Two soap bubbles coalesce.It noticed that, whilst joined together, the radii of the two bubbles are a and b where a>b.Then the radius of curvature of interface between the two bubbles will be

Options

A. a-b
B. a+b
C. `ab//(a-b)`
D. `ab//(a+b)`

Answer

To find the radius of curvature of the interface between two soap bubbles with radii \( a \) and \( b \) (where \( a > b \)), we can use the concept of excess pressure in soap bubbles.

### Step-by-Step Solution:

1. **Understand the Pressure Inside the Bubbles:**
- The pressure inside a soap bubble is given by the formula:
\[ P = P_{\text{atm}} + \frac{4T}{R} \]
where \( P_{\text{atm}} \) is the atmospheric pressure, \( T \) is the surface tension, and \( R \) is the radius of the bubble.

2. **Write the Pressure Equations for Both Bubbles:**
- For the larger bubble with radius \( a \):
\[ P_1 = P_{\text{atm}} + \frac{4T}{a} \]
- For the smaller bubble with radius \( b \):
\[ P_2 = P_{\text{atm}} + \frac{4T}{b} \]

3. **Calculate the Excess Pressure:**
- The difference in pressure across the interface (excess pressure) is given by:
\[ \Delta P = P_1 - P_2 \]
- Substituting the expressions for \( P_1 \) and \( P_2 \):
\[ \Delta P = \left( P_{\text{atm}} + \frac{4T}{a} \right) - \left( P_{\text{atm}} + \frac{4T}{b} \right) \]
- Simplifying this gives:
\[ \Delta P = \frac{4T}{a} - \frac{4T}{b} = 4T \left( \frac{1}{a} - \frac{1}{b} \right) \]

4. **Relate the Excess Pressure to the Radius of Curvature:**
- The excess pressure across a curved interface is also related to the radius of curvature \( R \) of the interface:
\[ \Delta P = \frac{2T}{R} \]

5. **Set the Two Expressions for Excess Pressure Equal:**
- Equating the two expressions for \( \Delta P \):
\[ 4T \left( \frac{1}{a} - \frac{1}{b} \right) = \frac{2T}{R} \]

6. **Solve for the Radius of Curvature \( R \):**
- Cancel \( T \) from both sides (assuming \( T \neq 0 \)):
\[ 4 \left( \frac{1}{a} - \frac{1}{b} \right) = \frac{2}{R} \]
- Rearranging gives:
\[ R = \frac{2}{4 \left( \frac{1}{a} - \frac{1}{b} \right)} = \frac{1}{2 \left( \frac{1}{a} - \frac{1}{b} \right)} \]
- Simplifying further:
\[ R = \frac{ab}{2(b - a)} \]
- Since \( a > b \), we can take the absolute value:
\[ R = \frac{ab}{a - b} \]

### Final Answer:
The radius of curvature of the interface between the two soap bubbles is:
\[ R = \frac{ab}{a - b} \]