A current of 2 amperes is flowing through a circular coil of radius 10 cm containing 100 turns. The magnetic flux density at the centre of the coil is

Options

A. `0.126 xx 10^(-2)`
B. `1.26 xx 10^(-2)`
C. `1.26 xx 10^(-4) `
D. `1.26 xx 10^(-5)`

Explanation

To find the magnetic flux density at the center of a circular coil, we can use the formula:

\[ B = \frac{N \mu_0 I}{2R} \]

where:
- \( B \) is the magnetic flux density,
- \( N \) is the number of turns in the coil,
- \( \mu_0 \) is the permeability of free space (\( 4\pi \times 10^{-7} \, \text{T m/A} \)),
- \( I \) is the current in amperes,
- \( R \) is the radius of the coil in meters.

### Step 1: Identify the given values
- Current, \( I = 2 \, \text{A} \)
- Radius, \( R = 10 \, \text{cm} = 0.1 \, \text{m} \)
- Number of turns, \( N = 100 \)

### Step 2: Substitute the values into the formula
We need to substitute the known values into the formula for magnetic flux density.

\[ B = \frac{N \mu_0 I}{2R} \]

Substituting the values:

\[ B = \frac{100 \times (4\pi \times 10^{-7}) \times 2}{2 \times 0.1} \]

### Step 3: Simplify the equation
First, calculate the denominator:

\[ 2R = 2 \times 0.1 = 0.2 \]

Now substitute this back into the equation:

\[ B = \frac{100 \times (4\pi \times 10^{-7}) \times 2}{0.2} \]

### Step 4: Calculate \( B \)
Now we can calculate \( B \):

\[ B = \frac{100 \times (4\pi \times 10^{-7}) \times 2}{0.2} \]

Calculating the numerator:

\[ 100 \times (4\pi \times 10^{-7}) \times 2 = 800\pi \times 10^{-7} \]

Now, divide by \( 0.2 \):

\[ B = \frac{800\pi \times 10^{-7}}{0.2} = 4000\pi \times 10^{-7} \]

Using \( \pi \approx 3.14 \):

\[ B \approx 4000 \times 3.14 \times 10^{-7} = 12560 \times 10^{-7} = 1.256 \times 10^{-3} \, \text{T} \]

### Step 5: Convert to appropriate units
To express this in terms of \( 10^{-2} \):

\[ B = 0.126 \times 10^{-2} \, \text{T} \]

### Final Answer
Thus, the magnetic flux density at the center of the coil is:

\[ B \approx 0.126 \times 10^{-2} \, \text{T} \]