# 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.

• Please use MathJax and edit the equations @Anwesh Panda Commented Jul 8, 2020 at 11:00
• @Lelouch ok i will edit it Commented Jul 8, 2020 at 11:21
• That's better, but you are still not using MathJax. Commented Jul 8, 2020 at 11:27
• Hey where the option for Mathjax Commented Jul 8, 2020 at 11:28
• Just use LaTeX syntax for mathematical formula, including the \$ Commented Jul 8, 2020 at 12:04

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.