# Potential of a spherical conductor next to a point charge

Griffiths, when he's talking about the method of images, shows how to calculate the potential distribution when given a point charge that is outside of a spherical conductor which is held at a fixed potential.

If you are given a similar scenario, where a point is outside a spherical conductor but the conductor is not held at a fixed potential, how would you calculate the potential of the conductor?

It's basically solving laplace's equation when the boundary conditions aren't known. But intuitively, it seems like a solution would exist, so I'm guessing the boundary condition can be extracted somehow.

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What is known about the potential of the conductor? For example, did it start out at a known potential before the point charge was moved into place? – David Z Feb 5 '12 at 6:25
This is my own problem, add whatever constraints you like. :) – StuartHa Feb 5 '12 at 15:18

Now remove the grounding line and add -Q to the sphere to satisfy our boundary condition. Since the point charge and the induced charge result in a net zero potential for the sphere, the only contributor will be the -Q charge we introduce. Thus it is clear the potential of the conductor sphere is $$\varphi=-\frac{Q}{4\pi\varepsilon_0 R}$$