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