Question

A circular coil of radius R=10 cm having 500 turns and total resistance $2\Omega$ is placed initially perpendicular to the earth's magnetic field of $3\times 10^{-5}T$. The coil is rotated about its vertical diameter by an angle $2\pi$ in $0.5$ seconds. The induced current in the coil is

• A. 0.5 mA
• B. 1.0 mA
• C. 1.5 mA
• D. 3.0 mA

Answer 1

Answer: (B)

Here, Radius, $R=10cm=10\times 10^{-2}m$
Number of turns, $N=500$, Resistance $=2\Omega$
Magnetic field, $B=3\times 10^{-5}\cdot T$ Initial flux through the coil, ${\varphi }_{B_{\text{initial }}}=BA\cos \theta$
$=3\times 10^{-5}\times \pi \times {\left(10\times 10^{-2}\right)}^2\cos 0^{\circ }=3\pi \times 10^{-7}Wb$
Final flux after rotation,
${\varphi }_{B_{\text{final }}}=3\times 10^{-5}\times \pi \times {\left(10\times 10^{-2}\right)}^2\times \cos 180^{\circ }$
$=-3\pi \times 10^{-7}Wb$
Emf induced in the coil, $\epsilon =-N\frac{\Delta {\varphi }_B}{\Delta t}=-N\frac{\left({\varphi }_{B_{\text{final }}}-{\varphi }_{B_{\text{initial }}}\right)}{\Delta t}$
$=-\frac{500\times \left(-3\pi \times 10^{-7}-3\pi \times 10^{-7}\right)}{0.5}$
$=\frac{500\times 6\pi \times 10^{-7}}{0.5}=2\times 10^{-3}V$
Induced current in the coil, $I=\frac{2\times 10^{-3}V}{2\Omega }=10^{-3}A=1mA$