If the radius of a spherical liquid (of surface tension S) drop increases from r to `r +Deltar` , the corresponding increase in the surface energy is

Options

A. `8 pi r Delta rS`
B. `4pi r DeltarS`
C. `16 pi r DeltarS`
D. `8 pi r DeltarS`

Explanation

To find the increase in surface energy when the radius of a spherical liquid drop increases from \( r \) to \( r + \Delta r \), we can follow these steps:

### Step 1: Understand the formula for surface energy
The surface energy \( E \) of a spherical drop is given by the formula:
\[ E = 4\pi r^2 S \]
where \( S \) is the surface tension and \( r \) is the radius of the drop.

### Step 2: Calculate the initial surface energy
For the initial radius \( r \), the surface energy \( E_1 \) is:
\[ E_1 = 4\pi r^2 S \]

### Step 3: Calculate the final surface energy
When the radius increases to \( r + \Delta r \), the new surface energy \( E_2 \) is:
\[ E_2 = 4\pi (r + \Delta r)^2 S \]

### Step 4: Expand the expression for \( E_2 \)
Using the binomial expansion, we can expand \( (r + \Delta r)^2 \):
\[ (r + \Delta r)^2 = r^2 + 2r\Delta r + (\Delta r)^2 \]
Thus, we can rewrite \( E_2 \) as:
\[ E_2 = 4\pi (r^2 + 2r\Delta r + (\Delta r)^2) S \]
\[ E_2 = 4\pi r^2 S + 8\pi r \Delta r S + 4\pi (\Delta r)^2 S \]

### Step 5: Calculate the increase in surface energy
The increase in surface energy \( \Delta E \) is given by:
\[ \Delta E = E_2 - E_1 \]
Substituting the values we found:
\[ \Delta E = (4\pi r^2 S + 8\pi r \Delta r S + 4\pi (\Delta r)^2 S) - 4\pi r^2 S \]
This simplifies to:
\[ \Delta E = 8\pi r \Delta r S + 4\pi (\Delta r)^2 S \]

### Step 6: Consider the case where \( \Delta r \) is very small
If \( \Delta r \) is very small, the term \( 4\pi (\Delta r)^2 S \) becomes negligible compared to \( 8\pi r \Delta r S \). Therefore, we can approximate:
\[ \Delta E \approx 8\pi r \Delta r S \]

### Conclusion
The corresponding increase in the surface energy when the radius of the spherical liquid drop increases from \( r \) to \( r + \Delta r \) is:
\[ \Delta E \approx 8\pi r \Delta r S \]

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