Question

A small soap bubble of radius 4 cm is trapped inside another bubble of radius 6 cm without any contact. Let $P_2$ be the pressure inside the inner bubble and $P_0$, the pressure outside the outer bubble. Radius of another bubble with pressure difference $P_2$ –$P_0$, between its inside and outside would be
A 6cm
B 12cm
C 4.8cm
D 2.4cm

Answer 1

For the bubble to be stable and not collapse, the pressure inside the bubble must be higher than the pressure on the outside. The force due to the pressure difference must balance the force from the surface tension. The force to the pressure difference is ($P_i$ - $P_o$)πr.
For a soap bubble with two surfaces $P_i$ - $P_o$ = 4T/r.

Complete step by step answer:
As we enter the soap bubble the pressure increases by $\dfrac{{4T}}{r}$, where T is the surface tension and r is the radius of the bubble.
The pressure inside the inner bubble can be written as $P_2$ = $P_0$ + $\dfrac{{4T}}{{{r_1}}} + \dfrac{{4T}}{{{r_2}}}$
Where $r_1$ is the radius of the inner bubble=4cm and $r_2$ is the radius of the outer bubble=6cm.
Substituting the values of $r_1$ and $r_2$, we get
$P_2$=$P_0$+$\dfrac{{4T}}{4} + \dfrac{{4T}}{6}$
$P_2$-$P_0$=$\dfrac{{12T + 8T}}{{12}} = \dfrac{{20T}}{{12}}$
$P_2$-$P_0$=$\dfrac{{5T}}{3}$
For a soap bubble with two surfaces $P_i$ - $P_o$ = 4T/R, where R is the unknown radius.
Therefore,$\dfrac{{4T}}{R} = \dfrac{{5T}}{3}$
$\Rightarrow R = \dfrac{{12}}{5} = 2.4cm$
The required radius is 2.4cm.

So, the correct answer is “Option D”.

Additional Information:
We have two surfaces, the inner and the outer surface of the bubble. The force from surface tension is F = 2TL = 2T2πr = 4Tπr. For the bubble to be stable and not collapse, the pressure inside the bubble must be higher than the pressure on the outside. The force due to the pressure difference must balance the force from the surface tension. The force to the pressure difference is ($P_i$ - $P_o$)πr$^2$.

Note:
As we enter into the soap bubble the pressure increases by 4T/r.
For a single spherical surface balance is achieved if $P_i$ - $P_o$ = 2T/r.
For a single spherical surface balance is achieved if $P_i$ - $P_o$ = 2T/r. This is known as Laplace's law for a spherical membrane.