Magnetic Dipole Moment

Section 10.7 derived the torque on a current-carrying coil in the compact form $\tau=(NIA)B\sin\theta$. This particular combination of quantities, $NIA$, recurs so often, and plays such a central organising role, that it is given its own name and its own vector notation: the MAGNETIC DIPOLE MOMENT of the coil, usually written $\vec{P}$ (though $\vec{m}$ is also a very common alternative notation). Its magnitude is defined as

where $N$ is the number of turns of the coil, $I$ the current passing through it, and $A$ the area enclosed by each turn. Its DIRECTION is defined to be along $\hat{n}$, the unit vector normal (perpendicular) to the plane of the coil -- with the specific SENSE of $\hat{n}$ (out of the two possible perpendicular directions) fixed by the same right-hand curl rule used throughout this chapter: curl the right-hand fingers in the sense the current actually circulates around the loop, and the outstretched thumb then gives the direction of both $\hat{n}$ and $\vec{P}$.

With this single vector $\vec{P}$ defined, the torque expression of Section 10.7 collapses into an extremely compact vector form:

where the magnitude of this cross product, $PB\sin\theta$, exactly reproduces the earlier scalar formula (since $P=NIA$), and $\theta$ is now understood simply as the angle between the two vectors $\vec{P}$ and $\vec{B}$. This is worth comparing directly with a result already met in Class XI, for an ELECTRIC dipole of moment $\vec{p}$ in an external electric field $\vec{E}$: $\vec{\tau}=\vec{p}\times\vec{E}$. The two expressions are structurally identical -- a dipole moment vector crossed with the corresponding field vector gives the torque -- and this parallel between electric and magnetic dipoles is not a coincidence; it runs throughout this chapter (it reappears, for instance, in the potential-energy formula of the very next section, and again in the far-field, "magnetic dipole," behaviour of a current loop discussed in Section 10.14). …