A soap bubble of radius 'r' is blown up to form a bubble of radius 2r under isothemal conditions. If `sigma` be the surface tension of soap solution , the energy spent in doing so is

Options

A. `6pisigma r^2`
B. `24 pi sigmar^2`
C. `12 pi sigmar^2`
D. `9 pisigma r^2`

Answer

To find the energy spent in blowing up a soap bubble from radius 'r' to radius '2r', we can follow these steps:

### Step 1: Calculate the initial and final surface areas of the bubble.
- The surface area \( A_1 \) of the initial bubble (radius \( r \)) is given by:
\[ A_1 = 4\pi r^2 \]
- The surface area \( A_2 \) of the final bubble (radius \( 2r \)) is given by:
\[ A_2 = 4\pi (2r)^2 = 4\pi \cdot 4r^2 = 16\pi r^2 \]

### Step 2: Determine the change in surface area.
- The change in surface area \( \Delta A \) is:
\[ \Delta A = A_2 - A_1 = 16\pi r^2 - 4\pi r^2 = 12\pi r^2 \]

### Step 3: Relate work done to change in surface area.
- The work done \( W \) against the surface tension \( \sigma \) is given by:
\[ W = \sigma \cdot \Delta A \]
- Since the bubble has two surfaces (inner and outer), the effective work done is:
\[ W = \sigma \cdot 2 \cdot \Delta A = 2\sigma \cdot 12\pi r^2 = 24\pi \sigma r^2 \]

### Conclusion:
The energy spent in blowing up the soap bubble from radius \( r \) to radius \( 2r \) is:
\[ W = 24\pi \sigma r^2 \]