If T is the surface tension of soap solution, the amount of work done in blowing a soap bubble from a diameter D to 2D is

Options

A. `2piD^(2)T`
B. `4piD^(2)T`
C. `6piD^(2)T`
D. `8piD^(2)T`

Answer

To find the amount of work done in blowing a soap bubble from a diameter \( D \) to \( 2D \), we can follow these steps:

### Step-by-Step Solution:

1. **Understanding Surface Tension**:
Surface tension \( T \) is defined as the force per unit length acting along the surface of a liquid. The work done in creating a new surface area is given by the formula:
\[ W = T \times \Delta A \]
where \( \Delta A \) is the change in surface area.

2. **Identifying the Initial and Final Diameters**:
- The initial diameter of the bubble is \( D \).
- The final diameter of the bubble is \( 2D \).

3. **Calculating the Radii**:
- The initial radius \( r_1 \) is \( \frac{D}{2} \).
- The final radius \( r_2 \) is \( \frac{2D}{2} = D \).

4. **Calculating the Surface Areas**:
The surface area \( A \) of a sphere is given by:
\[ A = 4 \pi r^2 \]
- Initial surface area \( A_1 \):
\[ A_1 = 4 \pi \left(\frac{D}{2}\right)^2 = 4 \pi \left(\frac{D^2}{4}\right) = \pi D^2 \]
- Final surface area \( A_2 \):
\[ A_2 = 4 \pi (D)^2 = 4 \pi D^2 \]

5. **Calculating the Change in Surface Area**:
The change in surface area \( \Delta A \) is:
\[ \Delta A = A_2 - A_1 = 4 \pi D^2 - \pi D^2 = 3 \pi D^2 \]

6. **Calculating the Work Done**:
Since a soap bubble has two surfaces (inner and outer), the work done \( W \) is:
\[ W = 2T \times \Delta A = 2T \times (3 \pi D^2) = 6 \pi D^2 T \]

### Final Answer:
Thus, the amount of work done in blowing a soap bubble from a diameter \( D \) to \( 2D \) is:
\[ W = 6 \pi D^2 T \]