A wire of fixed length is wound on a solenoid of length`l` and redius `r`. Its self-inductance is found to be `L`. Now, if the same wire is wound on a solenoid of length `l//2` and redius `r//2` then the self-inductance will be

Options

A. `2L`
B. `L`
C. `4L`
D. `8L`

Explanation

To determine the self-inductance of a solenoid when the dimensions are changed, we can use the formula for the self-inductance \( L \) of a solenoid:

\[ L = \mu_0 \frac{N^2 A}{l} \]

Where:
- \( L \) is the self-inductance,
- \( \mu_0 \) is the permeability of free space,
- \( N \) is the number of turns,
- \( A \) is the cross-sectional area of the solenoid, and
- \( l \) is the length of the solenoid.

### Step 1: Determine the original self-inductance
For the first solenoid with length \( l \) and radius \( r \):
- The cross-sectional area \( A_1 = \pi r^2 \).
- The self-inductance is given as \( L = \mu_0 \frac{N^2 \pi r^2}{l} \).

### Step 2: Determine the new dimensions
Now, the wire is wound on a new solenoid with:
- Length \( l' = \frac{l}{2} \)
- Radius \( r' = \frac{r}{2} \)

### Step 3: Calculate the new cross-sectional area
The new cross-sectional area \( A_2 \) is:
\[ A_2 = \pi (r')^2 = \pi \left(\frac{r}{2}\right)^2 = \pi \frac{r^2}{4} \]

### Step 4: Determine the number of turns
Since the length of the wire remains constant, the number of turns \( N' \) in the new solenoid will be different. The number of turns is proportional to the length of the solenoid:
\[ N' = \frac{l'}{d} = \frac{\frac{l}{2}}{d} = \frac{N}{2} \]
where \( d \) is the distance between adjacent turns.

### Step 5: Calculate the new self-inductance
Now we can find the new self-inductance \( L' \):
\[ L' = \mu_0 \frac{(N')^2 A_2}{l'} \]
Substituting the values:
\[ L' = \mu_0 \frac{\left(\frac{N}{2}\right)^2 \left(\pi \frac{r^2}{4}\right)}{\frac{l}{2}} \]
\[ L' = \mu_0 \frac{\frac{N^2}{4} \cdot \pi \frac{r^2}{4}}{\frac{l}{2}} = \mu_0 \frac{N^2 \pi r^2}{l} \cdot \frac{1}{4} \cdot 2 \]
\[ L' = \mu_0 \frac{N^2 \pi r^2}{l} \cdot \frac{1}{2} = \frac{L}{2} \]

### Conclusion
Thus, the self-inductance of the new solenoid is:
\[ L' = \frac{L}{2} \]