Question

When two soap bubbles of radii a and b (b > a) coalesce, then find the radius of curvature of the common surface.
A. \[\left( {\dfrac{{b - a}}{{ab}}} \right)\]
B. \[\left( {\dfrac{{ab}}{{b - a}}} \right)\]
C. \[\left( {\dfrac{{ab}}{{a + b}}} \right)\]
D. \[\left( {\dfrac{{a + b}}{{ab}}} \right)\]

Answer 1

Hint: Before we start addressing the problem, we need to know about the excess pressure. Excess pressure is caused by the liquid or soap bubble due to surface tension. The net force that is experienced by the bubble or liquid drop is inward. Its direction is normal to the surface or liquid.

Formula Used:
To find the excess pressure in the soap bubble the formula is,
\[P = \dfrac{{4T}}{r}\]
Where, $T$ is surface tension and $r$ is radius of the soap bubble.

Complete step by step solution:
Consider a soap bubble of radius a and another soap bubble of radius b, here radius of b is greater than radius a. When these two soap bubbles combine, it forms a common surface as shown in the figure. From the formula we have,
\[P = \dfrac{{4T}}{r}\]……….. (1)
The excess pressure is inversely proportional to the radius that is, if the radius is small the pressure will be more and vice versa.

The excess pressure of the soap bubble having radii a and b is,
\[{P_{excess}} = {P_1}\] (Excess pressure of soap bubble of radius a)
\[\Rightarrow {P_{excess}} = {P_2}\] (Excess pressure of soap bubble of radius b)
We can say that,
\[{P_1} > {P_2}\]
\[ \Rightarrow {P_1} - {P_2} = P\]…….. (2)
Here, P is the excess pressure of a virtual soap bubble that exists by the common surface.

Now, from equation (1) we can write,
\[{P_1} = \dfrac{{4T}}{a}\] and \[{P_2} = \dfrac{{4T}}{b}\]
Substitute the above value in equation (2), we get
\[\dfrac{{4T}}{a} - \dfrac{{4T}}{b} = \dfrac{{4T}}{R}\]
\[\Rightarrow \dfrac{1}{a} - \dfrac{1}{b} = \dfrac{1}{R}\]
\[\therefore R = \left( {\dfrac{{ab}}{{b - a}}} \right)\]
Therefore, the radius of curvature of the common surface is \[\left( {\dfrac{{ab}}{{b - a}}} \right)\].

Hence, option B is the correct answer

Note:Remember that, when the radius of the soap bubble increases then the excess pressure of that bubble goes on decreases and vice-versa and in the soap bubble, the inside pressure is more than compared to outside.