What is the correct definition of magnetic moment? I found myself tired of by searching what is the exact definition of magnetic moment.
I have also come across two formulas for same quantity
$\vec{M} = \vec{m} \times \vec{d}$
where $\vec{M}$ is the magnetic moment, $\vec{m}$ is the magnetic pole strength and $\vec{d}$ is the distance between two poles
Another formula
$\vec{M} = \vec{i}\times\vec{A}$
Where $\vec{M}$ is magnetic moment, $\vec{i}$ is electric current and $\vec{A}$ is area vector
So please suggest me which formula is accurate one and give detailed description of this.
 A: Both are correct formulae for ${\boldsymbol \mu}$, although it is hard to find magnetic monopoles so the first is of little use.  Interestingly the two formulae  give different formulae for the force on a magnetic dipole in an inhomogeneous field. The monopole-pair formula gives
$$
{\bf F}_{\rm monopole-pair}=({\boldsymbol \mu} \cdot \nabla) {\bf B} 
$$
while the current loop gives
$$ 
{\bf F}_{\rm current-loop}=  \nabla ({\boldsymbol \mu}\cdot {\bf B}).
$$
These are different in a region with a non-zero current, so in principle one can determine whether a dipole is of the monopole-pair or the current-loop variety. I believe that experiments on neutrons suggest that their magnetic dipole moment obeys  the current-loop force equation.
The usual and practical definition of ${\boldsymbol \mu}$ comes from the torque on the dipole being
$$
{\boldsymbol \tau}= {\boldsymbol \mu}\times {\bf B}.
$$
For a general current-distribution one has
$$
{\boldsymbol \mu}= \frac 12 \int d^3x \left({\bf r}\times {\bf J}({\bf r})\right).
 $$
If the current is in a planar loop this last equation reduces to $|{\boldsymbol \mu}|= I 
\times {\rm Area}.$ (Here the "$\times$" is just  multiplication of numbers; in the previous formulae it was the cross-product of vectors.) For a long solenoid one can aproximate the integral by the  length of the solenoid by a pole strength, as in the monopole formula, but this is only a approximation as there are no actual poles.
