Question

$1000$ small water drops each of capacitance C join together to form one large spherical drop. The capacitance of a bigger sphere is

• A. $C$
• B. $10C$
• C. $100C$
• D. $1000C$

Answer 1

The correct option is B $10C$

$\text{Step 1: Volume conservation}$

The combined volume of $n$ smaller drops is equal to the volume of large sphere.

$V_i=V_f$

$\begin{array}{c}N\times \frac{4}{3}\pi r^3\end{array}=\begin{array}{c}\frac{4}{3}\pi R^3\end{array}$ (Here "r" is radius of each small sphere and "R" is radius of big sphere)

$1000\times \begin{array}{c}\frac{4}{3}\pi r^3\end{array}$ = $\begin{array}{c}\frac{4}{3}\pi R^3\end{array}$

$\therefore R=10r$

$\text{Step 2: Comparision of capcitances}$

Capacitance of sphere$=4\pi {\in }_0.\left(radius\right)$

Capcitance of smaller sphere

$C=4\pi {\in }_{0}r$ $....\left(1\right)$

Capacitance of larger sphere

$C_f=4\pi {\in }_{0}R=40\pi {\in }_{0}r=10C$

$\Rightarrow C_f=10C$ (Using Equation $1$)

Hence the capacitance of larger sphere is $10C$.

Option (B) correct