There is a wonderful theorem of mathematical physics which says that the effect of a mass uniformly distributed over a spherical surface is the same as if the mass were concentrated at the the center of the sphere at all points outside of the sphere. This theorem is easily extended to a mass uniformly distributed over a spherical ball. However the theorem does not extend to mass not uniformly distributed over a spherical surface or over a non-spherical surface or solid. In particular. it does apply to a system of discrete particles.

What is true is that as the distance from the center of mass of the particles increases without bound the effect asymptotically approaches the the effect the particles would have if they were concentrated at the center of mass of the particles. An equivalent form is that as the scale of the distribution relative to the distance from the center of mass goes to zero the effect asymptotically approaches the effect the particle would have if they were concentrated at the center of mass.

The Effect as the Distance from the
Center of Mass Increases Without Bound

Let i, y_i, z_i, m_i} be the position coordinates and mass of the i-th particle, with the origin being at the center of mass of the particles. This latter means that

Σm_ix_i = 0, Σm_iy_i = 0, Σm_iz_i = 0

Consider a point on the x-axis at a distance X from the origin; i.e., a point at coordinates (X, 0, 0). The distance from the i-th particle to the point (X, 0, 0) is

s_i = [(X-x_i)² + y_i² + z_i²]^½

Note that s_i can be expressed as

s_i = X[(1-x_i/X)² + (y_i/X)² + (z_i/X)²]^½

The magnitude of the force at (X, 0,0) due to the system of particles is

ΣGm_i/s_i²

where G is the gravitational constant.

The component of the force at (X, 0, 0) directed along the x-axis is the sum of the forces due to each particle multiplied times the cosine of the angle ψ_i between that force and the x-axis. That cosine is equal to

cos(ψ_i) = (X−x_i)/s_i
or, equivalently,
cos(ψ_i) = (1−x_i/X)/[(1-x_i/X)² + (y_i/X)² + (z_i/X)²]^½

Note that the limit of cos(ψ_i) as X→∞ is 1.

Therefore the force directed along the x-axis at (X, 0, 0) is

F_x = Σ (Gm_i)/(X²[(1-x_i/X)² + (y_i/X)² + (z_i/X)²])cos(ψ_i)

If the mass of the particles were concentrated at the origin the force (X, 0, 0) would be

F₀ = G(Σm_j)/X²

and it would be directed along the x-axis.

The ratio of the two forces is then

F_x/F₀ = Σ(m_i/Σm_j)cos(ψ_i)/[(1-x_i/X)² + (y_i/X)² + (z_i/X)²]

The limit of this ratio at X increases without bound is

lim_X→∞ (F_x/F₀) = (Σm_i/Σm_i)(1)/(1-0-0) = 1

The Limit as the Scale of the Particle Arrangement
to the Distance from the Origin Goes to Zero

Now consider the coordinates being given by

x_i = Rr_icos(θ_i)
y_i = Rr_isin(θ_i)cos(φ_i)
z_i = Rr_isin(θ_i)sin(φ_i)

where R is the scale of the system as measured by the maximum radius of a sphere centered on the center of mass and enclosing all the particles and r_i≤1 for all i. The above transformation of coordinates is based upon a spherical coordinate system in which the x-axis coincides with the axis of the sphere and θ is a latitude angle. The angle φ corresponds to longitude, the angle in the yz plane between the radial for the point and the z-axis.

Now

cos(ψ_i) = (1−(R/X)r_icos(θ_i)

The limit of cos(ψ_i) as (R/X) goes to zero is 1 for all i.

Let ζ_i be defined as

ζ_i = [(1-(R/X)r_icos(θ_i)² + [(R/X)r_isin(θ_i)cos(φ_i)]² + [(R/X)r_isin(θ_i)sin(φ_i)]²]

Note that

lim_{R/X → 0} ζ_i = 1
for all i

The ratio F_x/F₀ is now

F_x/F₀ = Σ(m_i/Σm_j)cos(ψ_i)/ζ_i

Therefore

lim_{R/X → 0}(F_x/F₀) = Σ(m_i/Σm_j)·1/1 = (Σm_i/Σm_j) = 1

Thus the effect of a distributed system of particles asymptotically approaches the effect that would prevail if all of the mass were concentrated at the center of mass of the system as the ratio of the scale of the system to the distance from the center of mass approaches zero.

Some Empirical Comparisons

The radius of our Milky Way galaxy is about 50,000 light years. The typical distance between galaxies in a galactic cluster is about forty times greater. Thus R/X for the distances between galaxies in a galactic cluster is about 1/40=0.025.

The radius of a Uranium nucleus is about 7.5 fermi (f). The radius of the first electron orbit is about 53,000 f. Thus at the distance of the electrons from the nucleus the ratio R/X is 1.4×10^-4. Thus considering the electrostatic charge as being concentrated at the center of the nucleus is a good approximation for distances comparable to the electrons.

(To be continued.)