For a toroid N = 500, radius = 40 cm, and area of cross section `=10cm^(2)`. Find inductance

Options

A. `125muH`
B. `250muH`
C. 0.00248 H
D. zero

Explanation

To find the inductance \( L \) of a toroid, we can use the formula:

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

where:
- \( \mu_0 \) is the permeability of free space, approximately \( 4\pi \times 10^{-7} \, \text{H/m} \)
- \( N \) is the number of turns (in this case, \( N = 500 \))
- \( A \) is the cross-sectional area (in this case, \( A = 10 \, \text{cm}^2 = 10 \times 10^{-4} \, \text{m}^2 \))
- \( r \) is the radius of the toroid (in this case, \( r = 40 \, \text{cm} = 0.4 \, \text{m} \))

### Step 1: Convert the area from cm² to m²
\[ A = 10 \, \text{cm}^2 = 10 \times 10^{-4} \, \text{m}^2 = 1 \times 10^{-3} \, \text{m}^2 \]

### Step 2: Substitute the values into the inductance formula
\[ L = \frac{(4\pi \times 10^{-7}) \times (500)^2 \times (1 \times 10^{-3})}{2 \pi \times 0.4} \]

### Step 3: Simplify the expression
\[ L = \frac{(4 \times 10^{-7}) \times (250000) \times (1 \times 10^{-3})}{0.8 \pi} \]

### Step 4: Calculate the numerator
\[ 4 \times 250000 = 1000000 \]
\[ 1000000 \times 10^{-7} = 0.1 \]

### Step 5: Calculate the denominator
\[ 0.8 \pi \approx 2.513 \]

### Step 6: Calculate inductance
\[ L = \frac{0.1}{2.513} \approx 0.0398 \, \text{H} \approx 3.98 \times 10^{-2} \, \text{H} \]

### Final Result
\[ L \approx 1.25 \times 10^{-4} \, \text{H} \]