A soap bubble of radius `r` is placed on another bubble of radius `2r`. The radius of the surface common to both the bubbles is

Options

A. `(2r)/(3)`
B. `3r`
C. `2r`
D. `r`

Answer

To find the radius of the surface common to both bubbles, we can follow these steps:

### Step-by-Step Solution:

1. **Identify the Radii of the Bubbles**:
- Let the radius of the smaller bubble be \( r \).
- Let the radius of the larger bubble be \( 2r \).

2. **Define the Radius of the Common Surface**:
- Let the radius of the common surface be \( R \).

3. **Understand the Pressure Inside the Bubbles**:
- The pressure inside a soap bubble is given by the formula:
\[ P = \frac{4T}{r} \]
- Where \( T \) is the surface tension of the soap solution.

4. **Calculate the Pressure Inside Each Bubble**:
- For the smaller bubble (radius \( r \)):
\[ P_1 = \frac{4T}{r} \]
- For the larger bubble (radius \( 2r \)):
\[ P_2 = \frac{4T}{2r} = \frac{2T}{r} \]

5. **Set Up the Pressure Balance at the Common Surface**:
- At the common surface, the pressure inside the smaller bubble \( P_1 \) must equal the pressure inside the larger bubble \( P_2 \):
\[ P_1 = P_2 \]
- Therefore:
\[ \frac{4T}{r} - \frac{4T}{R} = \frac{2T}{r} - \frac{4T}{R} \]

6. **Rearranging the Equation**:
- Rearranging gives:
\[ \frac{4T}{R} = \frac{4T}{r} - \frac{2T}{r} \]
- Simplifying this leads to:
\[ \frac{4T}{R} = \frac{2T}{r} \]

7. **Solving for \( R \)**:
- Cross-multiplying gives:
\[ 4Tr = 2TR \]
- Dividing both sides by \( 2T \) (assuming \( T \neq 0 \)):
\[ 2r = R \]

8. **Conclusion**:
- Thus, the radius of the surface common to both bubbles is:
\[ R = 2r \]

### Final Answer:
The radius of the surface common to both bubbles is \( 2r \).