Two soap bubble of radii 3 mm and 4 mm are in contact radius of curvature of interface between those two bubbles is

Options

A. 1mm
B. 7mm
C. 12mm
D. `(12)/(7)`mm

Explanation

To find the radius of curvature of the interface between two soap bubbles with radii 3 mm and 4 mm, we can follow these steps:

### Step-by-Step Solution:

1. **Identify the Radii of the Bubbles**:
- Let \( R_1 = 3 \, \text{mm} \) (radius of the first bubble).
- Let \( R_2 = 4 \, \text{mm} \) (radius of the second bubble).

2. **Understand the Concept of Excess Pressure**:
- The excess pressure inside a soap bubble is given by the formula:
\[ \Delta P = \frac{4T}{R} \]
- Here, \( T \) is the surface tension of the soap solution, and \( R \) is the radius of the bubble.

3. **Determine the Excess Pressures**:
- For the first bubble (smaller radius):
\[ \Delta P_1 = \frac{4T}{R_1} = \frac{4T}{3} \]
- For the second bubble (larger radius):
\[ \Delta P_2 = \frac{4T}{R_2} = \frac{4T}{4} = T \]

4. **Calculate the Pressure Difference**:
- The pressure difference at the interface between the two bubbles is given by:
\[ \Delta P = \Delta P_1 - \Delta P_2 = \frac{4T}{3} - T \]
- Simplifying this:
\[ \Delta P = \frac{4T}{3} - \frac{3T}{3} = \frac{T}{3} \]

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

6. **Set the Two Expressions for Pressure Difference Equal**:
- Equating the two expressions for pressure difference:
\[ \frac{T}{3} = \frac{4T}{R} \]

7. **Solve for the Radius of Curvature \( R \)**:
- Cancel \( T \) from both sides (assuming \( T \neq 0 \)):
\[ \frac{1}{3} = \frac{4}{R} \]
- Rearranging gives:
\[ R = 12 \, \text{mm} \]

### Final Answer:
The radius of curvature of the interface between the two soap bubbles is \( R = 12 \, \text{mm} \).

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