Although it is seldom articulated there is a third theorem that says

• at a point on the sphere the force is the same as if one half of the charge were concentrated at the center of the sphere

Analysis

This is an attempt to consider the general problem. Let r be the radius of the sphere and ρ the charge density. Let ρf(y) be the force from an infinitesimal element of area r²dφdθ at a point y distance from it.

For the ring between φ-dφ/2 and φ+dφ/2 the force is

dF = ρf(y)[2πz]rdφ
and the component in the x direction
dFcos(φ) = ρf(y)[2πz]rcos(&phil)dφ

where he radius of the ring, z, is equal to rsin(φ). The distance y is given by

y² = x² + r² - 2xrcos(φ)

The component of the force in the x-direction is

F(x,r) = ∫₀^πρf(y)(2πr²)sin(φ)cos(φ)dφ
which reduces to
F(x,r) = 2πr²∫₀^πρf(y)sin(φ)cos(φ)dφ

The variable of integration can be changed from φ to y. Thus

2ydy = 2xrsin(φ)dφ
and hence
dφ = ydy/(xrsin(φ))

Therefore, since y=x-r when φ=0 and y=x+r when φ=π and cos(φ)=(x²+r²−y²)/(2xr)

F(x,r) = 2πr²∫_x-r^x+r[ρf(y)y(x²+r²−y²)/(2(xr)²)]dy
which reduces to
F(x,r) = (π/x²)∫_x-r^x+r[ρf(y)y(x²+r²−y²)]dy

For convenience let ρf(y)y be denoted as g(y). Then

F(x,r) = (π/x²)∫_x-r^x+r[g(y)(x²+r²−y²)]dy

Reduction of a Distributed Force to a Point Force

For an inverse distance squared force F(x,r) reduces to the form K/x² where K is proportional to the amount of charge on the sphere. In particular, for an inverse distance squared force, F is independent of r.

Now consider ∂F/∂r which is

∂F/∂r = (π/x²){[g(x+r)(2xr) − g(x-r)(2xr)] + 2r∫_x-r^x+rg(y)dy}

Let the indefinite integral ∫g(y)dy be denoted as G(y). Thus the above equation can represented as

∂F/∂r = (π/x²){[G'(x+r)(2xr) − G'(x-r)(2xr)] + 2r[G(x+r)−G(x-r)]}

If ∂F/∂r is identically equal to zero then

x[G'(x+r)−G'(x-r)/(2r)] = −[G(x+r)−G(x-r)/(2r)]
this holds for all r and hence
it holds for all r as r goes to zero
Thus
lim_r→0 {x[G'(x+r)−G'(x-r)/(2r)]} = − lim_r→0 [G(x+r)−G(x-r)/(2r)]
which is
xG"(x) = −G'(x)
or, equivalently
xg'(x) = −g(x)

This equation can be solved as

dg/g = −dx/x

The solution to this equation is

g(x) = k/x

where k is an arbitrary constant.

Since g(x)=xρf(x) this means the solution for f(x) is

f(x) = h/x²

where h is an arbitrary constant.

Thus it is found that the only force function that allows a force distributed over a sphere to be reduced to a point force at the center of the sphere is the inverse distance squared form.

Analysis for Nonreductive Force Forms

Concider the case in which f(y)=Ke^-y/y², the form that applies when the particle which carries the force decays.

(To be continued.)