Question

Q1. A spherical drop of mercury of radius R has a Capacitance given by C = 4πϵ0 R. If two such drops combine to form a single larger drop, what is its capacitance?

Answer 1

Given:

A spherical drop of mercury of radius = R.

Capacitance =  4πϵ0 R

Two such drops combine to form a single larger drop.

To Find:

The capacitance of the final drop.

Solution:

We can see that,

the capacitance of the drop of radius R is C = 4π∈0 R; i.e.,

• Capacitance is proportional to  the radius.

Now, we combine two droplets, then,

• the volume of the final combined droplet will be sum of the volume of the two other drops.
• Therefore, volume  will be twice that of the smaller  droplets

• We know, volume = 4/3πR³

Let Vf be the volume of the bigger drop and V be the volume of smaller drop.

Let Rf be the radius of the bigger drop.

• Vf = 2V
• 4/3πRf³ = 2 x 4/3πR³
• Rf³ = 2R³
• Rf = r∛2 = 1.26  R
• Let Cf be the capacitance of the bigger drop.

Therefore , Cf = 4π∈0 Rf = 4π∈0 ( 1.26) R = 1.26(C)

Capacitance of larger drop will be 1.26 times that of the smaller drops.