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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
where H and s_{0} 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. The name for this force law is a negative exponentially weighted inverse distance squared law.
Consider a charge of Q uniformly distributed over a sphere (spherical surface) 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 on 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 σ on 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 the direction of the move is then
The force dF_{−} due to the charge on the intersection of the cone with the sphere in the opposite direction reduces to
Therefore the net force on the charge q is equal to
where Γ is equal to [Hqσ·exp(−R/s_{0})]. Since q and σ are of the same sign the term Γ is positive and the net force on q is a repulsion. Furthermore the greater the value of δ the greater is the net repulsion force.
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 2π 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 if 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 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.
Suppose the force law is of the form
where f(s) is a declining function. This means the force drops off fastern than inverse distance squared. This is thought to be the case with the nuclear strong force.
The previous argument would apply so long as f(s+δ) is less than f(s−δ.
It is again clear from symmetry that when the centers of the sphere coincide that the net force on the spheres due to each other is zero. (There are forces to compress the inner sphere and to expand the outer sphere, but not to move their centers.)
What has to be established is that when the inner sphere moves off of the center of the outer sphere the increase in repulsion due to those parts of the inner sphere that are closer to the outer sphere is greater than the decrease in repulsion due to those parts that are farther away from the outer sphere.
Consider a cone centered on the inner sphere of infintesimal solid angle dω. Let the radius of the inner sphere be denoted as r. The intersection of the cone of infinitesimal solid angle with the inner sphere is two infinitesimal patches of area r²dω, which may be considered as point charges.
Let δ represent the deviation of the center of the inner sphere from that of the outer sphere. The effect of the charge on the outer sphere on the infinitesimal point charge patch in the direction of the deviation is a function of δ+r·cos(θ) where θ is the angle between the axis of the cone and the deviation δ. The effect of the charge on the outer sphere on the infinitesimal point charge patch in the direction opposite to the deviation is a function of δ−r·cos(θ). By the proper selection of the direction of the cone the angle θ can always be taken as less than or equal to π/2 and hence cos(θ) is non-negative.
For the case considered previously the net force on the two infinitesimal point charge patches is directed toward the center of the outer sphere and of magnitude
Generally this is positive for each infinitesimal cone and hence over all is positive. The effect of the interaction between the charges on the two spheres is a force driving their centers to coincide. As in the case of a point charge this conclusion can be extended to any force that drops off more rapidly than the inverse distance squared.
It is remarkable that a situtation with separate spheres having the same charge is unstable due to their mutual repulsion. However an arrangement of concentric spheres having the same charge is stable.
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