why is potential constant inside a charged sphere?

Answer 1

This would only apply to a conducting sphere (or another sphere with a uniform charge distribution.)

It is easy to see in the center of the sphere that all of the field from the charge will cancel out, because you are equidistant from an equal charge density in all directions. Now, if you are close to one part on the inside, the amount of electric potential from a unit area near you will be more than that of a unit area further away, BUT, there is a lot more area that is further away. It still cancels out.

There might be a more intuitive explanation than that, but I don't know it.

To see this mathematically, divide up the shell into rings with their axis going throught the position of the point in question so that the forces perpindicular to that axis cancel out. Calculate the potential of each ring. Add them all up and it will cancel. If you know calc do the same thing with rings that are delta wide and integrate. You need to do this in cartesian rather than spherical coordinates, but it is not that hard if you are clever. You can show the same thing for a solid sphere except the rings become disks.

This is the same reason that within a uniform medium with non-zero density, only the stuff closer to the center than you exerts any net gravitational force.

Answer 2

If it is a conducting sphere, there is zero electric field within the sphere. If there were, it would induce current flow on the inner surface of the sphere which would dissipate the electric field. Zero electric field is another way of saying that the electric potential throughout that space is uniform.

If it's a solid conducting sphere, and no current is flowing, then there's also no electric field.

If it's not a conducting sphere, then there's no particular reason for the potential to be constant within it.

Answer 3

Well, I can't compete with Mr. Quark's thoroughness here, but there's a nice, concise explanation on this page:

http://physics.bu.edu/~duffy/semester2/d06_potential_spheres.html