A current of `(1)/(4pi)` A is flowing through a toroid. It has 1000 number of turns per meter then value of magnetic field (in `Wb//m^(2)`) along its axis is

Options

A. `10^(-2)`
B. `10^(-3)`
C. `10^(-4)`
D. `10^(-7)`

Answer

To find the magnetic field along the axis of a toroid with a given current and number of turns per meter, we can follow these steps:

### Step 1: Understand the Formula
The magnetic field \( B \) inside a toroid can be calculated using the formula:
\[ B = \frac{\mu_0 n I}{2 \pi r} \]
where:
- \( \mu_0 \) is the permeability of free space (\( 4\pi \times 10^{-7} \, \text{T m/A} \)),
- \( n \) is the number of turns per unit length (in turns/m),
- \( I \) is the current (in A),
- \( r \) is the radius of the toroid (in m).

### Step 2: Substitute Known Values
From the question:
- Current \( I = \frac{1}{4\pi} \, \text{A} \)
- Number of turns per meter \( n = 1000 \, \text{turns/m} \)

We can substitute these values into the formula. However, since we are looking for the magnetic field along the axis of the toroid, we can simplify the formula to:
\[ B = \mu_0 n I \]
This is because, along the axis, the factor of \( 2\pi r \) is not needed.

### Step 3: Calculate the Magnetic Field
Now, substituting the values:
\[ B = (4\pi \times 10^{-7}) \times (1000) \times \left(\frac{1}{4\pi}\right) \]

### Step 4: Simplify the Expression
The \( 4\pi \) in the numerator and denominator cancels out:
\[ B = 10^{-7} \times 1000 \]
\[ B = 10^{-4} \, \text{Wb/m}^2 \]

### Final Answer
Thus, the magnetic field along the axis of the toroid is:
\[ B = 10^{-4} \, \text{Wb/m}^2 \]