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Q.

A spherical bubble of radius $r_{1}$ is formed inside another bubble of radius $r_{2}$ . The radius of a soap bubble which maintain the same pressure difference as inside the smaller and outside the larger, is

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a

$\frac{r_{1} r_{2}}{r_{1} + r_{2}}$

b

$r_{1} + r_{2}$

c

$\frac{r_{1} r_{2}}{r_{1} - r_{2}}$

d

$r_{1} - r_{2}$

answer is C.

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Detailed Solution

$\begin{array}{l} P_{1} - P_{2} = \frac{4 T}{r_{1}} \\ P_{2} - P_{0} = \frac{4 T}{r_{2}} \\ ΔP = P_{1} - P_{0} \\ \Rightarrow P_{1} - P_{0} = 4 T (\frac{1}{r_{1}} + \frac{1}{r_{2}}) \\ \Rightarrow \frac{4 T}{r} = 4 T (\frac{r_{1} + r_{2}}{r_{1} r_{2}}) \end{array}$

$\Rightarrow r = \frac{r_{1} r_{2}}{r_{1} + r_{2}}$

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