A toroid of core of inner radius 0.25 m and outer radius 0.26 m around which 3500 turns of a wire are wound. If the current in the wire is 11 A, then magnetic field inside the core of the toroid is

Options

A. `3xx10^2T`
B. `3xx10^-2T`
C. `3xx10^-7T`
D. `3xx10^7T`

Answer

To find the magnetic field inside the core of a toroid, we can use the formula derived from Ampère's circuital law. Here’s a step-by-step solution to the problem:

### Step 1: Identify the given values
- Inner radius of the toroid, \( R_1 = 0.25 \, \text{m} \)
- Outer radius of the toroid, \( R_2 = 0.26 \, \text{m} \)
- Number of turns, \( N = 3500 \)
- Current, \( I = 11 \, \text{A} \)

### Step 2: Calculate the average radius of the toroid
The average radius \( R \) of the toroid can be calculated as:
\[ R = \frac{R_1 + R_2}{2} = \frac{0.25 \, \text{m} + 0.26 \, \text{m}}{2} = \frac{0.51 \, \text{m}}{2} = 0.255 \, \text{m} \]

### Step 3: Use the formula for the magnetic field inside a toroid
The magnetic field \( B \) inside a toroid is given by the formula:
\[ B = \frac{\mu_0 N I}{2 \pi R} \]
where \( \mu_0 \) is the permeability of free space, approximately \( 4\pi \times 10^{-7} \, \text{T m/A} \).

### Step 4: Substitute the values into the formula
Now, substituting the values we have:
\[ B = \frac{(4\pi \times 10^{-7} \, \text{T m/A}) \times 3500 \times 11}{2 \pi \times 0.255} \]

### Step 5: Simplify the expression
The \( \pi \) cancels out:
\[ B = \frac{(4 \times 10^{-7}) \times 3500 \times 11}{2 \times 0.255} \]

Calculating the numerator:
\[ 4 \times 3500 \times 11 = 154000 \]
Thus,
\[ B = \frac{154000 \times 10^{-7}}{2 \times 0.255} \]

Calculating the denominator:
\[ 2 \times 0.255 = 0.51 \]

Now substituting back:
\[ B = \frac{154000 \times 10^{-7}}{0.51} \]

### Step 6: Calculate the final value of B
Calculating the value:
\[ B = 3.02 \times 10^{-2} \, \text{T} \approx 3 \times 10^{-2} \, \text{T} \]

### Final Answer
The magnetic field inside the core of the toroid is approximately:
\[ B \approx 3 \times 10^{-2} \, \text{T} \]

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