A rectangular coil of N turns, area A , and resistance R rotates with an angular velocity ω in a uniform magnetic field B . The axis of rotation is perpendicular to the magnetic field. What is the ave

Question

A rectangular coil of N turns, area A , and resistance R rotates with an angular velocity ω in a uniform magnetic field B . The axis of rotation is perpendicular to the magnetic field. What is the average power dissipated in the coil over one full cycle?

TestHub · Accepted Answer

The magnetic flux through the coil at time $t$ is $Φ = N A B cos (ω t)$ . The induced EMF is $E = - \frac{d Φ}{d t} = N A B ω sin (ω t)$ . The instantaneous power dissipated is $P (t) = \frac{E ^{2}}{R} = \frac{( N A B ω ) ^{2} s i n ^{2} ( ω t )}{R}$ .

The average power over one cycle is $⟨ P ⟩ = \frac{( N A B ω ) ^{2}}{R} ⟨ sin^{2} (ω t)⟩$ . Since $⟨ sin^{2} (ω t)⟩ = 1/2$ over a full cycle, the average power is $\frac{( N A B ω ) ^{2}}{2 R}$ .

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