The rotor in a certain electric motor is a flat, rectangular coil with $80$ turns of wire and dimensions $2\mathrm{.}5$ cm by $4\mathrm{.}0$ cm$.$ The rotor rotates in a uniform magnetic field of $0\mathrm{.}800$ T$.$ When the plane of the rotor is perpendicular to the direction of the magnetic field, the rotor carries a current of $10\mathrm{.}0$ mA$.$ In this orientation, the magnetic moment of the rotor is directed opposite the magnetic field. The rotor then turns through one$-$half revolution. This process is repeated to cause the rotor to turn steadily at an angular speed of $3\mathrm{.}60103$rev$/$min$3\mathrm{.}60\backslash$times $10^3\backslash ,\backslash$text$\{$rev$/$min$\}3\mathrm{.}60103$rev$/$min$.$

$($a$)$ Find the maximum torque acting on the rotor.

$($b$)$ Find the peak power output of the motor.

$($c$)$ Determine the amount of work performed by the magnetic field on the rotor in every full revolution.

$($d$)$ What is the average power of the motor?