Question $1(20$ points$)$

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An infinitely long current$-$carrying wire with steady current I is wrapped around an infinitely long solenoid of radius $R,$ making $n$ turns per unit length. Find the emf generated through a sphere of radius $\frac{R}{2}$ inside the solenoid that is rotated at angular frequency ${}_0.$

${}_{0}IRcos({}_0)$

$4{}_{O}n{R}^{2}cos({}_{O}t)$

$4{}_{0}nl{R}^{2}cos({}_{0}t)$

zero

$4{}_{0}nIRcos({}_{0}t)$

${}_{0}nIRcos({}_{0}t)$