# Potential of a metal sphere with changing radius

• purple88hayes
In summary, the potential of the balloon changes by a factor of 1/2 when it is inflated to a radius of 2R.
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!

Are we talking about the potential relative to a point inside the sphere or outside the sphere?

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

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?

purple88hayes said:
I'm still confused though. Why the function has to be piecewise smooth?
Note that the electric field can be expressed as the gradient of the potential,

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

Therefore, the potential must be continuously differentiable (at least once) in order to be physically meaningful, i.e. in order to associate an electric field with the potential, we must be able to differentiate it at least once. Therefore, the potential must be [piecewise] smooth.

Alright, I think that makes sense. Thanks again for all your help!

## 1. What is the relationship between the potential and radius of a metal sphere?

The potential of a metal sphere is directly proportional to its radius. This means that as the radius increases, the potential also increases and vice versa.

## 2. How does the charge of the metal sphere affect its potential?

The charge of the metal sphere has no direct effect on its potential. However, if the charge is increased, it may cause a change in the distribution of charges on the surface of the sphere, which can affect the potential.

## 3. Can the potential of a metal sphere be negative?

Yes, the potential of a metal sphere can be negative. This occurs when the sphere is negatively charged, and the potential at a certain distance from the sphere is lower than the potential at infinity.

## 4. How does the potential of a metal sphere change with distance?

The potential of a metal sphere decreases as the distance from the sphere increases. This is known as the inverse square law, where the potential is inversely proportional to the distance squared.

## 5. What factors can affect the potential of a metal sphere?

The potential of a metal sphere can be affected by its charge, radius, and the distance from the sphere. It can also be influenced by external factors such as the presence of other charged objects and the dielectric constant of the surrounding medium.

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