Question

The self-energy of a conducting shell of radius $R$ and charge $Q$ is:
A. ${\displaystyle \frac{k{Q}^2}{R}}$
B. ${\displaystyle \frac{k{Q}^2}{2R}}$
C. ${\displaystyle \frac{2k{Q}^2}{R}}$
D. ${\displaystyle \frac{3k{Q}^2}{5R}}$

Answer 1

Hint: The work done to make a body charged is stored in the form of energy. This energy is known as self-energy. It is assumed that when a body is charged small charges are brought near the object from infinity and charges are brought one after others they don’t interact with the object in bulk.

Complete answer:
When an object is charged a charge is brought near the object from the infinity. Hypothetically it is assumed that when the first charge is brought from infinity to a non-charged it does not experience any kind of force (opposing force). When the other charge particle (carrier) is brought near to the object it experiences some force exerted by the charge already present on the object.

Let us suppose that at a time $t$ the instantaneous charge is brought towards the object be $dq$. So the work done by the body to make the object charged will be,
$\begin{array}{rl}& dW=dt\mathrm{\Delta }V\\ & \Rightarrow dW={\displaystyle \frac{kq}{R}}dq\end{array}$
Where,
$k$ is the constant whose value is ${\displaystyle \frac{1}{4\pi {\epsilon }_0}}or9\times {10}^9$
$R$ is the radius of the sphere
$q$ is the charge already present on the surface
$dq$ is the charge brought from infinity to the surface
Let the maximum charge that the spherical shell can gain in time $$t$$ is $Q$. By integrating both the sides of the above equations over the of time 0 to $t$ and charge 0 to $Q$. The work done by the charge is given by,
$\begin{array}{rl}& \Rightarrow W={\displaystyle \frac{k}{R}}{\int }_0^{Q}qdq\\ & \Rightarrow W={\displaystyle \frac{k}{R}}{\left[{\displaystyle \frac{q^2}{2}}\right]}_0^Q\\ & \Rightarrow W=k{\displaystyle \frac{Q^2}{2R}}\end{array}$
The work done is stored in the form of energy inside the sphere. The stored energy is termed as self-charge of the body.

So, the correct option showing the true value of the self-charge on the body is Option B.

Note:
When a charge does some work against some opposing force it develops certain potential differences between the object and the surroundings. This potential difference gives rise to a voltage. This potential difference is very necessary for the charge flow in a circuit. If there is no potential difference there will be zero charge flow in the circuit.