside the sphere? If so, then it seems the only problem that remains is determining what that charge would be.

 

[hikaru1221]

Since the conductor is an equipotential, doesn't that mean that I could place the added charge anywhere inside the sphere? If so, then it seems the only problem that remains is determining what that charge would be.

I think you are mixing the model and the real object. The real object is the conducting sphere, and more exactly, the surface charge layer on the sphere. The model is the image charges. You REPLACE the real object with your model, under the constraint that E-field at the places outside "the sphere" remains the same. I put the word "the sphere" inside the inverted commas as when you replace the real sphere with the image charges, the sphere is no longer there!

That is the reason why you have to put the second image charge at the center, not anywhere else inside "the sphere". Notice that when you replace the sphere with the image charges, E-field does exist inside "the sphere", but this doesn't violate with the constraint.