Question 171: Electrostatics & 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 ( $\mathrm{V}_1$ ) on the surface of one such small bubble is given by the formula for the potential of a charged sphere:
$$ \mathrm{V}_1=\frac{1}{4 \pi \epsilon_0} \frac{\mathrm{q}}{\mathrm{r}} $$
Three such identical bubbles coalesce to form one large bubble.
Volume of 3 small bubbles $=$ Volume of 1 large bubble
$$ 3 	imes\left(\frac{4}{3} \pi r^3ight)=\frac{4}{3} \pi R^3 $$
$$\Rightarrow $$ $$ 3 r^3=R^3 \Rightarrow R=3^{1 / 3} r $$
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.
$$ \mathrm{Q}=\mathrm{q}+\mathrm{q}+\mathrm{q}=3 \mathrm{q} $$
The electric potential ( $\mathrm{V}_2$ ) on the surface of the resultant bigger bubble is :
$$ V_2=\frac{1}{4 \pi \epsilon_0} \frac{Q}{R} $$
$\Rightarrow V_2=\frac{1}{4 \pi \epsilon_0} \frac{3 q}{3^{\frac{1}{3}} r}=3^{\frac{2}{3}}\left(\frac{1}{4 \pi \epsilon_0} \frac{q}{r}ight)$
$$ \Rightarrow \mathrm{V}_2=3^{\frac{2}{3}} \mathrm{~V}_1 $$
So, the ratio of the potential on one initial bubble ( $\mathrm{V}_1$ ) to that on the resultant bigger bubble ( $\mathrm{V}_2$ ),
$$ \frac{V_1}{V_2}=\frac{V_1}{3^{\frac{2}{3}} V_1}=\frac{1}{3^{\frac{2}{3}}} $$
So, the ratio is $1: 3^{2 / 3}$. Hence, the correct option is (A).

Answer

$1: 3^{2 / 3}$

Answer

$3^{2 / 3}: 1$

Answer

$1: 3^{1 / 3}$

Answer

$1: 2^{2 / 3}$

Electrostatics & 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: