Solving potential of electron inside the nucleus

[haseeb]

Homework Statement

I want to derive the following equation. It is the potential energy of an electron inside a nucleus assumed to be a uniformly charged sphere of R.

Homework Equations

V'(r) =( -Ze2/4∏ε0R)(3/2 - (1/2)(r/R)^2)

The Attempt at a Solution

I get E = Ze2r/(4∏ε0R3)

But I am having problem in integration limits and hence going towards final required result!

 

[mfb]

Something went wrong with the formatting.

The electric field should be linear with the distance - right. You can check if the electric field at the border of the nucleus agrees with the electric field of a point-charge. If yes, your result is right.

 

[haseeb]

Something went wrong with the formatting.

The electric field should be linear with the distance - right. You can check if the electric field at the border of the nucleus agrees with the electric field of a point-charge. If yes, your result is right.

First, Electrical field is not linearly dependent on distance! And I am not trying to find the field at the nucleus border but inside it. Can you help please to reach the desired formula?

 

[mfb]

Electrical field is not linearly dependent on distance!

Inside a homogeneously charged sphere, it is proportional to the radius, its magnitude is proportional to the distance to the center.

If your formula reproduces this relationship and agrees at the boundary, then it is right. Hence the suggestion to check if it fits at the boundary.

 

[haseeb]

Inside a homogeneously charged sphere, it is proportional to the radius, its magnitude is proportional to the distance to the center.

If your formula reproduces this relationship and agrees at the boundary, then it is right. Hence the suggestion to check if it fits at the boundary.

Yup! You are right... But How to drive it mathematically?

 

[mfb]

The shell theorem should help.