The Inverse Square Law

Suppose two point masses with masses m and M respectively are located a distance r apart. Sir Isaac Newton's inverse square law states that the magnitude | F| of the gravitational force between the two point masses is

| F| = G  Mm r2

(4)

where G is the universal gravitational constant. However, as Newton realized and struggled with for some time, objects in the real world are not point-masses and instead, the law (4) might need to be modified.

In particular, let's suppose that one of the bodies is not a ''point-mass,'' but instead is a sphere of radius R with uniform mass density m. Suppose that  r > R and that the sphere is centered at ( 0,0, r) . If the other body is a point-mass ''satellite'' of mass m located at the origin, then the gravitational force is directed along the z-axis. Suppose now that a small ''piece'' of the sphere is located at a point ( r,f,q) (in spherical coordinates), and suppose that it has a small mass dM. Then the distance between the small piece and the origin is r. so that by (4) the ''small'' magnitude d|F| of the gravitational force between the small ''piece'' and the satellite is

d|F| =  -Gm  dM r2

(5)

However, the symmetry of the sphere implies that we need only the gravitational pull in the vertical direction.  The amount of d|F| in the vertical direction is then given by cos(f) d| F| (see above).

Thus, the total gravitational force in the vertical direction is

| F| = ó õ ó õ ó õ S  cos( f) d| F| = ó õ ó õ ó õ S   -Gm  cos( f) r2 dM

where S is the sphere corresponding to the ''planet''. If dV denotes the volume of a small ''piece'' of the sphere, then dM = m dV, which leads to

| F| = -Gmm ó õ ó õ ó õ S   cos( f) r2 dV

(6)

In Cartesian coordinates, the sphere S is given by

x2+y2+( z-r) 2 = R2        or       x2+y2+z2-2rz+r2 = R2

In spherical coordinates this becomes

r2-2rrcos( f) +r2-R2 = 0

which by the quadratic formula leads to

r = rcos( f) ±   R2-r2( 1-cos2( f) ) = rcos( f) ±   R2-r2sin2( f)

Thus, the sphere is contained between

r1 = r cosf  -   R2-r2sin2( f)     and    r2 = r cosf  +   R2-r2sin2( f)

Let us also note that f ranges from 0 to sin^-1(R/r) while q ranges over [ 0,2p] .

Evaluating (6) in spherical coordinates leads to

| F| = -Gmm ó õ 2p 0  ó õ sin-1(R/r) 0  ó õ r2 r1   cos( f) r2   r2sin( f) drdfdq = -Gmm ó õ 2p 0  ó õ sin-1( R/r) 0  ó õ r2 r1  cos( f) sin( f) drdfdq = -Gmm ó õ 2p 0  ó õ sin-1( R/r) 0  ( r2-r1) cos( f) sin( f)drdfdq

Since r₂-r₁ = 2( R²-r²sin²( f)) ^1/2, this in turn leads to

| F| = -2Gmm ó õ 2p 0  ó õ sin-1(R/r) 0  ( R2-r2sin2( f) )1/2sin( f) cos( f) dfdq

If we let u( f) = R²-r²sin²( f) , then the limits of integration become

u( 0) = R2        and        u æ è sin-1 æ è  R r ö ø ö ø = R2-r2 æ è  R2 r2 ö ø = 0

Moreover, du = -2r²2sin( f) cos( f)df, so that

| F| =  Gmm r2 ó õ 2p 0  ó õ sin-1( R/r) 0  ( R2-r2sin2( f)) 1/2  ( -2r2sin( f) cos( f) ) dfdq =  Gmm r2 ó õ 2p 0  ó õ 0 R2  u1/2du  dq =  Gmm r2 ó õ 2p 0   u3/2 3/2 ê ê 0 R2    dq =  -Gmm r2 ó õ 2p 0   2R3 3 dq =  -Gm r2  m 4pR3 3

However, the volume of the sphere is V = 4pR³/3, so that the mass of the sphere is M = mV = m 4pR³/3. Thus, we have shown that

| F| =  -GMm r2

That is, a uniformly-dense spherical ''planet'' of mass M and a point-mass of mass M at the center of the sphere have the same gravitational attraction on a ''satellite'' point mass outside the sphere. Since the electromagnetic force also satisfies an inverse square law, this result also says that the electromagnetic force between spheres with uniform charge density is equivalent to the electromagnetic force between point-charges.