For a solenoid keeping the turn density constant its length makes halved and its cross section radius is doubled then the inductance of the solenoid increased by :-

Options

A. `200%`
B. `100%`
C. `800%`
D. `700%`

Answer

To solve the problem, we need to analyze how the inductance of a solenoid changes when its length is halved and its cross-sectional radius is doubled, while keeping the turn density constant.

### Step-by-Step Solution:

1. **Understand the formula for inductance of a solenoid**:
The inductance \( L \) of a solenoid is given by the formula:
\[ L = \mu_0 n^2 \pi r^2 L \]
where:
- \( \mu_0 \) is the permeability of free space,
- \( n \) is the number of turns per unit length,
- \( r \) is the radius of the solenoid,
- \( L \) is the length of the solenoid.

2. **Identify the changes in parameters**:
- The length of the solenoid is halved: \( L' = \frac{L}{2} \)
- The radius of the solenoid is doubled: \( r' = 2r \)

3. **Calculate the new number of turns per unit length**:
Since the turn density (number of turns per unit length) is kept constant, \( n \) remains the same.

4. **Substitute the new values into the inductance formula**:
The new inductance \( L' \) can be expressed as:
\[ L' = \mu_0 n^2 \pi (r')^2 (L') \]
Substituting the new values:
\[ L' = \mu_0 n^2 \pi (2r)^2 \left(\frac{L}{2}\right) \]
Simplifying this:
\[ L' = \mu_0 n^2 \pi (4r^2) \left(\frac{L}{2}\right) = \mu_0 n^2 \pi 4r^2 \cdot \frac{L}{2} = 2 \mu_0 n^2 \pi r^2 L \]

5. **Compare the new inductance with the original inductance**:
The original inductance \( L \) was:
\[ L = \mu_0 n^2 \pi r^2 L \]
The new inductance \( L' \) is:
\[ L' = 2 \mu_0 n^2 \pi r^2 L \]
This shows that:
\[ L' = 2L \]

6. **Calculate the increase in inductance**:
The increase in inductance is:
\[ \Delta L = L' - L = 2L - L = L \]
Therefore, the percentage increase in inductance is:
\[ \text{Percentage Increase} = \frac{\Delta L}{L} \times 100\% = \frac{L}{L} \times 100\% = 100\% \]

### Final Answer:
The inductance of the solenoid increased by **100%**.