What is a Van de Graaff generator? Explain its principle.

(N/A) Principle: The Van de Graaff generator is based on two electrostatic principles:
$1$. The property that charge given to a hollow conductor is transferred to its outer surface and spreads uniformly over it.
$2$. The property that the electric potential of a conductor is higher when it is smaller in size.
Mathematical derivation:
Let a charge $Q$ be placed on a spherical shell of radius $R$. The potential at the surface is $V(R) = \frac{1}{4 \pi \epsilon_{0}} \cdot \frac{Q}{R}$.
If a small sphere of radius $r$ carrying charge $q$ is placed inside the shell,the total potential at the surface of the small sphere is $V(r) = \frac{1}{4 \pi \epsilon_{0}} \left( \frac{Q}{R} + \frac{q}{r} \right)$ and at the surface of the large shell is $V(R) = \frac{1}{4 \pi \epsilon_{0}} \left( \frac{Q}{R} + \frac{q}{R} \right)$.
The potential difference is $V(r) - V(R) = \frac{q}{4 \pi \epsilon_{0}} \left( \frac{1}{r} - \frac{1}{R} \right)$.
Since $r < R$,the term $(\frac{1}{r} - \frac{1}{R})$ is positive,meaning the inner sphere is at a higher potential than the outer shell. When connected,charge flows from the inner sphere to the outer shell,allowing the accumulation of a very high potential on the outer shell.