A tightly wound 90 turn coil of radius 15 cm has a magnetic field of `4 xx 10^(-4)T` at its centre. The current flowing through it is

Options

A. 1.06A
B. 2.44A
C. 3.44A
D. 4.44A

Explanation

To find the current flowing through a tightly wound coil with a given magnetic field at its center, we can use the formula for the magnetic field produced by a circular coil:

\[ B = \frac{\mu_0 n I}{2r} \]

Where:
- \( B \) is the magnetic field at the center of the coil,
- \( \mu_0 \) is the permeability of free space (\( 4\pi \times 10^{-7} \, \text{T m/A} \)),
- \( n \) is the number of turns per unit length,
- \( I \) is the current flowing through the coil,
- \( r \) is the radius of the coil.

Given:
- Number of turns, \( N = 90 \)
- Radius, \( r = 15 \, \text{cm} = 0.15 \, \text{m} \)
- Magnetic field, \( B = 4 \times 10^{-4} \, \text{T} \)

### Step 1: Rearranging the formula to find current \( I \)

We can rearrange the formula to solve for \( I \):

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

### Step 2: Substitute the known values into the equation

Now we will substitute the known values into the equation:

\[ I = \frac{(4 \times 10^{-4} \, \text{T}) \cdot (2 \cdot 0.15 \, \text{m})}{4\pi \times 10^{-7} \, \text{T m/A} \cdot 90} \]

### Step 3: Calculate the numerator

Calculating the numerator:

\[ 2 \cdot 0.15 = 0.30 \, \text{m} \]
\[ (4 \times 10^{-4}) \cdot (0.30) = 1.2 \times 10^{-4} \, \text{T m} \]

### Step 4: Calculate the denominator

Calculating the denominator:

\[ 4\pi \times 10^{-7} \cdot 90 \approx 1.131 \times 10^{-5} \, \text{T m/A} \]

### Step 5: Substitute the values into the equation

Now substituting back into the equation for \( I \):

\[ I = \frac{1.2 \times 10^{-4}}{1.131 \times 10^{-5}} \approx 10.61 \, \text{A} \]

### Step 6: Final calculation

Calculating the final value:

\[ I \approx 1.06 \, \text{A} \]

Thus, the current flowing through the coil is approximately \( 1.06 \, \text{A} \).

### Final Answer:
The current flowing through the coil is \( 1.06 \, \text{A} \).

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