A conducting circular loop is placed in X - Y plane in presence of magnetic field B → = 3 t 3 j ^ + 3 t 2 k ^ in SI unit. If the radius of the loop is 1 m , the induced emf in the loop, at time, t = 2

Question

A conducting circular loop is placed in X - Y plane in presence of magnetic field B → = 3 t 3 j ^ + 3 t 2 k ^ in SI unit. If the radius of the loop is 1 m , the induced emf in the loop, at time, t = 2 s is n π V . The value of n is _____ .

TestHub · Accepted Answer

By the faraday's law in electromagnetic induction
$ε_{i n d} = - \frac{d ϕ}{d t} = - \frac{d (\vec{B} \cdot \vec{A})}{d t}$
Magnetic flux is $ϕ = \vec{B} \cdot \vec{A}$ $= (3 t^{3} \hat{j} + 3 t^{2} \hat{k}) \cdot (π {(1)}^{2} \hat{k})$
So, $ϕ = 3 t^{2} π$
Now, induced emf is $ε_{i n d} = |\frac{d ϕ}{d t}| = 6 t π$
At $t = 2, ε_{i n d} = 6 (2) π = 12 π V$ .
Thus, the value of $n = 12$ .

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