Question

A coil has 1000 turns and 500 $c{m^2}$ as its area. The plane of the coil is placed at right angles to a magnetic induction field of $2 \times {10^{ - 5}}web/{m^2}$. The coil is rotated through $180^\circ $ in 0.2 seconds. The average emf induced in the coil (in millivolts) is
$
  {\text{A}}{\text{. 5}} \\
  {\text{B}}{\text{. 10}} \\
  {\text{C}}{\text{. 15}} \\
  {\text{D}}{\text{. 20}} \\
$

Answer 1

Hint: The emf induced in a given coil is equal to the time rate of change of magnetic flux in the coil. The magnetic flux in a coil is equal to the product of magnetic field and the cross-sectional area of the coil multiplied by the number of turns in the coil.

Formula used: The emf through a coil due to changing flux is given as
$\varepsilon = N\dfrac{{d\phi }}{{dt}}{\text{ }}...\left( i \right)$
Here $\varepsilon $ is the emf induced in the coil which has N turns while $\phi $ is the flux through the coil.
The magnetic flux in a coil is given as
$\phi = BA\cos \theta $
Here B is the magnetic field through the coil while A is the cross-sectional area of the given coil. The angle between the magnetic field and area vector is $\theta $.

Complete step-by-step answer:
The given coil has number of turns equal to
$N = 1000$
The cross-sectional area of the coil is given as
$A = 500c{m^2} = 500 \times {10^{ - 4}}{m^2} = 5 \times {10^{ - 2}}{m^2}$
The magnetic field through the coil is given as
$B = 2 \times {10^{ - 5}}web/{m^2}$
The coil is rotated through $180^\circ $ in 0.2 seconds. Initial flux can be calculated as follows:
${\phi _1} = BA\cos 0^\circ = BA$
The angle is zero because the magnetic field and area vector are along the same direction. The final flux is given as
${\phi _2} = BA\cos 180^\circ = - BA$
Therefore the change in flux can be calculated as follows:
$d\phi = {\phi _1} - {\phi _2} = BA - \left( { - BA} \right) = 2BA$
Also $dt = 0.2s$
Now with all the known values we can calculate the emf through the coil using equation (i) in the following way.
$
  \varepsilon = N\dfrac{{d\phi }}{{dt}} = \dfrac{{2NBA}}{{dt}} \\
   = \dfrac{{2 \times 1000 \times 2 \times {{10}^{ - 5}} \times 5 \times {{10}^{ - 2}}}}{{0.2}} \\
   = 0.01V = 10mV \\
$
This is the required answer.

So, the correct answer is “Option B”.

Note: It is Faraday's law which states that when the magnetic flux in a coil changes then an emf is induced in the coil. The emf induced is directly proportional to the amount of change in flux and direction of emf is such that it opposes the change that induces it.