Question

The induced current is found to be highest when
$\left( A \right)$ The direction of motion of the coil is at ${0^o}$to the magnetic field.
$\left( B \right)$ The direction of motion of the coil is in a straight line with the magnetic field.
$\left( C \right)$ The direction of motion of the coil is the right angle to the magnetic field.
$\left( D \right)$ All of the above.

Answer 1

Hint – In this question use the property that induced current is the rate of change of magnetic flux w.r.t. time and also use the property that magnetic flux density is the ratio of magnetic flux to the area perpendicular to the magnetic flux direction, so use these concepts to reach the solution of the question.

Complete step-by-step answer:
Let induced current = I
Magnetic flux = $\phi $
So magnetic flux density (B) is the ratio of magnetic flux to the area (A) perpendicular to the magnetic flux direction.
$ \Rightarrow B = \dfrac{\phi }{A}$ Wb/${m^2}$.
Therefore, $\phi = BA$
Now let the time dependent magnetic flux density is given as $B = {B_m}\cos \left( {\omega t} \right)$
Where, ${B_m}$ is the maximum value or the magnitude of the magnetic flux density and $\left( {\omega t} \right)$ is the angle between the magnetic field and the normal (perpendicular) to area (A).
$ \Rightarrow \phi = {B_m}A\cos \left( {\omega t} \right)$
Now as we know that induced current is the rate of change of magnetic flux w.r.t. time.
$ \Rightarrow I = \dfrac{d}{{dt}}\left( \phi \right)$
Now substitute the values we have,
$ \Rightarrow I = \dfrac{d}{{dt}}\left( {{B_m}A\cos \left( {\omega t} \right)} \right)$
Now as we know that ${B_m}$ and A are constant w.r.t. time and the differentiation of cos is – sin therefore we have,
$ \Rightarrow I = - {B_m}A\omega \sin \left( {\omega t} \right)$
Now as we have to find the maximum induced current.
So we have to maximize the $\sin \left( {\omega t} \right)$ which is only possible when $\omega t = {90^o}$ , as $\sin \left( {{{90}^o}} \right) = 1$
Therefore, ${I_{\max }} = - {B_m}A\omega $
So the induced current is highest when the direction of motion of the coil is the right angle to the magnetic field.
So this is the required answer.
Hence option (C) is the correct answer.

Note – In physics, specifically electromagnetism, the magnetic flux (often denoted by $\phi $) through a surface is the integral surface of the normal magnetic field flux density B component that passes through that surface. The magnetic flux SI unit is the Weber (Wb; in derived units, volt – seconds), and the Maxwell is the CGS unit. Magnetic flux is usually measured with a flux meter containing measuring coils and electronics which evaluate the voltage change in the measuring coils to calculate the magnetic flux measurement.