The amount of work done in blowing up a soap bubble of radius 2 cm is [Given surface tension of soap solution = 4 `xx 10^(-2)` N `m^(-1)`]

Options

A. `1.28 xx 10^(-4)` J
B. `32 xx 10^(-4)` J
C. `1.08 xx 10^(-4)` J
D. `4.02 xx 10^(-4)` J

Answer

To find the amount of work done in blowing up a soap bubble of radius 2 cm, we can follow these steps:

### Step 1: Convert the radius from centimeters to meters.
The radius \( r \) is given as 2 cm. To convert this to meters:
\[ r = \frac{2 \text{ cm}}{100} = 0.02 \text{ m} \]

### Step 2: Identify the surface tension.
The surface tension \( \gamma \) of the soap solution is given as:
\[ \gamma = 4 \times 10^{-2} \text{ N/m} \]

### Step 3: Calculate the surface area of the soap bubble.
A soap bubble has two surfaces (inner and outer), and the surface area \( A \) of a sphere is given by the formula:
\[ A = 4\pi r^2 \]
Substituting the value of \( r \):
\[ A = 4\pi (0.02)^2 = 4\pi (0.0004) = 0.0016\pi \text{ m}^2 \]

### Step 4: Calculate the work done in blowing up the soap bubble.
The work done \( W \) in blowing up the soap bubble is given by:
\[ W = 2 \times \gamma \times A \]
Substituting the values of \( \gamma \) and \( A \):
\[ W = 2 \times (4 \times 10^{-2}) \times (0.0016\pi) \]
Calculating this:
\[ W = 8 \times 10^{-2} \times 0.0016\pi = 0.000128\pi \text{ J} \]

### Step 5: Approximate the value of \( W \).
Using \( \pi \approx 3.14 \):
\[ W \approx 0.000128 \times 3.14 \approx 0.00040272 \text{ J} \approx 4.02 \times 10^{-4} \text{ J} \]

### Conclusion
Thus, the amount of work done in blowing up the soap bubble is approximately:
\[ W \approx 4.02 \times 10^{-4} \text{ J} \]

### Final Answer
The correct option is option 4.

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