Electromagnetic Induction

JEE Main level, numerical answer, medium

18 of 153

Question 18 of 153

A conducting circular loop is placed in X−YX-Y plane in presence of magnetic field B⃗=(3t3j^+3t2k^)\vec{B}=\left(3{t}^{3}\hat{j}+3{t}^{2}\hat{k}\right) in SI unit. If the radius of the loop is 1m1m, the induced emf in the loop, at time, t=2st=2s is nπVn\pi V. The value of nn is _____ .

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Answer: 12

By the faraday's law in electromagnetic induction
ϵind=−dϕdt=−d(B⃗⋅A⃗)dt{\epsilon }_{ind}=-\dfrac{d\phi }{dt}=-\dfrac{d\left(\vec{B}\cdot \vec{A}\right)}{dt}
Magnetic flux is ϕ=B⃗⋅A⃗\phi =\vec{B}\cdot \vec{A}=(3t3j^+3t2k^)⋅(π(1)2k^)=\left(3{t}^{3}\hat{j}+3{t}^{2}\hat{k}\right)\cdot \left(\pi {\left(1\right)}^{2}\hat{k}\right)
So, ϕ=3t2π\phi =3{t}^{2}\pi
Now, induced emf is ϵind=∣dϕdt∣=6tπ{\epsilon }_{ind}=\left|\dfrac{d\phi }{dt}\right|=6t\pi
At t=2,ϵind=6(2)π=12πVt=2,{\epsilon }_{ind}=6\left(2\right)\pi =12\pi V.
Thus, the value of n=12n=12.

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