# Potential of a metal sphere with changing radius

purple88hayes

## Homework Statement

An inflatable metal balloon assumed to be spherical with radius R is charged to a potential of 1000 V. After all the wires and batteries are disconnected, the balloon is inflated to a new radius 2R. Does the potential of the balloon change as it is inflated? If so, by what factor? If not, why not?

## Homework Equations

V(pt. charge) = kQ/R

## The Attempt at a Solution

I think the answer should be that yes, V does change by a factor of 1/2 since R increases by 2 and V is proportional to 1/R. However, I also want to think the potential is infinite at a point on the sphere. I think I understand that we can treat the sphere as a point charge, but what I don't understand is what happens when a charged particle is on the sphere. Why doesn't potential go to infinity? It seems that since the distance between some bit charge dQ of the sphere and the test charge is 0 this would blow up to infinity. I'm probably over-thinking the question but I seem to have dug myself into a hole of thorough confusion. Can someone help explain this to me? Any help is appreciated!

Staff Emeritus
Gold Member
Are we talking about the potential relative to a point inside the sphere or outside the sphere?

purple88hayes
I'm assuming when they say 1000V that's at point a point on R relative to infinity. So outside.

purple88hayes
Cool, thanks for your help, I really appreciate it! Those pictures were very useful! I'm still confused though. Why the function has to be piecewise smooth?

Staff Emeritus
$$\underline{E} = \nabla V$$