Question

A spherical soap bubble of radius $1 \mathrm{~cm}$ is formed inside another soap bubble of radius $3 \mathrm{~cm} .$ The radius of a single soap bubble which maintains the same pressure difference as inside the smaller and outside the larger soap bubble as
A. $0.75 \mathrm{~cm}$
B. $0.75 \mathrm{~m}$
C. $7.5 \mathrm{~cm}$
D.$7.5 \mathrm{~m}$

Answer 1

Hint: The pressure inside the bubble must be greater than the pressure outside in order for it to remain stable and not collapse. The surface tension force must be balanced by the force due to the pressure difference. The power of the pressure difference is $\left(P_{i}-P_{o}\right) \pi r .$
For a soap bubble with two surfaces $P_{i}-P_{o}=4 \mathrm{~T} / \mathrm{r}$.

Complete answer:
The air inside a soap bubble appears to have a higher pressure than the air around it. The sound bubbles make when they burst, for example, demonstrate this. A soap bubble is a hollow sphere with an iridescent surface formed by an extremely thin film of soapy water enclosing air. Soap bubbles usually only last a few seconds before bursting, either on their own or when they come into contact with something else.
Excess pressure inside a soap bubble $=\dfrac{4T}{r}$
where $r$ is the radius of the bubble.
Hence $\dfrac{4 \mathrm{~T}}{3}+\dfrac{4 \mathrm{~T}}{1}=\dfrac{4 \mathrm{~T}}{\mathrm{R}}$
$\text{R}=\dfrac{3\times 1}{3+1}~\text{cm}$
$\text{R}=0.75~\text{cm}$
The correct answer is $\text{R}=0.75~\text{cm}$.
Hence, option A is right.

Note: The bubble has two surfaces: an inner and an outer surface. Surface tension produces a force of $\text{F}=2\text{TL}=2~\text{T}\left( 2\pi r \right)=4T\pi r$. The pressure inside the bubble must be greater than the pressure outside in order for it to remain stable and not collapse. The surface tension force must be balanced by the force due to the pressure difference. $\left(P_{i}-P_{o}\right) \pi r^{2}$ is the force applied to the pressure difference. As we enter into the soap bubble the pressure increases by $\dfrac{4T}{r}$.