Nuclear Model - Expression for Total Energy [Modern Physics]

[twotaileddemon]

Homework Statement

Derive an expression for the total energy required to asemble a sphere of charge corresponding to a nucleus of atomic number Z and radius R. Assume the nucleus is a sphere of uniform volume charge density $$\rho$$

Homework Equations

$$\rho$$ = mass / volume = 3Am / 4*pi*R³

The Attempt at a Solution

I know the solution is U = 3k * (Ze)² / 5R, so I need to work on deriving this.

I also know that when an alpha particle collides with the nucleus, the initial kinetic energy is equal to the electrical potential energy of the system and is given by
.5mv² = kq₁q₂/r = k(2e)(Ze)/d where d = 4kZe² / mv²

Not sure how this helps though, but it seems to be in a relevant section in the textbook. There is also info on the binding energy, but it doesn't seem applicable in this case.

Any tips on how to approach the problem, please?

 

[Matterwave]

Have you ever learned of the gravitational analog? Wherein the gravitational potential energy of a sphere of mass M and radius R is -3/5GM^2/R?

You can apply that analog perfectly to this case.

If you haven't seen that. Consider a spherical shell (infinitessimal thickness), what is the potential energy of this shell? How would I go about adding up all these spherical shells to form a sphere?

 

[twotaileddemon]

Have you ever learned of the gravitational analog? Wherein the gravitational potential energy of a sphere of mass M and radius R is -3/5GM^2/R?

You can apply that analog perfectly to this case.

If you haven't seen that. Consider a spherical shell (infinitessimal thickness), what is the potential energy of this shell? How would I go about adding up all these spherical shells to form a sphere?

I think I remember learning about that in a different course, let me check my notes and I'll get back to you. Thanks for the tip!

EDIT 4/19: I was able to figure out how to derive the proof. Thank you!