What will hold a greater amount of charge, a hollow metal sphere or a solid metal sphere?
Asked by: Matthew Lammens
Answer
All else being equal, a hollow and a solid conducting sphere hold the same amount of charge.
The amount of charge held by a capacitor is given by
Q = CV,
where Q is the amount of charge, C is the 'capacitance' of the capacitor and V is the voltage applied between the conducting plates or surfaces of the capacitor.
In the case of an isolated conducting sphere (hollow or solid), one plate consists of the outer surface of the sphere. The other plate can be imagined to correspond to the inner surface of a second conducting sphere, larger in radius and concentric with the first, in the limit in which the radius of the second sphere goes to infinity. Generally, the potential of the second sphere at infinity is taken to be zero. [This is only a convention, which does not
affect the final answer because all of the physically significant parameters involved in the problem depend only upon the difference between the potentials at various points in space, not upon their absolute values.]
The qualification 'all else being equal' above means that the same voltage V, relative to zero at infinity, is assumed to be applied to the hollow and solid spheres. Likewise, it is assumed that the outside, or surface, radii of the two spheres are identical. The significance of the latter is that if the outside radii of the two spheres are identical, both will exhibit the same value of capacitance C.
Since the same voltage V is assumed applied to each sphere, and since the capacitance of each sphere is C, the charge held by each sphere is one and the same value Q.
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.
With the charge distributed uniformly over the surface of the sphere, there is no potential difference between any two points on the surface, including diametrically opposite points. This means that the electric field within the sphere is everywhere zero. [Recall that the strength of the electric field is given by the difference of the potential at two points in space divided by the distance between the points.] Zero electric field means, in turn, that no electric current flows within the sphere, even if it is a solid conductor. Therefore, it makes no difference if the interior of the sphere is filled with a perfect conductor, or if
it is hollowed out and empty. In other words, the capacitance C does not depend upon whether the conducting sphere is hollow or solid.
Answered by: Warren Davis, Ph.D., President, Davis Associates, Inc., Newton, MA USA