Radius of a soap bubble is increased from R to 2R work done in this process in terms of surface tension is

The correct option is 24π R²S.

Underlying Concept:
A soap bubble has two free surfaces in contact with air (one inner surface and one outer surface). Therefore, when calculating the total surface area of a soap bubble, we must multiply the area of a single sphere by 2.
The work done ($W$) in expanding a soap bubble is equal to the product of its surface tension ($S$) and the increase in its total surface area ($\Delta A$):
$W=S\times \Delta A$

Step-by-Step Derivation:

1. Initial Surface Area ($A_1$):
For an initial radius $R$, the total initial surface area is:
$A_1=2\times \left(4\pi R^2\right)=8\pi R^2$

2. Final Surface Area ($A_2$):
When the radius is increased to $2R$, the total final surface area becomes:
$A_2=2\times \left[4\pi {\left(2R\right)}^2\right]=2\times \left(16\pi R^2\right)=32\pi R^2$

3. Change in Surface Area ($\Delta A$):
The increase in surface area is:
$\Delta A=A_2-A_1$

Substituting the values:
$\Delta A=32\pi R^2-8\pi R^2=24\pi R^2$

4. Work Done ($W$):
Using the relation for work done:
$W=S\times \Delta A$

Substituting the change in area:
$W=S\times 24\pi R^2=24\pi R^{2}S$

Therefore, the work done in this process is 24π R²S.