A current carrying coil (I = 5A, R = 10 cm.) having 50 number of turns find field at its centre-

Options

A. 1.57 mT
B. 3.14 mT
C. 1 mT
D. 2 mT

Explanation

To find the magnetic field at the center of a current-carrying coil, we can use the formula:

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

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

**Step 1: Identify the given values.**
- Current \(I = 5 \, \text{A}\)
- Radius \(R = 10 \, \text{cm} = 0.1 \, \text{m}\)
- Number of turns \(N = 50\)

**Step 2: Substitute the values into the formula.**
We need to calculate the magnetic field \(B\) using the formula:

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

Substituting the values:

\[ B = \frac{4\pi \times 10^{-7} \, \text{T m/A} \times 50 \times 5}{2 \times 0.1} \]

**Step 3: Calculate the numerator.**
Calculating the numerator:

\[ 4\pi \times 10^{-7} \times 50 \times 5 = 4\pi \times 2500 \times 10^{-7} = 10000\pi \times 10^{-7} \]

**Step 4: Calculate the denominator.**
Calculating the denominator:

\[ 2 \times 0.1 = 0.2 \]

**Step 5: Combine the results.**
Now, substituting back into the equation:

\[ B = \frac{10000\pi \times 10^{-7}}{0.2} \]

**Step 6: Simplify the expression.**
\[ B = 50000\pi \times 10^{-7} \]

**Step 7: Calculate the numerical value.**
Using \(\pi \approx 3.14\):

\[ B \approx 50000 \times 3.14 \times 10^{-7} = 157000 \times 10^{-7} = 1.57 \times 10^{-3} \, \text{T} \]

**Step 8: Convert to milliTesla.**
\[ B = 1.57 \, \text{mT} \]

Thus, the magnetic field at the center of the coil is:

\[ \boxed{1.57 \, \text{mT}} \]

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