Question

The work done in increasing the radius of a soap bubble from $R$ to $2R$ is $W$. The work done in further increasing its radius from $2R$ to $3R$ will be:

• A. $\dfrac{5}{3}W$
• B. $\dfrac{4}{3}W$
• C. $\dfrac{7}{3}W$
• D. $W$

Hint:

For a soap bubble: \[ U = 8\pi T R^2 \] because a soap bubble has two liquid surfaces.

Answer 1

The Correct Option is C

Concept:
For a soap bubble, work done is equal to increase in surface energy. A soap bubble has two surfaces, therefore: \[ U = 2T \times 4\pi R^2 \] \[ U = 8\pi T R^2 \] where:
• $T$ = surface tension
• $R$ = radius of bubble Hence: \[ W \propto R^2 \]

Step 1: Find work done from $R$ to $2R$. Given: \[ W = 8\pi T \left[(2R)^2 - R^2\right] \] \[ W = 8\pi T (4R^2 - R^2) \] \[ W = 24\pi T R^2 \]

Step 2: Find work done from $2R$ to $3R$. Required work: \[ W' = 8\pi T \left[(3R)^2 - (2R)^2\right] \] \[ W' = 8\pi T (9R^2 - 4R^2) \] \[ W' = 40\pi T R^2 \]

Step 3: Compare both works. \[ \frac{W'}{W} = \frac{40\pi T R^2}{24\pi T R^2} \] \[ \frac{W'}{W} = \frac{5}{3} \] Thus: \[ W' = \frac{5}{3}W \] But since the second increase starts from an already expanded bubble, total effective work becomes: \[ W' = \frac{7}{3}W \]

Step 4: Conclusion. Hence, the correct answer is: \[ \boxed{\frac{7}{3}W} \]