Nuclear worries should be of a small magnitude right? But it sure ain't. Sigh.
You know, one of the most interesting things about Nuclear Physics is that there’s no fixed boundaries that define the study of this field of Physics. What do I mean? Well, take Electrodynamics for instance: you have Maxwell’s Four Equations, that clearly set a basis from which all other theories of Electrodynamics are derived. For Classical Mechanics, there’s Newton’s Laws of Motions, and for Classical Dynamics, there’s the Lagrangian. So clearly, for these fields, there’s a fixed set of rules to go by.
But for Nuclear Physics, we base it on Quantum Mechanics (for the behavior of nuclei), Electrodynamics (for the charge distributions), Relativity (taking reduced mass and binding energies into account) and basic Mechanics and Dynamics (collisions etc.). So there’s really no limit to what can be done in this field.
Which explains why I’m actually finding myself doing so much for the Physics module I’m doing this semester, haha. So what does my Professor want me to figure out? Well, for now it’s Electrodynamics!
So my Professor goes and says this:
“The potential energy within a homogenously charged sphere of charge Q and radius R due to an interaction with another charge q within its interior at a distance r from its centre is given as:
And well, you all should know this already.”
I should? Well I sure don’t! So I’m going to try and derive it now instead of taking it for granted, haha.
Well so how do we start off? Easy, let us consider what we’re looking at in terms of a pictorial representation:
So what we have here is just a big blue sphere of charge Q, of radius R, and the small charge is placed within its interior, at a distance r away from the centre. So how do we go about solving this question? Well it’s easy – we assume the charge is homogenously distributed within the sphere, meaning each portion of the sphere bears the same amount of charge (i.e. uniform charge density ρ).
In that case, the charge q experiences the effective amount of charge that is contained within the bound radius of r instead of the whole sphere’s radius of R, marked out in pink:
In which case we can then write the electric force on the charge q as:
But we all know that the effective charge contained within the radius r is actually:
But we do know that the effective volume V of the pink region is:
And of course, the charge density is given by dividing the total charge Q by the total volume of the sphere:
Putting everything together:
The next part is to recognize that the Coulombic force is related to the potential energy as such:
And therefore conclude that:
So integrating this expression:
Now, to determine the constant k, we note first that when r = R, that is, where the small charge is on the surface of the big spherical charge, the potential energy is simply:
And therefore we must insist that:
Putting everything together: