The radius of a soap bubble is increased from `(1)/(sqrtpi)` cm to `(2)/(sqrtpi)` cm. If the surface tension of water is `30dynes` per cm, then the work done will be

Options

A. 180 ergs
B. 360 ergs
C. 720 ergs
D. 960 ergs

Explanation

To find the work done when the radius of a soap bubble increases from \(\frac{1}{\sqrt{\pi}}\) cm to \(\frac{2}{\sqrt{\pi}}\) cm, we can follow these steps:

### Step 1: Understand the concept of work done in terms of surface energy
The work done in expanding the soap bubble is equal to the change in surface energy. The surface energy \(E\) of a soap bubble is given by the formula:
\[ E = \text{Surface Tension} \times \text{Surface Area} \]
Since a soap bubble has two surfaces (inside and outside), we will consider the total surface area as \(2 \times \text{Surface Area}\).

### Step 2: Calculate the initial and final surface areas
The surface area \(A\) of a sphere is given by:
\[ A = 4\pi r^2 \]
For the initial radius \(r_1 = \frac{1}{\sqrt{\pi}}\):
\[ A_1 = 4\pi \left(\frac{1}{\sqrt{\pi}}\right)^2 = 4\pi \cdot \frac{1}{\pi} = 4 \text{ cm}^2 \]
For the final radius \(r_2 = \frac{2}{\sqrt{\pi}}\):
\[ A_2 = 4\pi \left(\frac{2}{\sqrt{\pi}}\right)^2 = 4\pi \cdot \frac{4}{\pi} = 16 \text{ cm}^2 \]

### Step 3: Calculate the change in surface area
The change in surface area \(\Delta A\) is given by:
\[ \Delta A = A_2 - A_1 = 16 \text{ cm}^2 - 4 \text{ cm}^2 = 12 \text{ cm}^2 \]

### Step 4: Calculate the change in surface energy
The change in surface energy \(\Delta E\) can be calculated using the surface tension \(\gamma\):
\[ \Delta E = \text{Surface Tension} \times \Delta A \]
Given that the surface tension \(\gamma = 30 \text{ dyn/cm}\):
\[ \Delta E = 30 \text{ dyn/cm} \times 12 \text{ cm}^2 = 360 \text{ dyn-cm} \]

### Step 5: Conclusion
The work done in increasing the radius of the soap bubble is:
\[ \text{Work Done} = 360 \text{ dyn-cm} \]