Mathematical Tools for Physics

9—Vector Calculus 1 272

9.11 More Complicated Potentials
The gravitational field from a point mass is~g=−Gmr/rˆ^2 , so the potential for this point mass isφ=−Gm/r.
This satisfies

~g=−∇φ=−∇

−Gm

r

=ˆr

∂

∂r

Gm

r

=−

Gmˆr

r^2

For several point masses, the gravitational field is the vector sum of the contributions from each mass. In the
same way the gravitational potential is the (scalar) sum of the potentials contributed by each mass. This is almost
always easier to calculate than the vector sum. If the distribution is continuous, you have an integral.

φtotal=

∑

−

Gmk

rk

or −

∫

Gdm

r

~r′

~r

x′

y′

z′

This sort of very abbreviated notation for the sums and integrals is normal once you have done a lot of them, but
when you’re just getting started it is useful to go back and forth between this terse notation and a more verbose
form. Expand the notation and you have

φtotal(~r) =−G

∫

dm

∣

∣~r−~r′

∣

∣ (44)

This is still not very explicit, so expand it some more. Let

~r′=xxˆ ′+ˆyy′+ˆzz′ and ~r=xxˆ +yyˆ +ˆzz

then φ(x,y,z) =−G

∫

dx′dy′dz′ρ(x′,y′,z′)

1

√

(x−x′)^2 + (y−y′)^2 + (z−z′)^2

whereρis the volume mass density so thatdm=ρdV, and the limits of integration are such that this extends
over the whole volume of the mass that is the source of the potential. The primed coordinates represent the