Under isothermal condition, energy E is supplied to a soap bubble of surface tension `sigma` and radius r, to double the radius of the soap bubble. The value of E is

Options

A. `16 pi r^(2) sigma`
B. `24 pi r^(2)sigma`
C. `8 pi r^(2) sigma`
D. `12 pi r^(2) sigma`

Explanation

To find the energy \( E \) supplied to a soap bubble to double its radius under isothermal conditions, we can follow these steps:

### Step 1: Understand the energy of a soap bubble
The energy \( E \) of a soap bubble is given by the formula:
\[ E = \text{Surface Area} \times \text{Surface Tension} \]
For a soap bubble, which has two surfaces (inside and outside), the surface area \( A \) is:
\[ A = 4\pi r^2 \]
Thus, the energy \( E \) can be expressed as:
\[ E = 4\pi r^2 \sigma \]
where \( \sigma \) is the surface tension.

### Step 2: Calculate the initial energy
For a soap bubble of radius \( r \), the initial energy is:
\[ E_{\text{initial}} = 4\pi r^2 \sigma \]

### Step 3: Determine the new radius
If the radius of the soap bubble is doubled, the new radius \( r' \) is:
\[ r' = 2r \]

### Step 4: Calculate the new energy
Now, we need to find the energy when the radius is \( r' \):
\[ E_{\text{new}} = 4\pi (r')^2 \sigma = 4\pi (2r)^2 \sigma \]
Calculating this gives:
\[ E_{\text{new}} = 4\pi (4r^2) \sigma = 16\pi r^2 \sigma \]

### Step 5: Find the energy supplied
The energy supplied \( E \) to double the radius is the difference between the new energy and the initial energy:
\[ E = E_{\text{new}} - E_{\text{initial}} = (16\pi r^2 \sigma) - (4\pi r^2 \sigma) \]
Simplifying this gives:
\[ E = 12\pi r^2 \sigma \]

### Final Answer
Thus, the energy \( E \) supplied to the soap bubble to double its radius is:
\[ E = 12\pi r^2 \sigma \]

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