# What is the potential of an uncharged sphere?

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1. Dec 15, 2016

### Zarif

1. The problem statement, all variables and given/known data
Find the potential of an uncharged metal sphere provided that a point charge q is located at a
distance r from its center

2. The attempt at a solution
As far as the charges are concerned , some negative charges will build up at the side of the charge because of induction .. Where does it lead to ? How can i calculate the potential?

2. Dec 15, 2016

### zhouhao

Could you find a formula to get the induced charge density?
If you just want to know the potential,then the potential is the $k\frac{q}{d}$,
d is the distance from the somewhere in the shpere to the point charge q.

3. Dec 15, 2016

### Zarif

I know that potential is given by that equation . I don't have a problem in that

4. Dec 15, 2016

### haruspex

Are you familiar with the method of images, as it applies to spherical surfaces?
Given a spherical surface (just as a notional surface in space, not a conductor) and a point charge outside, it is not hard to show that there is a position inside the sphere and a charge value such that adding that charge at that position you would get a zero potential everywhere on the sphere.
Can you see how that solves your problem?

5. Dec 16, 2016

### Zarif

I don't know about "method of images" .. It would be very very helpful if you give me some idea

6. Dec 16, 2016

### haruspex

It's easiest with a grounded conducting infinite plate. Suppose we put a point charge q at distance x above it. We know the plate remains at zero potential. Suppose we take away the plate and put a point charge -q at distance 2x below the +q. By symmetry, in the plane where the plate was, we still have zero potential. So in the half space above the plate, the induced charges on the plate had exactly the same effect as putting the -q as the mirror image of the +q.
It turns out that we can do the same with spherical surfaces. If we put a point charge q near a grounded sphere, the induced charges on the sphere are such that the resulting field outside the sphere is exactly the same as if you were to replace the sphere by a certain point charge at a certain location inside it. The difference is that the point charge is no longer simply -q, and the position it must be placed is not quite obvious.
By considering the points on the sphere closest to and furthest from the +q, and that the potential must be zero there, you can determine the charge and its position. It is not hard to show that this does indeed result in zero net potential at the boundary of the sphere.

See e.g. http://web.hep.uiuc.edu/home/serrede/P435/Lecture_Notes/P435_Lect_06.pdf

7. Dec 16, 2016

### ehild

The potential of the metal sphere is the same both on the inner and the outer surfaces. The charge is inside, so there are some induced charge distribution on the inner surface. What is the whole charge on the inner surface?
The metal sphere is neutral, so there should be charge on the outer surface. The electric field inside the metal wall is zero, so the outer charges do not know about the charge distribution on the inner surface. How is the outer charge distributed? What is the potential in the case of such charge distribution?

8. Dec 16, 2016

### haruspex

I don't think it specifies. I took it as outside, but the same methods should work.

9. Dec 16, 2016

### ehild

Well, the problem did not specify the position of the q charge, but the OP did not hear about the method of images, So I took it, that the charge was inside. In this case, the electric field inside would be obtained using an appropriate mirror charge outside, but the field outside is that of a charged sphere, and so is the potential of the sphere.
In case when the charge is outside, not only the charge q and mirror charge q' have to be taken into account, but also a charge -q' in the centre of the sphere, as it is not grounded. But the original charge and its mirror result in zero potential of the sphere, so only the potential of -q' counts.

10. Dec 16, 2016

### ehild

Zarif, was the charge q inside or outside of the sphere? Was r < R or r>R (R is the radius of the sphere)?

11. Dec 19, 2016

### Zarif

It is not clear from the question .. could be outside or inside

12. Dec 19, 2016

### ehild

Presumably the potential of the sphere is meant with respect to infinity. If the charge was outside, you need to find the mirror charge q'. The potential of the sphere due to the charge and its mirror is zero, but you have to place a charge -q' at the centre of the sphere, so as to compensate the added charge. The potential of the sphere is that of the potential due to -q'.
If the charge is inside, you can do what I explained in post #7.

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