A rectangular coil has $N=16$ turns and sides of length $a=20cm$ and $c=50cm.$ The coil is pivoted about the $z$ axis and its plane is at $30$ to a magnetic field $B=0\mathrm{.}5iT,$ as shown in Fig. $2.$ Take $I=10A.(30$ points$)$

$($a$)$ What is the magnetic dipole moment of the coil?

$($b$)$ What is the torque on the coil?

$($c$)$ What is the change in potential energy when the coil's plane is rotated from the original position to the perpendicular position of the magnetic field due to the torque?