Question 28: Electrostats & Capacitance

Question

Three small identical bubbles of water having same charge on each coalesce to form a bigger bubble. Then the ratio of the potentials on one initial bubble and that on the resultant bigger bubble is :

Accepted Answer

When small liquid bubbles (or drops) coalesce to form a larger one, two fundamental quantities are conserved: the total charge and the total volume.Let the radius of each small initial bubble be $r$ and the charge on each be $q$.The electric potential ($V1$) $on the surface of one such small bubble is given by the formula for the potential of a charged sphere:$$V_1 = \frac{1}{4\pi\epsilon_0} \frac{q}{r} 	ext{}$$Three such identical bubbles coalesce to form one large bubble.Volume of 3 small bubbles = Volume of 1 large bubble$$3 	imes \left( \frac{4}{3}\pi r^3 ight) = \frac{4}{3}\pi R^3 	ext{}$$$$\Rightarrow 3r^3 = R^3 \Rightarrow R = 3^{1/3}r 	ext{}$$Using conservation of charge, the total charge ($Q$) on the new bigger bubble is simply the algebraic sum of the charges of the three initial bubbles.$$Q = q + q + q = 3q 	ext{}$$The electric potential ($V2$) on the surface of the resultant bigger bubble is:$$V_2 = \frac{1}{4\pi\epsilon_0} \frac{Q}{R} 	ext{}$$$$\Rightarrow V_2 = \frac{1}{4\pi\epsilon_0} \frac{3q}{3^{1/3}r} = 3^{\frac{2}{3}} \left( \frac{1}{4\pi\epsilon_0} \frac{q}{r} ight) 	ext{}$$$$\Rightarrow V_2 = 3^{\frac{2}{3}} V_1 	ext{}$$So, the ratio of the potential on one initial bubble ($V_1$) to that on the resultant bigger bubble ($V_2$),$$\frac{V_1}{V_2} = \frac{V_1}{3^{\frac{2}{3}}V_1} = \frac{1}{3^{\frac{2}{3}}} 	ext{}$$  $So, the ratio is $1 : 3^{2/3}$. Hence, the correct option is (A).

Answer

A. $1 : 3^{2/3}$

Answer

B. $3^{2/3} : 1$

Answer

C. $1 : 3^{1/3}$

Answer

D. $1 : 2^{2/3}$

Electrostats & Capacitance

Three small identical bubbles of water having same charge on each coalesce to form a bigger bubble. Then the ratio of the potentials on one initial bubble and that on the resultant bigger bubble is :

Choose the correct answer: