Calculate gravitational potential due to a spherical shell of radius R at a point P.

Phi(g) = -G sigma integral dA/s, where

• sigma is mass/area = M/(4 pi R^2)
• s is the distance from the field point P to where we are integrating

• integrate over a spherical shell of radius R
• strip of thickness R d theta, width 2 pi R sin theta
• r is the position P where we calculate the potential
• s is the distance between the strip and P, which is on the z-axis
• s^2 = R^2 +r^2 - 2rR cos theta
• Phi(g) = -G M/(4 pi R^2) 2 pi R^2 integral sin theta d theta /s
• if we let x = cos theta, then dx = - sin theta d theta and
• the integral becomes G sigma 2 pi R^2 integral dx/s, or
• Phi(g) = -G M/2 integral dx/sqrt(R^2 +r^2 - 2rR x)
• this integrates to -GM/(-2rR)[ |R-r|-(R+r)]
• if r<R we are inside and the potential becomes -GM/R, independent of r
• inside, the potential is the same everywhere, so there is no force on a mass inside a shell from the shell
• outside, we get -GM/r, and the shell of mass M acts like a point source at the origin
• if we take the differential of s^2 = R^2 +r^2 - 2rR cos theta [see text p. 194] we get
• 2 s ds = 2rR sin theta d theta and the integral immediately simplifies to
• Phi(g) = -G M/(2rR) integral s ds/ s) = -G M/(2Rr) integral ds
• if we are inside ds integrates to R+r -(R-r) = 2r and Phi(g) = -G M/R
• if we are outside we get Phi(g) = -GM/r