The amount of work done in blowing a soap bubble such that its diameter increases from d to D is (T=surface tension of the solution)

Options

A. `2 pi (D^(2)-d^(2))S`
B. `pi(D^(2)-d^(2))S`
C. `4pi (D^(2)-d^(2))S`
D. `8pi(D^(2)-d^(2))S`

Explanation

To solve the problem of calculating the work done in blowing a soap bubble such that its diameter increases from \(d\) to \(D\), we can follow these steps:

### Step 1: Understand the relationship between diameter and radius
The radius of a soap bubble is half of its diameter. Thus, we can express the radii as:
- Initial radius, \(r_1 = \frac{d}{2}\)
- Final radius, \(r_2 = \frac{D}{2}\)

### Step 2: Calculate the change in surface area
The surface area \(A\) of a sphere (which is the shape of a soap bubble) is given by the formula:
\[ A = 4\pi r^2 \]
We need to find the change in surface area (\(\Delta A\)) as the bubble expands from diameter \(d\) to \(D\):
\[ \Delta A = A_2 - A_1 = 4\pi r_2^2 - 4\pi r_1^2 \]
Substituting the values of \(r_1\) and \(r_2\):
\[ \Delta A = 4\pi \left(\frac{D}{2}\right)^2 - 4\pi \left(\frac{d}{2}\right)^2 \]
\[ = 4\pi \left(\frac{D^2}{4} - \frac{d^2}{4}\right) \]
\[ = \pi (D^2 - d^2) \]

### Step 3: Calculate the work done
The work done (\(W\)) in expanding the bubble is equal to the change in surface area multiplied by the surface tension (\(T\)):
\[ W = T \cdot \Delta A \]
Substituting the expression for \(\Delta A\):
\[ W = T \cdot \pi (D^2 - d^2) \]

### Final Result
Thus, the amount of work done in blowing the soap bubble such that its diameter increases from \(d\) to \(D\) is:
\[ W = \pi T (D^2 - d^2) \]

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