The radius of a spherical soap bubble is r and surface tension of soap water is T respectively. Find the charge uniformly distributed over the outer surface of the bubble is required to double its radius. (Atmospheric pressure is $P_{o}$ and inside temperature of the bubble during expansion remain constant)

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$8 πr {[ϵ_{0} r (7 P_{0} r + 16 T)]}^{1 / 2}$

$πr {[\in_{0} r (7 P_{0} r + 12 T)]}^{1 / 2}$

$8 πr {[ϵ_{0} r (4 P_{0} r + 12 T)]}^{1 / 2}$

$4 πr {[\in_{0} r (7 P_{0} r + 8 T)]}^{1 / 2}$

answer is A.

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Detailed Solution

Before charging the bubble pressure inside the bubble $P_{1} = P_{0} + \frac{4 T}{r}$
After charging electrostatic pressure will be $\frac{σ^{2}}{2 ϵ_{0}}$
When radius becomes 2r then $\frac{P_{1}}{8} + \frac{σ^{2}}{2 ϵ_{0}} = P_{o} + \frac{4 T}{2 r}$
Where $σ = \frac{q}{4 π (2 r)^{2}}$