Given a current loop, the quantity IA is called the magnetic dipole moment (M) where I is the electric current flowing in the loop and A is the area enclosed by the loop. This is called the far-field approximation. Classically, magnetic flux density generated by the loop depends only on the area and not on its shape: M = IA.
On the other hand, if a magnetic dipole experiences an infinitesimal near local turning force (torque) in a non-uniform or time-varying magnetic field then the magnetic dipole moment is defined as the ratio of the torque (T) over the magnetic flux density (B): M = T/B. It is then logical that the far-field approximation is the integral of the near-field equation IA = ∫T/B. As a classical experimental constant, B can be taken outside the integral: ∫∂T = IAB. Torque is defined as the vector cross product of the radius vector r and a local infinitesimal force F: ∂T = r × F. Since r× F = - F× r, two possible distinct orientations exist such that quantized torque is made equivalent to quantized spin. Although torque is a vector, the scalar inner dot product of torque is equivalent to a square of energy.


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