(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.

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

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