Question

A soap bubble, having radius $1\text{mm}$, is blown from a detergent solution having a surface tension of $2.5\times {10}^{-2}\text{N/m}$. The pressure inside the bubble equals at a point $Z_o$ below the free surface of water in a container. Taking $g=10{\text{m/s}}^2$, density of water $={10}^3{\text{kg/m}}^3$, the value of $Z_0$ is
[Take $g=10{\text{m/s}}^2$]

• A. $5\text{cm}$
• B. $8\text{cm}$
• C. $1\text{cm}$
• D. $3\text{cm}$

Answer 1

The correct option is C $1\text{cm}$

Given,
Radius of soap bubble, $R=1\text{mm}={10}^{-3}\text{m}$
Surface Tension, $T=2.5\times {10}^{-2}\text{N/m}$
Value of $g=10{\text{m/s}}^2$
Density of water, $\rho =1000{\text{kg/m}}^3$
Depth of water at which pressure is equal to pressure inside bubble $=Z_o=?$
Let us suppose, atmospheric pressure is $p_o$.
We know pressure inside soap bubble is given by
$p=p_o+\frac{4T}{R}$
$=p_o+\frac{4\times 2.5\times {10}^{-2}}{{10}^{-3}}$
$=p_o+100$ ...........(1)
Pressure at depth $Z_o$ below free surface of water
$p^{{}^{\prime }}=p_0+\rho g{Z}_0$
$=p_0+1000\times 10\times Z_o$
$=p_0+{10}^4{Z}_o$ ...........(2)
Given, pressure at depth $Z_o$ is equal to pressure inside soap bubble.
From (1) and (2),
$p_o+100=p_0+{10}^4{Z}_o$
or $Z_o={10}^{-2}\text{m}=1\text{cm}$