Find the work done in blowing a soap bubble of surface tension `0.06 Nm^(-1)` from 2 cm radius to 5 cm radius.

Options

A. 3.1mJ
B. 1.25 mJ
C. 2.51 mJ
D. 4.55 mJ

Explanation

To find the work done in blowing a soap bubble from a radius of 2 cm to a radius of 5 cm, we can follow these steps:

### Step 1: Understand the Concept
The work done in blowing a soap bubble is related to the change in surface area of the bubble and the surface tension of the soap solution. The formula for work done (W) can be expressed as:
\[ W = \text{Surface Tension} \times \Delta A \]
where \(\Delta A\) is the change in surface area.

### Step 2: Calculate the Surface Areas
The surface area \(A\) of a sphere is given by the formula:
\[ A = 4\pi r^2 \]
We need to calculate the surface area for both the initial radius (2 cm) and the final radius (5 cm).

- For the initial radius \(r_1 = 2 \text{ cm} = 0.02 \text{ m}\):
\[ A_1 = 4\pi (0.02)^2 = 4\pi (0.0004) = 0.005024 \text{ m}^2 \]

- For the final radius \(r_2 = 5 \text{ cm} = 0.05 \text{ m}\):
\[ A_2 = 4\pi (0.05)^2 = 4\pi (0.0025) = 0.031416 \text{ m}^2 \]

### Step 3: Calculate the Change in Surface Area
Now, we can find the change in surface area \(\Delta A\):
\[ \Delta A = A_2 - A_1 = 0.031416 - 0.005024 = 0.026392 \text{ m}^2 \]

### Step 4: Calculate the Work Done
Now, we can substitute the values into the work done formula. Given that the surface tension \(\sigma = 0.06 \text{ N/m}\):
\[ W = \sigma \times \Delta A = 0.06 \times 0.026392 \]

Calculating this gives:
\[ W = 0.06 \times 0.026392 = 0.00158552 \text{ J} \]

### Step 5: Convert to Appropriate Units
To express this in a more manageable form, we can convert it to milliJoules:
\[ W = 1.58552 \text{ mJ} \]

### Final Answer
Thus, the work done in blowing the soap bubble from a radius of 2 cm to 5 cm is approximately:
\[ W \approx 1.59 \text{ mJ} \]

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