- #1
v0id
- 45
- 0
Homework Statement
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?
Homework 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]
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.