Self Energy of Uniformly Charged Thin Spherical Shell

Self Energy

The work that is done in charging a thin spherical shell is stored in the form of energy. This energy stored is called self-energy.

Determination of Self Energy of Uniformly Charged Thin Spherical Shell – Method 1

Let us assume that small charges are brought from infinity to the thin spherical shell. When the first charge is brought from infinity to the spherical shell without any charge, it will not experience any opposing force. When the second charge is brought near the spherical shell, it will experience an opposing force due to the charge already present in the shell. As more and more charges are added to the spherical shell, the opposing force will increase. Therefore, more work has to be done to add a charge to the spherical shell. This work done gets stored as energy.

At a time t, let the instantaneous charge of the spherical shell be q. So, the work done to bring a charge ‘dq’ from infinity to the spherical shell is taken as dW. When work is done against an opposing force, a certain potential difference is developed between the spherical shell and the surrounding area. Therefore,

Work done, dW = dq.dV ——–(1)

Here, dV is the potential difference.

dV = V_R – V_∝

= Kq/R – 0

= Kq/R ————–(2)

V_R is the potential on the surface of the spherical shell of radius R.

V_∝  is the potential at infinity.

Substituting the value of dV in equation (1), we get

dW = dq.(Kq/R)

k is a constant [ k = (1/4πε₀)]

q is the charge already present in the spherical shell at time t.

dq is the charge brought from infinity.

To determine the total work done in charging the spherical shell from 0 to Q, the above equation must be integrated.

This work done is stored in the form of self-energy in the spherical shell.

Determination of Self Energy of Uniformly Charged Thin Spherical Shell – Method 2

We are going to use a different approach to determine the self-energy of the spherical shell. Let us consider the spherical shell of radius R. The charge on the shell is taken as Q. The electrostatic energy will be stored inside the shell and outside the spherical shell. Therefore,

Self Energy, U_self = U_inside + U_outside—————–(1)

U_inside = Energy inside the spherical shell

U_outside = Energy outside the spherical shell

(Since the electric field (E) inside a uniformly charged spherical shell is zero, the energy inside the shell will also be equal to zero)

Putting these values in equation (1), we get

Substituting the value of the electric field outside the spherical shell and the volume of the spherical shell, we get

Therefore, the value of self-energy is given by

Video Lesson – Self Energy of Uniformly Charged Thin Spherical Shell