The magnetic flux through a coil of resistance 4 varies with time according to the equation = 12t - 2t^2 , where is in webers and t is in seconds. The total heat dissipated in the coil from t = 0 until the induced current becomes zero is:

Step-by-step solution

The induced emf in the coil is given by Faraday's law: = | d dt | = d dt (12t - 2t^2) = 12 - 4t The induced current is: I = R = 12 - 4t 4 = 3 - t The current becomes zero when 3 - t = 0 , which gives t = 3 s . The instantaneous power dissipated as heat is: P = I^2 R = (3 - t)^2 4 The total heat dissipated from t = 0 to t = 3 s is the integral of power over time: H = ₀³ P , dt = ₀³ 4(9 - 6t + t^2) , dt H = 4 [ 9t - 3t^2 + t^3 3 ]₀³ H = 4 ( 27 - 27 + 9 ) = 36 J Answer: 36 J