A solenoid having 500 turns and length 2 m has radius of 2 cm , the self inductance of solenoid

Options

A. `4xx10^(-4)H`
B. `2xx10^(-4)H`
C. `8xx10^(-4)H`
D. `16xx10^(-4)H`

Answer

To find the self-inductance of the solenoid, we can use the formula for the self-inductance \( L \) of a solenoid, which is given by:

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

Where:
- \( L \) is the self-inductance,
- \( \mu_0 \) is the permeability of free space (\( 4\pi \times 10^{-7} \, \text{H/m} \)),
- \( N \) is the total number of turns,
- \( A \) is the cross-sectional area of the solenoid,
- \( l \) is the length of the solenoid.

### Step 1: Identify the given values
- Total number of turns, \( N = 500 \)
- Length of the solenoid, \( l = 2 \, \text{m} \)
- Radius of the solenoid, \( r = 2 \, \text{cm} = 0.02 \, \text{m} \)

### Step 2: Calculate the cross-sectional area \( A \)
The cross-sectional area \( A \) of the solenoid can be calculated using the formula for the area of a circle:

\[ A = \pi r^2 \]

Substituting the radius:

\[ A = \pi (0.02)^2 = \pi \times 0.0004 = 0.0004\pi \, \text{m}^2 \]

### Step 3: Substitute the values into the inductance formula
Now, we can substitute all the values into the self-inductance formula:

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

Substituting the known values:

\[ L = (4\pi \times 10^{-7}) \frac{(500)^2 (0.0004\pi)}{2} \]

### Step 4: Simplify the expression
Calculating \( N^2 \):

\[ N^2 = 500^2 = 250000 \]

Now substituting \( N^2 \) into the equation:

\[ L = (4\pi \times 10^{-7}) \frac{250000 \times 0.0004\pi}{2} \]

Calculating the area term:

\[ = (4\pi \times 10^{-7}) \frac{100\pi}{2} \]

\[ = (4\pi \times 10^{-7}) \times 50\pi \]

### Step 5: Calculate the final value
Now, we simplify further:

\[ L = 200\pi^2 \times 10^{-7} \]

Using \( \pi \approx 3.14 \):

\[ L \approx 200 \times (3.14)^2 \times 10^{-7} \]

Calculating \( (3.14)^2 \):

\[ \approx 9.8596 \]

Thus:

\[ L \approx 200 \times 9.8596 \times 10^{-7} \approx 1971.92 \times 10^{-7} \, \text{H} \approx 1.97192 \times 10^{-4} \, \text{H} \]

### Final Answer
\[ L \approx 2 \times 10^{-4} \, \text{H} \]