Question

A coil having n turns and resistance R$\Omega$ is connected with a galvanometer of resistance 4R$\Omega$. This combination is moved in time t second from a magnetic field $w_1$ Weber to $w_2$. The induced current in the circuit is
(A) $\dfrac{-(w_2 - w_1)}{5Rt}$
(B) $\dfrac{-n(w_2 - w_1)}{5Rt}$
(C) $\dfrac{-(w_2 - w_1)}{Rnt}$
(D) $\dfrac{-n(w_2 - w_1)}{Rt}$

Answer 1

Hint: The rate of change of magnetic flux through the coil gives the EMF induced through the coil. This happens so as the magnetic field changes, flux changes and the current that is produced by this emf can be obtained by using Ohm's law.
Formula used:
The emf induced will be given by:
$e = -\dfrac{d \phi}{dt}$.
The expression for current from Ohm's law is:
$I = \dfrac{V}{R}$.

Complete answer:
We are given that the coil has n turns, we assume that the area of the coil is a. Then, we know that flux passing through the same coil can be expressed as:
$\phi = n B a$.
The magnetic field is given in two regions to us. The flux change through the coils when it moves in t time from one region to other is expressed as:
$e = - \dfrac{a (w_2 - w_1)}{t} $.
The total resistance in the circuit is given by 5R.
The current due to this emf in the coil is given by:
$I = - \dfrac{a (w_2 - w_1)}{5Rt}$
This is the total current in the circuit.

Therefore, the correct answer appears to be option (B).

Additional Information:
Electromagnetic induction is caused in a coil by changing magnetic flux through it. If the current through the coil changes, the magnetic flux through it changes. Another way of producing this change in magnetic flux is by changing the area of the coil somehow or by changing the magnitude of the magnetic field.

Note:
The formula for emf everywhere involves a differentiation of flux with respect to time but we subtracted the flux and divided it by the amount of time given to us. This might not occur to someone as easily. Differentiation is taken when we are dealing with very small quantities.