In a coil of resistance $8 \Omega$, the magnetic flux due to an external magnetic field varies with time as $\phi=\frac{2}{3}\left(9-t^{2}\right)$. The value of total heat produced in the coil, till the flux becomes zero, will be ______ J.

JEE Main · 2022 · 26 Jul 2022 · Shift 2 · Electromagnetic Induction

In a coil of resistance

8Ω

, the magnetic flux due to an external magnetic field varies with time as

ϕ = \frac{2}{3} (9 - t^{2})

. The value of total heat produced in the coil, till the flux becomes zero, will be ______ J.

Answer: 2

Solution

The flux becomes zero when

\frac{2}{3} (9 - t^{2}) = 0

, giving

t = 3

s.

The induced emf is:

e = - \frac{d ϕ}{d t} = - \frac{2}{3} (- 2 t) = \frac{4 t}{3}

Heat produced:

H = \int_{0}^{3} \frac{e ^{2}}{R} d t = \int_{0}^{3} \frac{( 4 t /3 ) ^{2}}{8} d t = \int_{0}^{3} \frac{16 t ^{2}}{72} d t = \frac{2}{9} [\frac{t ^{3}}{3}]_{0}^{3} = \frac{2}{9} \times 9 = 2 J

JEE Main · 2022 · 26 Jul 2022 · Shift 2 · Integer answer

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