Question

A spherical drop of capacitance $1\mu F$ is broken into eight drops of equal radius. Then, the capacitance of each drop will be
A) $\dfrac{1}{2}\mu F$
B) $\dfrac{1}{4}\mu F$
C) $8\mu F$
D) $8\mu F$

Answer 1

Hint: Use the formula of capacitance of a spherical conductor to find the radius of the drop in the first case. Now, find the radius of the drop when it is divided into eight parts. We can use the fact that the volume of the drop remains conserved before and after splitting. Then, again use this new radius to find the capacitance of the smaller drop.

Complete step by step answer:
Let the radius of the bigger drop be $R$.
Capacitance of a spherical shell is given by the formula,
$C = 4\pi {\varepsilon _0}R$
Where ${\varepsilon _0}$ is the permeability of an electric field in vacuum.
But, in this question, we can assume that the inside of the capacitor is filled with water,
So,
The capacitance will become,
$C = 4\pi {\varepsilon _W}R$
Where,
${\varepsilon _W}$ is the electrical permeability in water.
The capacitance of the larger drop is given $1\mu F$,
$1 = 4\pi {\varepsilon _W}R$
$R = \dfrac{1}{{4\pi {\varepsilon _W}}}$
Now,
The drop is divided into eight smaller drops,
Let the radius of each smaller drop be $r$
On conserving volume,
$\dfrac{4}{3}\pi {R^3} = 8 \times \dfrac{4}{3}\pi {r^3}$
$r = \dfrac{R}{2}$
Putting the value of $R = \dfrac{1}{{4\pi {\varepsilon _W}}}$
$r = \dfrac{1}{{8\pi {\varepsilon _W}}}$

Capacitance of smaller drop,
$C = 4\pi {\varepsilon _W} \times \dfrac{1}{{8\pi {\varepsilon _W}}}$
$C = \dfrac{1}{2}\mu F$
So, A is correct.

Note: The formula we used to find the capacitance of the drop is actually the formula to find the capacitance of a shell.
We have used this formula even in the case of drop because the rain drops are very good conductors. If we give charge to a body which is a conductor, all of the charge travels to the surface and makes a shell itself. This is because charge will repel each other inside the conductor and try to maximize the distance between them hence making a shell.