Question

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

• A. \( \frac{ab}{b - a} \)
• B. \( \frac{ab}{a + b} \)
• C. \( \frac{b - a}{ab} \)
• D. \( \frac{a + b}{ab} \)

Hint:

When two soap bubbles merge, the radius of curvature of the common interface is determined by the difference in pressures inside the bubbles, which is related to their respective radii and surface tension.

Answer 1

The Correct Option is A

The pressure inside a soap bubble is given by the formula: \[ P = P_0 + \frac{4T}{R}, \] where \( T \) is the surface tension and \( R \) is the radius of the bubble. Step 1: For the two bubbles with radii \( a \) and \( b \), the pressures are: \[ P_1 = P_0 + \frac{4T}{a}, \quad P_2 = P_0 + \frac{4T}{b}. \] Step 2: The pressure difference across the common surface is: \[ P_1 - P_2 = \frac{4T}{R}. \] Step 3: Substitute the expressions for \( P_1 \) and \( P_2 \): \[ \frac{4T}{a} - \frac{4T}{b} = \frac{4T}{R}. \] Step 4: Simplify: \[ \frac{1}{a} - \frac{1}{b} = \frac{1}{R}. \] Step 5: Solve for \( R \): \[ \frac{1}{R} = \frac{b - a}{ab}, \quad R = \frac{ab}{b - a}. \] Final Answer: \[ \boxed{\frac{ab}{b - a}}. \]