Question

A spherical drop of capacitance $12\mu F$ is broken into eight drops of equal radius. What is the capacitance of each small drop in $\mu F$?
A. 6
B. 4
C. 3
D. 1

Answer 1

In this question, we need to determine the capacitance of each drop-in micro-farads such that a spherical drop of $12\mu F$ is broken. For this, we will use the relation between voltage, charge and the radius of the sphere.

Complete step by step answer:
Electrical potential on a charged sphere is given by the formula
$V = \dfrac{{kQ}}{R}$, where $Q$ is the charge on the sphere, $R$ is the radius of that sphere and $k$ is the constant which is equal to $9 \times {10^9}N$.
The capacitance of spherical drop $C = 12\mu F$ .We know that the electrical potential on a charged sphere is given by the formula
$V = \dfrac{{kQ}}{R} - - (i)$
The charge on a capacitor plate is determined by the formula
$Q = CV - - (ii)$
Now we substitute the equation (ii) in equation (i) to find the capacitance on the sphere; hence we get

$$V = \dfrac{{k\left( {CV} \right)}}{R} \\\ 1 = \dfrac{{kC}}{R} \\\ C = \dfrac{R}{k} - - (iii) \\\$$

Since the capacitance on the spherical drop is given as $C = 12\mu F$ hence, we can write equation (iii) as
$\dfrac{R}{k} = 12\mu F$
Now it is said that spherical drop is broken into eight drops of equal radius, let us assume that the radius of the small drop is$r$, so by comparing the volume of the smaller drops and the larger drop we can write
$8\left( {\dfrac{4}{3}\pi {r^3}} \right) = \dfrac{4}{3}\pi {R^3}$
Hence by solving we get

$$8\left( {\dfrac{4}{3}\pi {r^3}} \right) = \dfrac{4}{3}\pi {R^3} \\\ \Rightarrow {\left( {\dfrac{R}{r}} \right)^3} = 8 \\\ \therefore \dfrac{R}{r} = 2 - - (iv) \\\$$

So the capacitance on each smaller drop will be equal to
$V = \dfrac{r}{k} - - (iv)$
Now we substitute the value of r from equation (iv), we get
$V = \dfrac{R}{{2k}}$
Where $\dfrac{R}{k} = 12\mu F$, hence the value of capacitance on smaller drop will be

$$V = \dfrac{{12}}{2} \\\ = 6\mu F \\\$$

So, the correct answer is Option A.

Note:
Students must note that the electric potential on a charged sphere reduces when that charged sphere is divided into smaller spheres.
It is said that a spherical drop is broken into eight drops of equal radius, let us assume that the radius of the small drop.