1m long wire is folded in form of a circular coil and 100 mA electric current is flowing in it, then magnetic field at a point 1m away from its centre on its axis

Options

A. `(10^(-5))/(4pi)T`
B. `(10^(-8))/(2pi)T`
C. `(10^(-5))/(2pi)T`
D. `(10^(-8))/(4pi)T`

Answer

To find the magnetic field at a point 1m away from the center of a circular coil formed by a 1m long wire carrying a current of 100 mA, we can follow these steps:

### Step 1: Determine the radius of the circular coil
The length of the wire is given as 1m. When the wire is folded into a circular shape, the circumference of the circle is equal to the length of the wire.

\[ 2\pi r = 1 \text{ m} \]

From this, we can solve for the radius \( r \):

\[ r = \frac{1}{2\pi} \text{ m} \]

### Step 2: Convert the current into standard units
The current \( I \) is given as 100 mA. We need to convert this into amperes:

\[ I = 100 \text{ mA} = 100 \times 10^{-3} \text{ A} = 0.1 \text{ A} \]

### Step 3: Use the formula for the magnetic field on the axis of a circular coil
The formula for the magnetic field \( B \) at a distance \( x \) from the center of a circular coil on its axis is given by:

\[ B = \frac{\mu_0 I r^2}{2 (x^2 + r^2)^{3/2}} \]

Where:
- \( \mu_0 = 4\pi \times 10^{-7} \text{ T m/A} \) (permeability of free space)
- \( I = 0.1 \text{ A} \)
- \( r = \frac{1}{2\pi} \text{ m} \)
- \( x = 1 \text{ m} \)

### Step 4: Substitute the values into the formula
Now we substitute the values into the formula:

\[ B = \frac{(4\pi \times 10^{-7}) \times (0.1) \times \left(\frac{1}{2\pi}\right)^2}{2 \left(1^2 + \left(\frac{1}{2\pi}\right)^2\right)^{3/2}} \]

Calculating \( r^2 \):

\[ r^2 = \left(\frac{1}{2\pi}\right)^2 = \frac{1}{4\pi^2} \]

Now substituting \( r^2 \) into the equation:

\[ B = \frac{(4\pi \times 10^{-7}) \times (0.1) \times \left(\frac{1}{4\pi^2}\right)}{2 \left(1 + \frac{1}{4\pi^2}\right)^{3/2}} \]

### Step 5: Simplify the expression
Now we simplify the expression step by step. The numerator becomes:

\[ \frac{4\pi \times 10^{-7} \times 0.1}{4\pi^2} = \frac{10^{-7}}{\pi} \]

The denominator can be simplified as follows:

\[ 1 + \frac{1}{4\pi^2} \approx 1 \quad \text{(since \( \frac{1}{4\pi^2} \) is very small)} \]

Thus, the denominator becomes approximately:

\[ 2 \left(1\right)^{3/2} = 2 \]

### Step 6: Final calculation
Putting it all together:

\[ B \approx \frac{10^{-7}/\pi}{2} = \frac{10^{-7}}{2\pi} \]

### Final Result
Thus, the magnetic field at a point 1m away from the center of the coil on its axis is:

\[ B \approx \frac{10^{-7}}{2\pi} \text{ T} \]