What is the magnetic dipole moment for a coil? Write its $SI$ unit and dimensional formula,and explain its stable and unstable equilibrium.

(A) The magnetic dipole moment of a current-carrying coil is defined as the product of the current flowing through the coil and its area vector.
If a current $I$ flows through a coil with area $A$,the magnetic dipole moment $\vec{m}$ is given by $\vec{m} = I\vec{A}$.
For a coil with $N$ turns,the magnetic dipole moment is $\vec{m} = NI\vec{A}$.
The $SI$ unit of magnetic dipole moment is $A \cdot m^2$ (Ampere-meter squared).
The dimensional formula is $[M^0 L^2 T^0 A^1]$.
The torque $\vec{\tau}$ acting on a magnetic dipole in an external magnetic field $\vec{B}$ is given by $\vec{\tau} = \vec{m} \times \vec{B}$,which has a magnitude $\tau = mB \sin \theta$.
Stable Equilibrium: When the magnetic dipole moment $\vec{m}$ is parallel to the magnetic field $\vec{B}$ $(\theta = 0^\circ)$,the torque is zero. This is the state of minimum potential energy $(U = -mB \cos 0^\circ = -mB)$,representing stable equilibrium.
Unstable Equilibrium: When the magnetic dipole moment $\vec{m}$ is anti-parallel to the magnetic field $\vec{B}$ $(\theta = 180^\circ)$,the torque is zero. This is the state of maximum potential energy $(U = -mB \cos 180^\circ = +mB)$,representing unstable equilibrium.