If work `W` is done in blowing a bubble of radius `R` from a soap solution. Then the work done is blowing a bubble of radius `2R` from the same solution is

Options

A. `(W)/(2)`
B. `2W`
C. `4W`
D. `(7)/(3)W`

Explanation

To solve the problem of calculating the work done in blowing a bubble of radius \( 2R \) from a soap solution, we can follow these steps:

### Step-by-Step Solution:

1. **Understand the Concept of Work Done**:
The work done in blowing a bubble is related to the surface energy of the bubble. The surface energy is given by the formula:
\[ \text{Work Done} = \text{Surface Tension} \times \text{Total Surface Area} \]

2. **Calculate the Surface Area of a Bubble**:
A soap bubble has two surfaces (inside and outside). The surface area \( A \) of a sphere is given by:
\[ A = 4\pi R^2 \]
Therefore, for a soap bubble of radius \( R \), the total surface area \( A_{total} \) is:
\[ A_{total} = 2 \times 4\pi R^2 = 8\pi R^2 \]

3. **Calculate the Work Done for Radius \( R \)**:
The work done \( W \) in blowing a bubble of radius \( R \) can be expressed as:
\[ W = \text{Surface Tension} \times A_{total} = S \times 8\pi R^2 \]
Thus, we have:
\[ W = 8\pi R^2 S \]

4. **Calculate the Work Done for Radius \( 2R \)**:
Now, we need to find the work done \( W' \) when the radius of the bubble is \( 2R \). The total surface area for a bubble of radius \( 2R \) is:
\[ A'_{total} = 2 \times 4\pi (2R)^2 = 2 \times 4\pi \times 4R^2 = 32\pi R^2 \]
Therefore, the work done \( W' \) is:
\[ W' = S \times A'_{total} = S \times 32\pi R^2 \]

5. **Relate \( W' \) to \( W \)**:
We can express \( W' \) in terms of \( W \):
\[ W' = 32\pi R^2 S = 4 \times (8\pi R^2 S) = 4W \]

### Final Answer:
The work done in blowing a bubble of radius \( 2R \) from the same soap solution is:
\[ W' = 4W \]