Question

The magnetic flux through a coil varies with time as $\phi = 5{t^2} + 6t + 9$ . The ratio of emf at t=3s to t=0s will be
(A) 1:9
(B) 1:6
(C) 6:1
(D) 9:1

Answer 1

Hint: The number of magnetic field lines that passes through the given magnetic field of the given closed surface is known as the Magnetic flux. It also provides the total magnetic field that passes through the given area. The magnetic flux measurement is tied to the specific chosen area.

Formula used: To solve this type of question we use the following formula.
$\varepsilon = - \dfrac{{d\phi }}{{dt}}$ ; Here, $\varepsilon$ is the induced emf, $\phi$is the flux.

Complete step by step answer:
Following is given in the question $\phi = 5{t^2} + 6t + 9$. We have to find the ratio of emf at t=3s and t=0s.
Let us find the magnitude of induced emf at t=3s.
$\left| \varepsilon \right| = \dfrac{{d\phi }}{{dt}} = \dfrac{d}{{dt}}\left( {5{t^2} + 6t + 9} \right)$
We can differentiate the above equation.
$\left| \varepsilon \right| = 10t + 6$
Therefore at t=3s, ${\left| \varepsilon \right|_{t = 3s}} = 10 \times 3 + 6 = 36$ (1)
Let us now find the emf at t=0s.
${\left| \varepsilon \right|_{t = 0s}} = 10 \times 0 + 6 = 6$ (2)
Now, let us divide equation (1) by (2).
$\dfrac{{{{\left| \varepsilon \right|}_{t = 3s}}}}{{{{\left| \varepsilon \right|}_{t = 0s}}}} = \dfrac{{36}}{6} = \dfrac{6}{1}$
Hence, option (C) 6:1 is the correct option.

Additional information:
Electromotive force (emf) is induced when flux linking with the conductor or coil changes.
Magnetic flux is magnetic field strength multiplied by the area. $\phi = \vec B.\vec A = BA\cos \theta$ .
Expression for Faraday’s motional emf is $\varepsilon = Blv\sin \theta$

How much emf will be induced depends on the following three factors:
1. The number of turns of wire in the coil can be increased.. The sum of all the individual loops of the coil will depend on the amount of induced emf produced.
2. By increasing the speed of the relative motion between the magnetic coil.
3. Increasing the strength of the magnetic field.

Note: Faraday’s law states that the induced emf is equal to rate of change of magnetic flux.
Induced emf opposes the change in current, due to source of emf.