Electrostatic Self-Energy of a Uniform Density Sphere of Charge

[harshey]

Find an expression for the electrostatic self-energy of an arbitrary spherically symmetric charge density distribution p(r). You may not assume that p(r) represents any point charge, or that it is constant, or that it is piecewise constant, or that it does or does not cut off at any finite radius r. your expression must cover all possibilities. your expression may include an integral or integrals which cannot be evaluated without knowing the specific form of p(r).

I had no idea how to start this problem because i couldn't figure out what my professor meant by electrostatic self-energy of an arbitrary spherically symmetric charge density distribution.

 

[Fungiss]

Well, self-energy is potential energy of unbalanced charge system (in this case the sphere had just charged and not in equilibrium state).

Your answer:
Ws = 1/(4pi*epsilon0)*3/5*p(r)*V/R.

You could start from general definition formula of Ws = 1/2*q*V, then calculate for 3 case: r < R, r > R, and r = R. The cases r = R and r > R help define constant C in case r < R.

 

[ogb p]

I would first clarify with my professor the specific definition and context of the term "electrostatic self-energy." This term could refer to the energy stored in a charged object due to its own electric field, or it could have a different meaning in this particular problem.

Assuming that the electrostatic self-energy refers to the energy stored in a charged object due to its own electric field, I would approach the problem by considering the electric potential energy of a small element of charge at a distance r from the center of the sphere. This potential energy would be given by the equation:

dU = kq(r)q(r')/r dr'

Where k is the Coulomb constant, q(r) is the charge density at a distance r from the center of the sphere, and q(r') is the charge density at a distance r' from the center of the sphere. This equation can be used to calculate the potential energy of each small element of charge in the sphere.

To find the total electrostatic self-energy of the sphere, we would need to integrate this equation over the entire volume of the sphere, taking into account all possible values of r and r'. This integral would involve the charge density distribution p(r) and would cover all possibilities, including cases where p(r) is not constant, piecewise constant, or has a finite cutoff radius.

Therefore, an expression for the electrostatic self-energy of an arbitrary spherically symmetric charge density distribution p(r) would be:

U = ∫∫ k p(r)p(r')/|r-r'| dV

Where dV is the volume element and the integral is taken over the entire volume of the sphere. This expression covers all possibilities and includes an integral that cannot be evaluated without knowing the specific form of p(r).

In summary, as a scientist, I would first clarify the definition of electrostatic self-energy in this context and then approach the problem by using the concept of electric potential energy and integrating over the entire volume of the sphere to find a general expression for the electrostatic self-energy.