A spherical soap bubble has a radius \['r'\]. The surface tension of the soap film is T. The energy needed to double the diameter of the bubble at the same temperature is:
A. \[24\pi {r^2}\sigma \]
B. \[4\pi {r^2}\sigma \]
C. \[2\pi {r^2}T\sigma \]
D. \[12\pi {r^2}T\sigma \]

Hint: The bubble has two surfaces: inner and outer surface. Therefore, the surface area of the bubble should be taken twice. Calculate the potential energy possessed by the soap bubble in both the cases and then take the difference in the energy to determine the extra energy required to double the diameter.
Formula used:
\[W = T\left( {2 \times 4\pi {r^2}} \right)\]
Here, T is the surface tension and r is the radius of the bubble.

Complete step by step answer:
Expression for the energy of the soap bubble is the product of surface tension and surface area of the bubble.
Therefore,
\[W = T\left( {2 \times 4\pi {r^2}} \right)\]
Here, T is the surface tension and r is the radius of the bubble.
The factor 2 represents that the bubble has two surfaces.
Therefore, the energy of the soap bubble,
\[W = 8\pi {r^2}T\]
The radius of the bubble is half the diameter of the bubble. Therefore, the above equation becomes,
\[W = 8\pi {\left( {\dfrac{d}{2}} \right)^2}T\]
\[ \Rightarrow W = 2\pi {d^2}T\]
Now energy of the soap bubble of diameter twice the diameter of the former soap bubble is,
\[W' = 2\pi {\left( {2d} \right)^2}T\]
\[ \Rightarrow W' = 8\pi {d^2}T\]
Therefore, the extra energy required to double the radius of the soap bubble is,
\[\Delta W = W' - W\]
\[ \Rightarrow \Delta W = 8\pi {d^2}T - 2\pi {d^2}T\]
\[\therefore \Delta W = 6\pi {d^2}T\]
Again, substitute 2r for d in the above equation.
\[\Delta W = 6\pi {\left( {2r} \right)^2}T\]
\[\therefore \Delta W = 24\pi {r^2}T\]

So, the correct answer is “Option A”.

Note:
The given option (A) is false as it does not involve the term surface tension.
The correct expression for the extra energy required to double the diameter of the soap bubble is \[\Delta W = 24\pi {r^2}T\].