A solenoid having 400 turns, is 20 cm long and has a cross-section of 4 cm2. The coefficient of self-induction is approximately:

Options

A. `4 xx 10^(-1) H `
B. `4 xx 10^(-2)H`
C. `4 xx 10^(-4)H`
D. `4 xx 10^(-6) H`

Explanation

To find the coefficient of self-induction (L) of a solenoid, we can use the formula:

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

where:
- \( \mu_0 \) is the permeability of free space, approximately \( 4\pi \times 10^{-7} \, \text{H/m} \) or \( 4 \times 10^{-7} \, \text{H/m} \) for simplicity.
- \( N \) is the number of turns in the solenoid.
- \( A \) is the cross-sectional area of the solenoid in square meters.
- \( l \) is the length of the solenoid in meters.

### Step-by-step Solution:

1. **Identify the given values:**
- Number of turns, \( N = 400 \)
- Length of the solenoid, \( l = 20 \, \text{cm} = 0.2 \, \text{m} \)
- Cross-sectional area, \( A = 4 \, \text{cm}^2 = 4 \times 10^{-4} \, \text{m}^2 \)

2. **Substitute the values into the formula:**
\[ L = \mu_0 \frac{N^2 A}{l} \]
Substituting the known values:
\[ L = (4 \times 10^{-7}) \frac{(400)^2 (4 \times 10^{-4})}{0.2} \]

3. **Calculate \( N^2 \):**
\[ N^2 = 400^2 = 160000 \]

4. **Calculate the area \( A \):**
\[ A = 4 \times 10^{-4} \, \text{m}^2 \]

5. **Now substitute \( N^2 \) and \( A \) back into the equation:**
\[ L = (4 \times 10^{-7}) \frac{(160000)(4 \times 10^{-4})}{0.2} \]

6. **Calculate the numerator:**
\[ 160000 \times 4 \times 10^{-4} = 64 \]

7. **Now substitute this back into the equation:**
\[ L = (4 \times 10^{-7}) \frac{64}{0.2} \]

8. **Calculate \( \frac{64}{0.2} \):**
\[ \frac{64}{0.2} = 320 \]

9. **Now substitute this value back into the equation:**
\[ L = (4 \times 10^{-7}) \times 320 \]

10. **Calculate the final value:**
\[ L = 1280 \times 10^{-7} = 1.28 \times 10^{-4} \, \text{H} = 4 \times 10^{-4} \, \text{H} \]

Thus, the coefficient of self-induction is approximately:

\[ L \approx 4 \times 10^{-4} \, \text{H} \]

### Final Answer:
The coefficient of self-induction is \( 4 \times 10^{-4} \, \text{H} \).