A soap bubble of diameter a is produced using the soap solution of surface tension T. Find the energy required to double the radius of the bubble without change of temperature.

Options

A. `2 pi a^(2)T`
B. `6 pi a^(2)T`
C. `3 pi a^(2)T`
D. `12 pi a^(2)T`

Explanation

To find the energy required to double the radius of a soap bubble, we will follow these steps:

### Step 1: Understand the initial conditions
The initial diameter of the soap bubble is given as \( a \). Therefore, the initial radius \( r_1 \) is:
\[ r_1 = \frac{a}{2} \]

### Step 2: Calculate the initial surface area
The surface area \( A_1 \) of a soap bubble is given by the formula for the surface area of a sphere:
\[ A_1 = 4\pi r_1^2 = 4\pi \left(\frac{a}{2}\right)^2 = 4\pi \frac{a^2}{4} = \pi a^2 \]

### Step 3: Calculate the initial energy
The energy \( U_1 \) associated with the surface of the soap bubble is given by:
\[ U_1 = T \times \text{(total surface area)} \]
Since a soap bubble has two surfaces (inside and outside), the total surface area is \( 2A_1 \):
\[ U_1 = T \times 2A_1 = T \times 2(\pi a^2) = 2\pi a^2 T \]

### Step 4: Determine the new radius
If we double the radius of the bubble, the new radius \( r_2 \) becomes:
\[ r_2 = 2r_1 = 2 \left(\frac{a}{2}\right) = a \]

### Step 5: Calculate the new surface area
The new surface area \( A_2 \) is:
\[ A_2 = 4\pi r_2^2 = 4\pi a^2 \]

### Step 6: Calculate the new energy
The new energy \( U_2 \) associated with the surface of the soap bubble is:
\[ U_2 = T \times 2A_2 = T \times 2(4\pi a^2) = 8\pi a^2 T \]

### Step 7: Calculate the energy required to double the radius
The energy required to double the radius \( \Delta U \) is the difference between the new energy and the initial energy:
\[ \Delta U = U_2 - U_1 = (8\pi a^2 T) - (2\pi a^2 T) = 6\pi a^2 T \]

### Final Answer
The energy required to double the radius of the soap bubble is:
\[ \Delta U = 6\pi a^2 T \]

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