Question

In electromagnetic induction, the induced charge in a coil is independent of:
A. Time
B. Change in flux
C. Resistance in the circuit
D. None of the above

Answer 1

Hint: According to Faraday’s law of electromagnetic induction, whenever a coil is kept in a changing magnetic flux, an EMF is induced in the coil that tries to oppose the changing magnetic flux. The induced EMF gives rise to induced charge and current in the coil.

Formula used: According to Faraday’s law of electromagnetic induction, the EMF $E$ induced in the coil is given by,
$E=-\dfrac{d{{\phi }_{B}}}{dt}$
where ${{\phi }_{B}}$ is the magnetic flux passing through the coil and $t$ is the time interval for which the magnetic flux is changing.
The negative sign implies that the induced EMF opposes the changing flux.

Also, according to Ohm’s law
$E=iR$
where $i$ is the current in the circuit, $R$ is the resistance.
$i=\dfrac{dq}{dt}$
where $q$ is the charge passing through a cross section of the conductor

Complete step-by-step answer:
When a coil is placed in a changing magnetic flux, an EMF is induced in the coil according to Faraday’s law of electromagnetic induction.
According to Faraday’s law of electromagnetic induction, the EMF $E$ induced in the coil is given by,
$E=-\dfrac{d{{\phi }_{B}}}{dt}$ --(1)
where ${{\phi }_{B}}$ is the magnetic flux passing through the coil and $t$ is the time interval for which the magnetic flux is changing.
The negative sign implies that the induced EMF opposes the changing flux.
Also, according to Ohm’s law
$E=iR$ --(2)
where $i$ is the current in the circuit, $R$ is the resistance.
Putting (1) in (2), we get,
$-\dfrac{d{{\phi }_{B}}}{dt}=iR$ --(3)
$i=\dfrac{dq}{dt}$ --(4)
where $q$ is the charge passing through a cross section of the conductor.
Putting (4) in (3), we get,
$-\dfrac{d{{\phi }_{B}}}{dt}=\dfrac{dq}{dt}R$
$\therefore dq=-\dfrac{d{{\phi }_{B}}}{R}$ --(5)
Therefore, as seen from (5), the induced charge in the coil is independent of the time for which the flux is changing.
Hence, the correct option is A) time.

Note: Students can make the mistake of only considering Faraday’s law of electromagnetic induction and hence come to the result that the EMF and hence, charge induced is independent of the resistance of the circuit. However, they must remember that Faraday’s law relates the emf induced in the coil and not the charge. By using Ohm’s law and the definition of current, we will get the correct relation between the magnetic flux and the resistance of the circuit.