A wire shaped as a semicircle of radius o rotates about an axis OO' with an angular velocity ω in a uniform magnetic field of induction B [Fig. (5)]. The rotationaxis is perpendicular to the field direction. The total resistance of the circuit is equal to fl. Neglect the magnetic field of induced current. The mean amount of thermal power being generated into the loop during a rotation period will be
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a
12Rπωa2B22
b
12Rπωa2B2
c
12Rπω2aB2
d
12RπωaB22
answer is A.
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Detailed Solution
Φ=B⋅A=Bπα22cosωte=−dΦdt=Bπa22ωsinωtInduced current i=eR=Bπa22RωsinωtAt any moment t, the thermal power generated in the circuitPt=e×i=Bπa2ω221Rsin2ωtMean power (P) = Bπa2ω221R∫0T sin2ωt∫0T dt=12Rπωa2B22