The work done to split a liquid drop id radius `R` into `N` identical drops is (taken `sigma` as the surface tension of the liquid)

Options

A. `4piR^(2)(N^(-1//3)-1)sigma`
B. `4piR^(2)N sigma`
C. `4piR^(2)(N^(-1//2)-1)`
D. None of these

Explanation

To find the work done to split a liquid drop of radius \( R \) into \( N \) identical drops, we can follow these steps:

### Step 1: Determine the Volume of the Original Drop
The volume \( V \) of a single spherical drop of radius \( R \) is given by the formula:
\[ V = \frac{4}{3} \pi R^3 \]

### Step 2: Determine the Volume of One Smaller Drop
When the original drop is split into \( N \) identical smaller drops, the volume of one smaller drop \( v \) is:
\[ v = \frac{V}{N} = \frac{1}{N} \cdot \frac{4}{3} \pi R^3 = \frac{4}{3} \pi r^3 \]
where \( r \) is the radius of one smaller drop.

### Step 3: Relate the Volumes to Find the Radius of Smaller Drops
Setting the volumes equal gives:
\[ \frac{4}{3} \pi r^3 = \frac{4}{3} \pi \frac{R^3}{N} \]
From this, we can simplify to find \( r \):
\[ r^3 = \frac{R^3}{N} \implies r = R \cdot N^{-1/3} \]

### Step 4: Calculate the Surface Area of the Drops
The surface area \( A \) of a single spherical drop is given by:
\[ A = 4 \pi R^2 \]
The total surface area of \( N \) smaller drops is:
\[ A_{\text{final}} = N \cdot 4 \pi r^2 = N \cdot 4 \pi \left(R \cdot N^{-1/3}\right)^2 = N \cdot 4 \pi \frac{R^2}{N^{2/3}} = 4 \pi R^2 N^{1/3} \]

### Step 5: Calculate the Change in Surface Area
The change in surface area \( \Delta A \) when the drop is split is:
\[ \Delta A = A_{\text{final}} - A_{\text{initial}} = 4 \pi R^2 N^{1/3} - 4 \pi R^2 \]
Factoring out \( 4 \pi R^2 \):
\[ \Delta A = 4 \pi R^2 (N^{1/3} - 1) \]

### Step 6: Calculate the Work Done
The work done \( W \) to split the drop is given by:
\[ W = \sigma \Delta A \]
Substituting for \( \Delta A \):
\[ W = \sigma \cdot 4 \pi R^2 (N^{1/3} - 1) \]

### Final Result
Thus, the work done to split a liquid drop of radius \( R \) into \( N \) identical drops is:
\[ W = 4 \pi \sigma R^2 (N^{1/3} - 1) \]

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