A tightly wound \(100\) turns coil of radius \(10 cm\) carries a current of \( 7 A\). The magnitude of the magnetic field at the centre of the coil is (Take permeability of free space as \(4\pi×10^{-7}\) SI units):
- \(44mT\)
- \(4.4T\)
- \(4.4mT\)
- \(44 T\)
The Correct Option is C
Approach Solution - 1
To determine the magnitude of the magnetic field at the center of a tightly wound coil, we use the formula for the magnetic field at the center of a circular loop of wire:
\[B=\frac{\mu_0 n I}{2R}\]
where:
- \(B\) is the magnetic field,
- \(\mu_0 = 4\pi \times 10^{-7} \, \text{T}\cdot \text{m/A}\) (permeability of free space),
- \(n = 100\) is the number of turns,
- \(I = 7 \, \text{A}\) is the current,
- \(R = 0.1 \, \text{m}\) is the radius of the coil.
Substituting the given values into the formula:
\[B = \frac{(4\pi \times 10^{-7}) \cdot 100 \cdot 7}{2 \cdot 0.1}\]
Calculating the expression inside the parentheses first:
\[B = \frac{(4\pi \times 10^{-7}) \cdot 700}{0.2}\]
Further simplifying:
\[B = \frac{2800\pi \times 10^{-7}}{0.2}\]
\[B = 14000\pi \times 10^{-7}\]
Simplifying \(\pi \approx 3.1416\):
\[B = 14000 \times 3.1416 \times 10^{-7}\]
\[B \approx 43998.4 \times 10^{-7}\]
Converting the field into milliTesla (1T = 1000 mT):
\[B \approx 4.4 \, \text{mT}\]
Therefore, the magnitude of the magnetic field at the center of the coil is approximately \(4.4 \, \text{mT}\).
Approach Solution -2
Step 1: Use the formula for the magnetic field at the center of a circular coil. - The magnetic field at the center of a circular coil is given by: \[ B = \frac{\mu_0 N I}{2R} \] where N = 100 (number of turns), I = 7A (current), R = 0.1m (radius), and \(\mu_0 = 4\pi \times 10^{-7} Tm/A\) (permeability of free space). Step 2: Substitute the values into the formula. \[ B = \frac{(4\pi \times 10^{-7}) \cdot 100 \cdot 7}{2 \times 0.1} \] Simplify: \[ B = \frac{28\pi \times 10^{-7}}{0.2} \] \[ B = 4.4 \times 10^{-3} T = \mathbf{4.4mT} \]
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