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A wire shpaed as a semi circle of radius a rotates about axis OO' with an angular velocity ω in a uniform magnetic field of induction B. The rotation axis is perpendicular to the field direction. The total resistance of the circuit is equal to R. Neglecting the magnetic field of the induced current, find the mean amount of thermal power being generated in the loop during a rotation period.


A wire shpaed as a semi circle of radius a rotates about axis OO' with an angular velocity ω in a uniform magnetic field of induction B. The rotation axis is perpendicular to the field direction. The total resistance of the circuit is equal to R. Neglecting the magnetic field of the induced current, find the mean amount of thermal power being generated in the loop during a rotation period.


Grade:upto college level

1 Answers

ROSHAN MUJEEB
askIITians Faculty 829 Points
2 years ago
Flux ofB→B→, at an orbitarty moment of timett:
Φt=B→.S→=Bπa22cosωtΦt=B→.S→=Bπa22cosωt,
From Faraday's law, inducede.m.f.,ξIn=dΦdte.m.f.,ξIn=dΦdt
=−d(Bπa22cosωt)dt=Bπa2ω2sinωt=-d(Bπa22cosωt)dt=Bπa2ω2sinωt.
and induced current,iin=ξinR=Bπa22Rωsinωtiin=ξinR=Bπa22Rωsinωt.
Now thermal power, generated in the circuit, at the momentt=tt=t:
p(t)=ξin×iin=(Bπa2ω2)21Rsin2ωtp(t)=ξin×iin=(Bπa2ω2)21Rsin2ωt
and mean thermal power generated,
<P>=[Bπa2ω2]21R∫T0sin2ωtdt∫T0dt=12π(πa2B2)2

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