In a uniform magnetic field of induction B a wire in the form of semicircle of radius r rotates about the diameter of the circle with angular frequency ω. The axis of rotation is perpendicular to the field. If the total resistance of the circuit is R the mean power generated per period of rotation is

$fraction numerator Bπr squared space straight omega over denominator 2 straight R end fraction$
$fraction numerator left parenthesis Bπr squared straight omega right parenthesis squared over denominator 2 straight R end fraction$
$fraction numerator left parenthesis Bπrω squared right parenthesis over denominator 2 straight R end fraction$
$fraction numerator left parenthesis Bπrω squared right parenthesis squared over denominator 8 straight R end fraction$

Solution

fraction numerator left parenthesis Bπr squared straight omega right parenthesis squared over denominator 2 straight R end fraction

Magnetic space flux space equals space BA space cos space straight theta space equals space straight B. πr squared over 2 space cos space ωt therefore space straight P space equals straight epsilon squared subscript ind over straight R space equals space fraction numerator straight B squared straight pi squared straight r to the power of 4 straight omega squared space sin squared space ωt over denominator 4 straight R end fraction Now comma space less than sin squared space ωt greater than space equals space 1 divided by 2 space left parenthesis mean space value right parenthesis therefore space less than straight P greater than space equals space fraction numerator left parenthesis Bπr squared straight omega right parenthesis squared over denominator 8 straight R end fraction

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