Q: A conducing spherical bubble of radius r and thickness t (t >> r) is charged to a potential V. Now it collapses to form a spherical droplet. Find the potential of the droplet.

Sol. Here charge and mass are conserved. If R is the radius of the resulting drop formed and ρ is density of soap solution,

$\large \frac{4}{3}\pi R^3 \rho = 4 \pi r^2 t \rho $

$\large R = (3 r^2 t)^{1/3}$

Now potential of the bubble is

$\large V = \frac{1}{4 \pi \epsilon_0} \frac{q}{r}$

or , $\large q = 4 \pi \epsilon_0 r V $

Now potential of resulting drop is

$\large V’ = \frac{1}{4 \pi \epsilon_0} \frac{q}{R}$

$\large V’ = \frac{1}{4 \pi \epsilon_0} \frac{4 \pi \epsilon_0 r V}{(3 r^2 t)^{1/3}}$