Question

if 64 identical droplets merge to a single drop, find the ratio of their surface charge density and voltage​

Answer 1

64 × volume of a smaller drop = volume of a bigger drop

or, 64 × 4/3 πr³ = 4/3 πR³

or, (4r)³ = R³ => R = 4r......(1)

Let charge on each smaller drop is q

then, surface charge density on each smaller drop , $\sigma_s$= q/(4πr²) ...... (2)

then, total charge on 64 smaller drops = 64q

area of bigger drop, A = 4πR² = 4π(4r)²

= 64πr³

so, surface charge density on bigger drop, $\sigma_b$ = 64q/(64πr³)

= q/πr³ ........(3)

from equation (2) and (3),

$\frac{\sigma_s}{\sigma_b}=\frac{1}{4}$

hence, surface charge density on bigger drop is 4 times of surface charge density on smaller drop.

voltage on smaller drop, v = kq/r

voltage on bigger drop, V = k(64q)/(4r)

= 16kq/r

so, voltage on bigger drop is 16 times of voltage on smaller drop.