A plane of coil of 10 turns is tightly wound around a solenoid of diameter 2 cm having 400 turns per centimeter. The relative permeability of the core is 800. Calculate the inductance of solenoid

Answer

To calculate the inductance of the solenoid, we can follow these steps:

### Step-by-Step Solution:

1. **Identify Given Data**:
- Number of turns in the coil (N) = 10 turns
- Diameter of the solenoid (d) = 2 cm = 0.02 m
- Turns per centimeter (n) = 400 turns/cm = 40000 turns/m
- Relative permeability (μ_r) = 800

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:
\[ A = \pi \left(\frac{d}{2}\right)^2 \]
Substituting the diameter:
\[ A = \pi \left(\frac{0.02}{2}\right)^2 = \pi \left(0.01\right)^2 = \pi \times 0.0001 \approx 3.14 \times 10^{-4} \, \text{m}^2 \]

3. **Calculate the Permeability of the Core (μ)**:
The permeability of the core (μ) is given by:
\[ \mu = \mu_r \times \mu_0 \]
where \(\mu_0\) (the permeability of free space) is approximately \(4\pi \times 10^{-7} \, \text{H/m}\).
Thus,
\[ \mu = 800 \times (4\pi \times 10^{-7}) \approx 800 \times 1.25664 \times 10^{-6} \approx 1.0053 \times 10^{-3} \, \text{H/m} \]

4. **Calculate the Inductance (L)**:
The formula for the inductance of a solenoid is given by:
\[ L = \frac{\mu N^2 A}{l} \]
where \(l\) is the length of the solenoid. The length can be calculated from the number of turns per meter:
\[ l = \frac{N}{n} = \frac{10}{40000} = 0.00025 \, \text{m} \]
Now substituting the values into the inductance formula:
\[ L = \frac{(1.0053 \times 10^{-3}) \times (10^2) \times (3.14 \times 10^{-4})}{0.00025} \]
\[ L = \frac{(1.0053 \times 10^{-3}) \times 100 \times (3.14 \times 10^{-4})}{0.00025} \]
\[ L = \frac{(1.0053 \times 10^{-1}) \times (3.14 \times 10^{-4})}{0.00025} \]
\[ L = \frac{3.154682 \times 10^{-5}}{0.00025} \approx 0.1264 \, \text{H} \]

### Final Answer:
The inductance of the solenoid is approximately **0.1264 H**.