A soap bubble of diameter a is produced using the soap solution of surface tension T. Find the energy required to double the radius of the bubble without change of temperature.

Options

A. `2 pi a^(2)T`
B. `6 pi a^(2)T`
C. `3 pi a^(2)T`
D. `12 pi a^(2)T`

Explanation

To find the energy required to double the radius of a soap bubble, we can follow these steps:

### Step 1: Understand the initial and final conditions
- The initial diameter of the soap bubble is \( a \), which means the initial radius \( r_1 \) is \( \frac{a}{2} \).
- We are tasked with doubling the radius, so the final radius \( r_2 \) will be \( a \).

### Step 2: Calculate the initial and final surface areas
- The surface area \( A \) of a sphere is given by the formula:
\[ A = 4\pi r^2 \]
- Calculate the initial surface area \( A_1 \):
\[ A_1 = 4\pi \left(\frac{a}{2}\right)^2 = 4\pi \frac{a^2}{4} = \pi a^2 \]
- Calculate the final surface area \( A_2 \):
\[ A_2 = 4\pi (a)^2 = 4\pi a^2 \]

### Step 3: Determine the change in surface area
- The change in surface area \( \Delta A \) is given by:
\[ \Delta A = A_2 - A_1 = 4\pi a^2 - \pi a^2 = 3\pi a^2 \]

### Step 4: Calculate the work done to change the surface area
- The work done \( W \) to change the surface area of a soap bubble is given by:
\[ W = \text{Surface Tension} \times \text{Change in Surface Area} \]
- Since a soap bubble has two surfaces (inner and outer), the work done is:
\[ W = 2T \Delta A = 2T (3\pi a^2) = 6\pi a^2 T \]

### Conclusion
The energy required to double the radius of the soap bubble is:
\[ W = 6\pi a^2 T \]