The radius of a soap bubble is r. the surface tension of soap solution is T, keeping temperature constant, the radius of the soap bubble is doubled the energy necessary for this will be-

Options

A. `24pir^(2)T`
B. `8pir^(2)T`
C. `12pir^(2)T`
D. `16pir^(2)T`

Answer

To solve the problem of finding the energy necessary to double the radius of a soap bubble, we will follow these steps:

### Step 1: Understand the surface energy of a soap bubble
A soap bubble has two surfaces (inside and outside), and the surface energy (U) of a soap bubble is given by the formula:
\[ U = \text{Surface Tension} \times \text{Total Surface Area} \]

### Step 2: Calculate the initial surface energy for radius \( r \)
The total surface area \( A \) of a soap bubble with radius \( r \) is:
\[ A = 2 \times 4\pi r^2 = 8\pi r^2 \]
Thus, the initial surface energy \( U_1 \) when the radius is \( r \) is:
\[ U_1 = T \times 8\pi r^2 = 8\pi T r^2 \]

### Step 3: Calculate the surface energy for the new radius \( 2r \)
When the radius is doubled to \( 2r \), the new total surface area \( A' \) becomes:
\[ A' = 2 \times 4\pi (2r)^2 = 2 \times 4\pi \times 4r^2 = 32\pi r^2 \]
Thus, the new surface energy \( U_2 \) when the radius is \( 2r \) is:
\[ U_2 = T \times 32\pi r^2 = 32\pi T r^2 \]

### Step 4: Calculate the energy required to change the radius from \( r \) to \( 2r \)
The energy required \( E \) to increase the radius from \( r \) to \( 2r \) is the difference in surface energy:
\[ E = U_2 - U_1 \]
Substituting the values we calculated:
\[ E = 32\pi T r^2 - 8\pi T r^2 \]
\[ E = (32 - 8)\pi T r^2 \]
\[ E = 24\pi T r^2 \]

### Final Answer
The energy necessary to double the radius of the soap bubble is:
\[ E = 24\pi T r^2 \]

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