The inner and outer radius of a toroid core are 28 cm and 29 cm respectively and around the core 3700 turns of a wire are wounded. If the current in the wire is 10 A, then the magnetic field inside the core of the toroid is

Options

A. `2.60 xx 10^(-2)T`
B. `2.60 xx 10^(-3)T`
C. `4.52 xx 10^(-2)T`
D. `4.52 xx 10^(-3)T`

Explanation

To find the magnetic field inside the core of a toroid, we can follow these steps:

### Step 1: Identify the given values
- Inner radius of the toroid (a) = 28 cm = 0.28 m
- Outer radius of the toroid (b) = 29 cm = 0.29 m
- Total number of turns (N) = 3700 turns
- Current (I) = 10 A

### Step 2: Calculate the average radius (R_avg)
The average radius of the toroid can be calculated as:
\[ R_{\text{avg}} = \frac{a + b}{2} = \frac{0.28 \, \text{m} + 0.29 \, \text{m}}{2} = 0.285 \, \text{m} \]

### Step 3: Calculate the number of turns per unit length (n)
The number of turns per unit length (n) can be calculated using the formula:
\[ n = \frac{N}{2 \pi R_{\text{avg}}} \]
Substituting the values:
\[ n = \frac{3700}{2 \pi (0.285)} \approx \frac{3700}{1.791} \approx 2067.27 \, \text{turns/m} \]
Rounding this gives:
\[ n \approx 2067 \, \text{turns/m} \]

### Step 4: Calculate the magnetic field (B) inside the toroid
The magnetic field inside a toroid is given by the formula:
\[ B = \mu_0 n I \]
Where \(\mu_0\) (the permeability of free space) is approximately \(4\pi \times 10^{-7} \, \text{T m/A}\). Substituting the values:
\[ B = (4\pi \times 10^{-7}) \times (2067) \times (10) \]
Calculating this:
\[ B \approx (4 \times 3.14 \times 10^{-7}) \times 2067 \times 10 \approx 2.60 \times 10^{-2} \, \text{T} \]

### Final Answer
The magnetic field inside the core of the toroid is approximately:
\[ B \approx 0.026 \, \text{T} \, \text{or} \, 2.60 \times 10^{-2} \, \text{T} \]