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: \[ E=8\pi TR^2 \] because a soap bubble has two surfaces. For a liquid drop: \[ E=4\pi TR^2 \] because it has only one surface.

Answer 1

The Correct Option is A

Step 1: Understanding the Concept:
A soap bubble consists of a thin film of liquid with two free surfaces: one inner surface and one outer surface.
The surface energy of a soap bubble is given by the product of its surface tension (\(T\)) and its total surface area (\(A\)).
Work done in increasing the size of the bubble is equal to the change in its surface energy.
Because there are two surfaces, the effective area is \(2 \times 4\pi r^2 = 8\pi r^2\).
Step 2: Key Formula or Approach:
Work Done (\(W\)) = Surface Tension (\(T\)) \(\times\) Change in Area (\(\Delta A\)).
For a soap bubble of radius \(r\):
\(A = 2 \times 4\pi r^2 = 8\pi r^2\).
Change in area when radius goes from \(r_1\) to \(r_2\):
\(\Delta A = 8\pi(r_2^2 - r_1^2)\).
Therefore, \(W = 8\pi T (r_2^2 - r_1^2)\).
Step 3: Detailed Explanation:
**Part 1: Increasing radius from \(R\) to \(2R\).**
The work done is given as \(W\).
\[ W = 8\pi T [(2R)^2 - R^2] \]
\[ W = 8\pi T [4R^2 - R^2] \]
\[ W = 8\pi T [3R^2] = 24\pi R^2 T \quad \text{--- (Equation 1)} \]
**Part 2: Increasing radius from \(2R\) to \(3R\).**
Let the work done for this second step be \(W'\).
\[ W' = 8\pi T [(3R)^2 - (2R)^2] \]
\[ W' = 8\pi T [9R^2 - 4R^2] \]
\[ W' = 8\pi T [5R^2] = 40\pi R^2 T \quad \text{--- (Equation 2)} \]
Now, we need to express \(W'\) in terms of \(W\). Divide Equation 2 by Equation 1:
\[ \frac{W'}{W} = \frac{40\pi R^2 T}{24\pi R^2 T} \]
\[ \frac{W'}{W} = \frac{40}{24} \]
Both numbers are divisible by 8:
\[ \frac{W'}{W} = \frac{5}{3} \]
\[ W' = \frac{5}{3} W \]
This shows that the energy required to expand the bubble increases as the bubble gets larger, because the change in surface area per unit change in radius is greater for larger radii.
Step 4: Final Answer:
The work done in further increasing the radius from \(2R\) to \(3R\) is \(\frac{5}{3}W\).
Hence, the correct option is (A).