1. Key concept: work done against surface tension

For a soap bubble, there are two surfaces (inner and outer), so the excess surface energy is:

$U = 2T \cdot A$

where $T$ is the surface tension and $A = 4\pi r^2$ is the area of one spherical surface.

Thus,

$U = 2T(4\pi r^2) = 8\pi T r^2$

The work done in expanding the bubble from radius $r_1$ to $r_2$ is the increase in surface energy:

$W = 8\pi T (r_2^2 - r_1^2)$

2. Given data

$r_1 = 3\text{ cm} = 0.03\text{ m}$ $r_2 = 5\text{ cm} = 0.05\text{ m}$ $T = 0.03\,\text{N m}^{-1}$

3. Substitute into the formula

$W = 8\pi (0.03)\left[(0.05)^2 - (0.03)^2\right]$

First compute the squares:

$(0.05)^2 = 0.0025$ $(0.03)^2 = 0.0009$

So,

$r_2^2 - r_1^2 = 0.0025 - 0.0009 = 0.0016$

Now,

$W = 8\pi (0.03)(0.0016)$

$W = 8\pi (4.8 \times 10^{-5})$

$W = 3.84 \times 10^{-4}\pi\,\text{J}$

Convert to mJ:

$1\text{ mJ} = 10^{-3}\text{ J}$

Hence,

$W = 0.384\pi\,\text{mJ}$

This is nearly:

$0.4\pi\,\text{mJ}$

4. Option check

• A: $0.2\pi$ mJ ❌
• B: $2\pi$ mJ ❌
• C: $0.4\pi$ mJ ✅
• D: $4\pi$ mJ ❌

5. Final answer

The correct option is:

$\boxed{\text{C }\; 0.4\pi\,\text{mJ}}$