Air is pushed into a soap bubble of radius r to double its...

Air is pushed into a soap bubble of radius $r$ to double its radius. If the surface tension of the soap solution is $S$, the work done in the process is
(A) $8 \pi r^{2} S$
(B) $12 \pi r^{2} S$
(C) $16 \pi r^{2} S$
(D) $24 \pi r^{2} S$

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Key Concepts

Soap Bubble Physics

Soap bubbles are unique because they have two surfaces (an inner and an outer surface), both contributing to the overall surface energy. The pressure difference across a soap bubble is governed by the Young–Laplace equation, and any change in the radius of the bubble results in a change in surface area on both sides. This dual-surface aspect is important when calculating the work done in processes that alter the bubble’s size.

Surface Tension

Surface tension is a measure of the cohesive force at the surface of a liquid, which acts to minimize the surface area. It is a crucial factor in problems involving interfaces and is typically expressed in units of force per unit length or energy per unit area. In physical setups such as bubbles, this property dictates the energy necessary to create and expand an interface.

Work Done in Surface Expansion

When the surface area of a liquid interface increases, work is required to overcome the forces due to surface tension. This work is directly related to the change in the surface area and is given by the product of the surface tension and the incremental area added. This concept is central to energy considerations in processes such as the expansion of bubbles or droplets.

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