A soap bubble has a radius of 6cm. The work done in increasing the radius from 6 cm to 10 on will be (in mJ). (Surface tension of soap solution is 0.04 Nm.)

Options

A. `2.04pi`
B. `1.02pi`
C. `4.05pi`
D. `6.05pi`

Explanation

To solve the problem of calculating the work done in increasing the radius of a soap bubble from 6 cm to 10 cm, we will follow these steps:

### Step 1: Understand the Concept of Work Done
The work done (W) in increasing the radius of a soap bubble is given by the formula:
\[ W = \text{Surface Tension} \times \Delta \text{Surface Area} \]
Since a soap bubble has two surfaces (inside and outside), we will consider this in our calculations.

### Step 2: Identify Given Values
- Initial radius (R1) = 6 cm = 0.06 m (convert to meters)
- Final radius (R2) = 10 cm = 0.10 m (convert to meters)
- Surface tension (T) = 0.04 N/m

### Step 3: Calculate the Change in Surface Area
The surface area (A) of a sphere is given by the formula:
\[ A = 4\pi r^2 \]
We need to find the change in surface area (\(\Delta A\)):
\[ \Delta A = A_2 - A_1 = 4\pi R_2^2 - 4\pi R_1^2 \]
Substituting the values:
\[ \Delta A = 4\pi (0.10^2) - 4\pi (0.06^2) \]

### Step 4: Simplify the Change in Surface Area
Factor out \(4\pi\):
\[ \Delta A = 4\pi \left( (0.10)^2 - (0.06)^2 \right) \]
Calculating the squares:
\[ (0.10)^2 = 0.01 \]
\[ (0.06)^2 = 0.0036 \]
So,
\[ \Delta A = 4\pi (0.01 - 0.0036) = 4\pi (0.0064) \]

### Step 5: Calculate \(\Delta A\)
Now calculate:
\[ \Delta A = 4\pi \times 0.0064 \]
Using \(\pi \approx 3.14\):
\[ \Delta A \approx 4 \times 3.14 \times 0.0064 \approx 0.0805 \text{ m}^2 \]

### Step 6: Calculate the Work Done
Now substitute \(\Delta A\) into the work done formula:
\[ W = T \times \Delta A \]
\[ W = 0.04 \times 0.0805 \]
\[ W \approx 0.00322 \text{ J} \]

### Step 7: Convert to Millijoules
Since \(1 \text{ J} = 1000 \text{ mJ}\):
\[ W \approx 0.00322 \text{ J} = 3.22 \text{ mJ} \]

### Final Answer
The work done in increasing the radius of the soap bubble from 6 cm to 10 cm is approximately:
\[ \boxed{3.22 \text{ mJ}} \]