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 The Force Internally Due to a Charge Uniformly Distributed on a Sphere when the Force Law is not a Strictly Inverse-Distance-Squared Law

There is a beautiful and well-known theorem in mathematical physics that when a charge such as electrical or gravitational is uniformly distributed on a spherical surface its effect on a charge at a point interior to the sphere is zero and on a point exterior to the sphere is the same as if the charge were concentrated at the center of the sphere. This is true only if the force between two charges is strictly inversely proportional to the to the distance between them squared. This webpage is an investigation of what holds under more general force laws.

The force law which will be used initially is

#### F = HQq·exp(−s/s0/s²

where H and s0 are constants, Q and q are the charges and s is the distance between them. It is assumed that q and Q are of the same sign so the force is a repulsion. This force law is relevant when the force is carried by particles but those particles decay over time and hence with distance.

Consider a charge of Q uniformly distributed over a sphere of radius R and a point charge of q located at the center of the sphere. By symmetry the force on the point charge is zero. This is the relevant case.

Consider a cone of solid angle Ω centered the point charge q at the center of the sphere. Any force due to the charge on the surface of the sphere which is encompassed within the intersection of the cone with the sphere in one direction is exactly counterbalance by the insection of the cone with the sphere in the opposite direction. This is not necessarily true is the charge q is not located at the center of the sphere.

Consider now the situation in which the point charge is moved a small distance δ away from the center of the sphere. Let dΩ be the solid angle of a small cone centered on the point charge q. The charge density σon the sphere is (Q/(4πR²)). The charge on the surface of the intersection of the cone and the sphere is then σdΩ(R−δ)². The force due to the charge on the intersection of the cone with the sphere in one direction is then

#### dF+ = Hq[σdΩ(R−δ)²]exp(−(R−δ)/s0)/(R−δ)² which reduces to dF+ = Hσ·exp(−(R−δ)/s0)dΩ and further to dF+ = exp(δ/s0)[Hqσ·exp(−R/s0)]dΩ

The force dF due to the charge on the intersection of the cone with the sphere in the opposite direction reduces to

#### dF− = −exp(−δ/s0)[Hqσ·exp(−R/s0)]dΩ

Therefore the net force on the charge q is equal to

#### dF+ + dF− =[exp(δ/s0)−exp(−δ/s0)] ΓdΩ

where Γ is equal to [Hqσ·exp(−R/s0)]

For each small cone the effect to a deviation of δ from zero is a repulsion that pushes δ back to zero. The integration of dΩ over the range of 0 to π is therefore positive and hence a repulsion. So the location of the point charge q at the center of the charged sphere is stable. Furthermore is the point charge were located away from the center the net force on it would drive it to the center. Likewise the force on the spherical charge due to the point charge is such that it would drive to the sphere in a direction that would place the center of the sphere on the point charge.

Thus when the point charge is located at the center of the charged sphere the force on the point charge is zero, as is also the force on the spherical charge. Furthermore any deviation from that equilibrium arrangement induces forces that drive it back to that arrangement. In other words, the arrangement of the point charge at the center of the sphere is stable.

## Generalization

If the force law is of the form

#### F = HqQf(s)/s²

The previous argument would apply so long as f(δ) is less than f(−δ). This is the case when the force drops off with distance faster than inverse distance squared. This is thought to be the case with the nuclear strong force.