The work done in increasing the radius of a soap bubble from R to 3 R is

Options

A. `64pi SR^(2)`
B. `(18piSR^(2))/(3)`
C. `8piSR^(2)`
D. `16piSR^(2)`

Answer

To find the work done in increasing the radius of a soap bubble from \( R \) to \( 3R \), we can follow these steps:

### Step 1: Understand the structure of a soap bubble
A soap bubble consists of two layers of soap film, which means we need to consider the surface area of both layers.

### Step 2: Calculate the surface area of a soap bubble
The surface area \( A \) of a single sphere is given by the formula:
\[ A = 4\pi r^2 \]
Since the soap bubble has two layers, the total surface area \( A_{total} \) is:
\[ A_{total} = 2 \times 4\pi r^2 = 8\pi r^2 \]

### Step 3: Determine the initial and final radii
The initial radius \( R_i \) is \( R \) and the final radius \( R_f \) is \( 3R \).

### Step 4: Calculate the initial and final surface areas
- The initial surface area \( A_i \) when the radius is \( R \):
\[ A_i = 8\pi R^2 \]
- The final surface area \( A_f \) when the radius is \( 3R \):
\[ A_f = 8\pi (3R)^2 = 8\pi \times 9R^2 = 72\pi R^2 \]

### Step 5: Calculate the increase in surface area
The increase in surface area \( \Delta A \) is given by:
\[ \Delta A = A_f - A_i = 72\pi R^2 - 8\pi R^2 = 64\pi R^2 \]

### Step 6: Calculate the work done
The work done \( W \) in increasing the surface area is given by:
\[ W = \text{Surface Tension} \times \Delta A \]
Let \( S \) be the surface tension of the soap bubble. Thus, the work done is:
\[ W = S \times \Delta A = S \times 64\pi R^2 \]

### Final Answer
The work done in increasing the radius of the soap bubble from \( R \) to \( 3R \) is:
\[ W = 64\pi S R^2 \]

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