(adsbygoogle = window.adsbygoogle || []).push({}); 1. The problem statement, all variables and given/known data

I'm well aware of how to compute the gravitational [electric] potential [tex]\Phi[/tex] due to a spherical mass [charge] distribution of radius [tex]R[/tex] by using Newton's theorems for spherical shells. However, how does one find an analytic expression for [tex]\Phi[/tex] without invoking these theorems?

2. Relevant equations

[tex]\rho(r, \theta, \phi) = \rho(r) = M \delta(r - R)[/tex]

or equivalently:

[tex]\rho(\vec{x}) = M \delta(x_1^2 + x_2^2 + x_3^2 - R^2)[/tex]

[tex]\Phi(\vec{x}) = -\int_{\tau} \frac{G \rho(\vec{x}')}{|\vec{x}' - \vec{x}|} \; d^3x'[/tex]

3. The attempt at a solution

Now I can't simply transform the equation for potential into spherical coordinates because vector subtraction is involved. Furthermore, taking the line integral of gravitational force does not make things easier, since [tex]\vec{F} = \int_{\tau} \frac{G \rho(\vec{x}')(\vec{x}' - \vec{x})}{|\vec{x}' - \vec{x}|^3} \; d^3x'[/tex].

Any help is welcome.

**Physics Forums | Science Articles, Homework Help, Discussion**

Dismiss Notice

Join Physics Forums Today!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

# Computing Potential of a Spherical Shell w/o using Newton's theorems

**Physics Forums | Science Articles, Homework Help, Discussion**