A conducting circular loop is placed in $\mathrm{X}-\mathrm{Y}$ plane in presence of magnetic field $\overrightarrow{\mathrm{B}}=\left(3 \mathrm{t}^{3} \hat{\mathrm{j}}+3 \mathrm{t}^{2} \hat{\mathrm{k}}\right)$ in SI unit. If the radius of the loop is 1 m, the induced emf in the loop, at time, $\mathrm{t}=2 \mathrm{~s}$ is $n \pi$ V. The value of $n$ is ______.

JEE Main · 2022 · 27 Jul 2022 · Shift 2 · Electromagnetic Induction

A conducting circular loop is placed in

X - Y

plane in presence of magnetic field

B = (3 t^{3} \hat{j} + 3 t^{2} \hat{k})

in SI unit. If the radius of the loop is 1 m, the induced emf in the loop, at time,

t = 2 s

nπ

V. The value of

n

is ______.

Answer: 12

Solution

The loop is in the X-Y plane, so its area vector is

A = π r^{2} \hat{k} = π \hat{k}

(since

r = 1

m). The magnetic flux through the loop is

ϕ = B \cdot A = (3 t^{3} \hat{j} + 3 t^{2} \hat{k}) \cdot (π \hat{k}) = 3 t^{2} π

. The induced emf is

ε = - \frac{d ϕ}{d t} = - 6 t π

. At

t = 2

∣ ε ∣ = 12 π

V. Therefore

n = 12

JEE Main · 2022 · 27 Jul 2022 · Shift 2 · Integer answer

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