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

Options

A. `12 pi R^(2)T`
B. `64 pi R^(2)T`
C. `8 pi R^(2)T`
D. `16 piR^(2)T`

Explanation

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 Surface Area of the Soap Bubble
A soap bubble has two surfaces (inner and outer), so the effective surface area \( A \) of a soap bubble is given by:
\[ A = 2 \times \text{Surface Area of a sphere} = 2 \times 4\pi R^2 = 8\pi R^2 \]

### Step 2: Calculate the Initial Surface Area
For the initial radius \( R \):
\[ A_i = 8\pi R^2 \]

### Step 3: Calculate the Final Surface Area
For the final radius \( 3R \):
\[ A_f = 8\pi (3R)^2 = 8\pi \times 9R^2 = 72\pi R^2 \]

### Step 4: Determine the Change in Surface Area
The change 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 5: Calculate the Work Done
The work done \( W \) in increasing the surface area of the soap bubble is related to the surface tension \( T \) and the change in area:
\[ W = T \times \Delta A \]
Substituting the change in area:
\[ W = T \times 64\pi R^2 \]

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

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