Question

The magnetic flux linked in a coil, in webers, is given by the equations $\phi = 3{t^2} + 4t + 9$. Then the magnitude of induced e.m.f. at $t = 2$ second will be:
(A) 2 volt
(B) 4 volt
(C) 8 volt
(D) 16 volt

Answer 1

The magnetic flux and the induced e.m.f. of a coil are related by the equation, $E = N\dfrac{{d\phi }}{{dt}}$. So from the given equation of magnetic flux, we can find the induced e.m.f. by differentiating it with respect to time. By substituting the time as $t = 2$ seconds, we find the e.m.f. induced at that time.
Formula used: In the solution, we will be using the following formula,
$\Rightarrow E = N\dfrac{{d\phi }}{{dt}}$
where $E$ is the induced e.m.f
$N$ is the number of turns of the coil and
$\phi$ is the magnetic flux.

Complete step by step answer:
Whenever there is a magnetic flux linked with a coil an e.m.f. is induced in that coil. This e.m.f. is given by the formula
$\Rightarrow E = N\dfrac{{d\phi }}{{dt}}$
In this question, the calculation is given for a single coil. So we take the value of $N$ as 1. Therefore, the equation becomes,
$\Rightarrow E = \dfrac{{d\phi }}{{dt}}$
Now, the magnetic flux linked with the coil is a function of time, that is it changes with time. The value of this magnetic flux is given by the equation,
$\Rightarrow \phi = 3{t^2} + 4t + 9$
On differentiating this with respect to time we get,
$\Rightarrow \dfrac{{d\phi }}{{dt}} = \dfrac{d}{{dt}}\left( {3{t^2} + 4t + 9} \right)$
This is equal to,
$\Rightarrow \dfrac{{d\phi }}{{dt}} = 6t + 4$
So this is the induced e.m.f. Now the time in the question is given $t = 2$.
Therefor on substituting this value of the time we get,
$\Rightarrow E = 6 \times 2 + 4$
On doing the calculation we get,
$\Rightarrow E = 16$ volt.
So the correct answer is option (D); 16 volt.

Note:
The induced emf is equal to the rate of change of the magnetic flux. This equation is given by the laws of electromagnetic induction or the Faraday’s laws. This law is a fundamental relationship that serves as a summary of the ways that voltage can be created by changing the magnetic environment.