What is the potential of an uncharged sphere?

In summary: The charge is inside, so there are some induced charge distribution on the inner surface. What is the whole charge on the inner surface?
  • #1
Zarif
9
1

Homework Statement


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?
 
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  • #2
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
zhouhao said:
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.
I know that potential is given by that equation . I don't have a problem in that
 
  • #4
Zarif said:
How can i calculate the potential?
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
haruspex said:
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?
I don't know about "method of images" .. It would be very very helpful if you give me some idea
 
  • #6
Zarif said:
I don't know about "method of images" .. It would be very very helpful if you give me some idea
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
Zarif said:

Homework Statement


Find the potential of an uncharged metal sphere provided that a point charge q is located at a
distance r from its cente
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?
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?
upload_2016-12-16_22-10-41.png
 
  • #8
ehild said:
The charge is inside,
I don't think it specifies. I took it as outside, but the same methods should work.
 
  • #9
haruspex said:
I don't think it specifies. I took it as outside, but the same methods should work.
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
Zarif said:

Homework Statement


Find the potential of an uncharged metal sphere provided that a point charge q is located at a
distance r from its center
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
ehild said:
Zarif, was the charge q inside or outside of the sphere? Was r < R or r>R (R is the radius of the sphere)?
It is not clear from the question .. could be outside or inside
 
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  • #12
Zarif said:
It is not clear from the question .. could be outside or inside
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.
 

FAQ: What is the potential of an uncharged sphere?

What is the potential of an uncharged sphere?

The potential of an uncharged sphere is zero, as there is no net charge present to create an electric field.

Does the size of the sphere affect its potential?

Yes, the potential is directly proportional to the size of the sphere. A larger sphere will have a higher potential compared to a smaller sphere with the same charge distribution.

Can an uncharged sphere have a non-zero potential?

No, an uncharged sphere by definition has an equal number of positive and negative charges, resulting in a net charge of zero and therefore a potential of zero.

How is the potential of an uncharged sphere measured?

The potential of an uncharged sphere can be measured using a voltmeter, which measures the difference in potential between two points.

Does the material of the sphere affect its potential?

No, the potential of an uncharged sphere is determined solely by its size and charge distribution, not the material it is made of.

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