Question

Assume each oil drop consists of a capacitance of $C$. If combine $n$ drops to form a biggerdrop, then the capacitance of bigger drop $C$' would be

• A. $C^{\prime }=\frac{2n^{\frac{1}{3}}}{3}C$
• B. $C^{\prime }=\frac{5n^{\frac{1}{3}}}{4}C$
• C. $C^{\prime }=\frac{n^{\frac{1}{3}}}{5}C$
• D. $C^{\prime }=C\cdot n^{\frac{1}{3}}$

Answer 1

Answer: (D)

Capacitance of an oil drop of radius $r$ is
$C=4\pi \epsilon r$
when $n$ drops combine to form a large drop of radius $R$,
then $n\times$ volume of 1 small drop $=$ volume of 1 large drop
$n\times \frac{4}{3}\pi r^3=\frac{4}{3}\pi R^3$
$\Rightarrow R=n^{1/3}\cdot r$
So, capacitance of large drop is
$C^{\prime }=4\pi \epsilon R=4\pi \epsilon \cdot n^{\frac{1}{3}}\cdot r=n^{\frac{1}{3}}C$