Question

In the given figure, for wire I, \[\dfrac{{dI}}{{dt}} = a\] and for wire II, \[\dfrac{{dI}}{{dt}} = - a\]. Find the direction of induced current in the circular coil.

A. Clockwise
B. Anticlockwise
C. The current will be zero
D. Can’t be determined

Answer 1

Hint: The magnetic field due to an infinite wire carrying current $I$ at a distance $r$ from it is given by $B = \dfrac{{{\mu _0}I}}{{2\pi r}}$. The electric flux is given by and rate of change of this flux gives the induced emf. The direction of induced current will be determined by Lenz law which states that the direction of the current induced in a conductor by a changing magnetic field is such that the magnetic field created by the induced current opposes the initial changing magnetic field which produced it.

Complete step by step answer:
Let us first discuss the reason for induced current.
An emf and current is induced in a coil when the flux linked to it changes with respect to time.
The magnetic field due to an infinite wire carrying current $I$ at a distance $r$ from it is given by $B = \dfrac{{{\mu _0}I}}{{2\pi r}}$. The electric flux is given by and rate of change of this flux gives the induced emf i.e. $e = \dfrac{{d\phi }}{{dt}}$. As the area of the circular coil remains constant then the emf depends upon $\dfrac{{dB}}{{dt}}$ which further depends upon $r$ and $\dfrac{{dI}}{{dt}}$ .
As wire II is closer to the coil than wire I and given that the current in it is decreasing as $\dfrac{{dI}}{{dt}} = - a$. So, $\dfrac{{dB}}{{dt}}$ of wire II dominates over that of wire I and induced current will be according to the wire II.
The direction of induced current will be determined by Lenz law which states that the direction of the current induced in a conductor by a changing magnetic field is such that the magnetic field created by the induced current opposes the initial changing magnetic field which produced it.
So, the direction of induced current in the coil will be given by right hand thumb rule. As the net flux is decreasing inwards so the current will be clockwise to increase the flux inward.
Hence, option A is correct.

Note: Lenz’s law is primarily based on Faraday’s law of induction. Faraday’s law of Induction states that a current is induced in a conductor when the flux linked to it changes. Lenz’s law gives the direction of this induced current, which opposes the initial changing magnetic field which produced it. This can be signified in the formula for Faraday’s law as there is a negative sign (‘–’) in it i.e. $e = - \dfrac{{d\phi }}{{dt}}$ .