# Capacitance of an isolated sphere - solid vs hollow

• elisagroup
In summary, it is agreed among physics books that there is no difference in capacitance between a solid and hollow isolated sphere. This is because in a perfect conductor, like charges will move to the outer surface to achieve the energetically most favorable distribution. This is true even if the interior of the sphere is hollow. However, this explanation does not account for the scenario where the outer surface becomes crowded and charges redistribute on the inner layers of the sphere. This is because it is describing a perfect conductor, which does not exist in reality. For real world conductors that come close to being ideal, we can still calculate capacitance as if they were ideal. Under certain conditions, such as using an insulating material instead of a conductor, there
elisagroup
Seems the physics books agree that there is no difference in capacitance whether an isolated sphere is solid or hollow. And the reason mentioned for that always sounds something like the following:

"The reason that the capacitance C, and hence the charge Q, is not affected by whether or not the sphere is hollow or solid is because, in a perfect conductor, like charges are free to take up equilibrium positions in response to the mutual electrostatic (Coulomb) repulsion between them. This means that all of the charges will move to the outer surface of the sphere, and will be distributed uniformly over the surface of the sphere, in order to 'get as far away as possible' from their neighbors. This is the energetically most favorable distribution of the charge. Since the material of which the sphere is made is a conductor, all charges can find their way to the outer surface, whether the interior is hollow or solid. "

But does it account for the scenario where the sphere would be charged to such a degree that the outer surface would get crowded, with charges looking for a more relaxed state and redistributing themselves on the inner layers of the sphere as well. And If that is true, wouldn't that mean that under certain conditions there is a difference in capacitance between solid vs hollow configurations?

elisagroup said:
But does it account for the scenario where the sphere would be charged to such a degree that the outer surface would get crowded, with charges looking for a more relaxed state and redistributing themselves on the inner layers of the sphere as well.
It does not, because it is describing a perfect (the word “ideal” is used more often) conductor, and that cannot happen in a perfect conductor. In an ideal conductor the charge density at a point can be arbitrarily high, so the crowding you’re describing never happens. Of course ideal conductors don’t really exist, but many real world conductors come so close that we can calculate capacitance as if they were ideal.
And If that is true, wouldn't that mean that under certain conditions there is a difference in capacitance between solid vs hollow configurations?
Those “certain conditions” would be if we had a sphere made of insulating rather than conductive material - but here it specifically says we’re working with a conductive sphere.

elisagroup

## 1. What is the difference between the capacitance of a solid sphere and a hollow sphere?

The capacitance of an isolated sphere is determined by its size and shape, as well as the dielectric material surrounding it. In a solid sphere, the electric field lines are evenly distributed throughout the entire volume, resulting in a higher capacitance compared to a hollow sphere where the electric field lines are concentrated at the surface. This leads to a lower capacitance for a hollow sphere.

## 2. How does the radius of the sphere affect its capacitance?

The capacitance of an isolated sphere is directly proportional to its radius. This means that as the radius of the sphere increases, its capacitance also increases. This is because a larger sphere has a larger surface area, allowing for more electric field lines to be present and increasing the capacitance.

## 3. Does the dielectric material surrounding the sphere affect its capacitance?

Yes, the dielectric material surrounding the sphere can greatly affect its capacitance. A dielectric material with a higher permittivity will result in a higher capacitance, as it allows for more electric field lines to be present. On the other hand, a dielectric material with a lower permittivity will result in a lower capacitance.

## 4. Can the capacitance of an isolated sphere be calculated using a formula?

Yes, the capacitance of an isolated sphere can be calculated using the formula C = 4πε0r, where C is the capacitance, ε0 is the permittivity of free space, and r is the radius of the sphere. This formula applies to both solid and hollow spheres.

## 5. How does the shape of the sphere affect its capacitance?

The shape of the sphere does not affect its capacitance as long as it remains isolated. This means that both a solid and hollow sphere with the same radius will have the same capacitance. However, if the sphere is not isolated and is part of a larger circuit, its shape can affect its capacitance as well as other factors such as the distance between the sphere and other conductors.

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