When two soap bubbles of radii $$a$$ and $$b$$ ($$b > a$$) coalesce, the radius of curvature of common surface is:

We need to determine the radius of curvature of the common interface formed when two soap bubbles of different radii coalesce under isothermal conditions.

1. Understand Excess Pressure in a Soap Bubble

A soap bubble has two free surfaces (inner and outer) in contact with air. The excess pressure ($$\Delta P$$) inside a soap bubble of radius $$r$$ with surface tension $$T$$ is given by the formula:

$$\Delta P = \frac{4T}{r}$$

Let the two individual soap bubbles have radii $$a$$ and $$b$$, with given condition $$b > a$$.

• Excess pressure inside the smaller bubble ($$P_a$$): $$P_a = \frac{4T}{a}$$
• Excess pressure inside the larger bubble ($$P_b$$): $$P_b = \frac{4T}{b}$$

Since $$b > a$$, the internal pressure of the smaller bubble is greater than that of the larger bubble ($$P_a > P_b$$).

2. Analyze the Common Interface Surface

When the two bubbles coalesce and form a common internal boundary, the pressure difference across this shared wall determines its curvature.

Because the smaller bubble has a higher internal pressure, the common surface bulges into the larger bubble, becoming concave towards the smaller bubble.

The net excess pressure ($$\Delta P_{\text{common}}$$) acting across this common surface of radius $$R$$ is:

$$\Delta P_{\text{common}} = P_a - P_b$$

3. Deriving the Radius of Curvature ($$R$$)

Substitute the structural excess pressure formulas into the balance equation:

$$\frac{4T}{R} = \frac{4T}{a} - \frac{4T}{b}$$

Dividing the entire equation by $$4T$$ simplifies it to:

$$\frac{1}{R} = \frac{1}{a} - \frac{1}{b}$$

Find a common denominator on the right-hand side:

$$\frac{1}{R} = \frac{b - a}{ab}$$

Taking the reciprocal to isolate $$R$$ gives:

$$R = \frac{ab}{b - a}$$

Conclusion

The radius of curvature of the common surface is $$\frac{ab}{b - a}$$, which corresponds to Option A.