Does the Triple Integral Formula Apply to a Point-Mass Inside a Spherical Shell?

[cscott]

Does $-Gm\rho2\pi\left(R_2^2-R_1^2\right)$ make sense for the potential of a point-mass "m" inside a spherical shell of radii $R_1< R_2$ and density $\rho$?

Now I've already found the potential outside of a homogeneous sphere of same density. I'm now asked to use these two results to find the potential inside a homogeneous sphere of again indeity density sphere, using the previous two results.

Can I do this by considering the point on the inside of the shell with a sphere in the space in the middle? Does it make sense to add the two potential formulas using the correct radii?

 

[rl.bhat]

Expression for potential is GMm/R. So something is missing in the given expression

 

[ogb p]

Yes, it does make sense to use the triple integral formula for calculating the potential inside a spherical shell with a point-mass inside. The equation you have provided, -Gm\rho2\pi\left(R_2^2-R_1^2\right), is the correct formula for calculating the potential inside a spherical shell with a point-mass inside. This can be derived by considering the potential due to the point-mass and the potential due to the spherical shell separately and then adding them together. The radii used in the formula should correspond to the radii of the point-mass and the spherical shell, as you have correctly mentioned.

In order to find the potential inside a homogeneous sphere with the same density, you can use the same approach and add the potential due to the homogeneous sphere and the potential due to the point-mass separately. The resulting formula would be different from the one for a spherical shell, but it would still involve triple integration. This is because the potential due to a homogeneous sphere is not the same as the potential due to a spherical shell.

Therefore, it is possible to use the previous two results to find the potential inside a homogeneous sphere by considering the point on the inside of the shell as a separate point-mass and adding its potential to the potential of the homogeneous sphere. This approach is valid and makes sense from a mathematical perspective. However, it is important to note that this approach assumes that the point-mass is located at the center of the homogeneous sphere. If the point-mass is located at a different position, then the formula for the potential would be different.