In a coil of resistance 16Ω, the magnetic flux due to an external magnetic field varies with time as ϕ=23(9−t2). The value of total heat produced in the coil, till the flux becomes zero, will be ______J.

The magnetic flux given by ϕ=23(9−t2)When the flux becomes zero then ϕ=23(9−t2)=0t=3 secThe emf will be e=−dϕdt=−23(0−2t)=4t3Heat produced in 3sec=∫0te2Rdt=∫0316t29×16dt=1J

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In a coil of resistance $16 Ω,$ 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.

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Detailed Solution

The magnetic flux given by $ϕ = \frac{2}{3} (9 - t^{2})$

When the flux becomes zero then $ϕ = \frac{2}{3} (9 - t^{2}) = 0$

$t = 3 \sec$

The emf will be $e = \frac{- d ϕ}{d t} = - \frac{2}{3} (0 - 2 t) = \frac{4 t}{3}$

Heat produced in $3 \sec = \int_{0}^{t} \frac{e^{2}}{R} d t = \int_{0}^{3} \frac{16 t^{2}}{9 \times 16} d t = 1 J$

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