Question

# The magnetic field through a coil having $200$ turns and cross-sectional area $0.04{m^2}$ changes from $0.1Wb{m^{ - 2}}$ to $0.04Wb{m^{ - 2}}$ in $0.02\sec$ . Find the induced emf.

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Hint An emf will be induced when the magnetic field changes. In other words, when the flux associated with a coil or a conductor changes there will be an induced emf. Emf means electromotive force that is the driving force of the current. Here the change in the magnetic field is given and we have to find the induced emf.

Complete step by step solution
The number of turns of the coil is given as, $N = 200$
The cross-sectional area of the coil is given by, $A = 0.04{m^2}$
The magnetic field before changing is, ${B_1} = 0.1Wb{m^{ - 2}}$
The magnetic field is changed to ${B_2} = 0.04Wb{m^{ - 2}}$
The time is given as, $t = 0.02\sec$
The induced emf is proportional to the rate of change of flux,
i.e, $emf = \dfrac{{d\Phi }}{{dt}} = \dfrac{{ - d\left( \Phi \right)}}{{dt}}$
The flux can be written as,
$\Phi = NBA$
where $N$ stands for the number of turns, $B$ stands for the magnetic field, and $A$ stands for the cross-sectional area of the coil.
Substituting this value of $\Phi$ in the above equation,
$emf = - NA\dfrac{{dB}}{{dt}}$
The change in the magnetic field, $dB = {B_2} - {B_1}$
The value of ${B_1}$ is given as, ${B_1} = 0.1Wb{m^{ - 2}}$
The value of ${B_2}$ is given as, ${B_2} = 0.04Wb{m^{ - 2}}$
The total number of turns in the coil is given as, $N = 200$
The area of cross-section of the coil is given as, $A = 0.04{m^2}$
Substituting these values in the above equation,
The induced emf will be,
$emf = - 200 \times 0.04 \times \dfrac{{0.04 - 0.1}}{{0.02}} = 24V$
The answer is: $24V$ .

Note
The external energy required to drive the free electrons in a particular direction is called the electromotive force. The emf is the work done in moving a unit positive charge from one end to the other. The rate of flow of charge is the electric current. The unit of emf is volt $(V)$ or Joule/Coulomb $(J/C)$ .