Magnetic Dipole Moment

Definition

A current carrying coil can be described by a vector $\vec{\mu}$ (also written $\vec{m}$), its magnetic dipole moment. If $\hat{n}$ is the unit vector normal to the plane of the coil, the direction of $\vec{\mu}$ is that of $\hat{n}$, fixed by the right hand thumb rule: curl the fingers of the right hand along the current in the loop and the stretched thumb gives the direction of $\vec{\mu}$.

Its magnitude is

$$\mu = NIA$$

where $N$ is the number of turns of the coil, $I$ the current through it, and $A$ the area enclosed by each turn. SI unit: A m$^2$, equivalently J/T. For a single loop, $\vec{\mu} = I\vec{A}$, where $\vec{A}$ has magnitude $A$ and direction normal to the plane of the loop. For a circular coil of radius $r$, $A = \pi r^2$.

Diagram: draw the flat coil in the plane of the paper with the current circulating anticlockwise; the arrow $\vec{\mu}$ is drawn out of the plane of the coil through its centre, perpendicular to it.

Torque in terms of the dipole moment

If the coil is held in a uniform magnetic field $\vec{B}$, the torque responsible for its rotation is $\tau = NIAB\sin\theta$. Writing $NIA = \mu$,

$$\tau = \mu B \sin\theta$$

with $\theta$ the angle between $\vec{\mu}$ (that is, $\hat{n}$) and $\vec{B}$. In vector form,

$$\vec{\tau} = \vec{\mu}\times\vec{B}$$

or, in the notation the board uses for this question, $\vec{\tau} = \vec{m}\times\vec{B}$, where $\vec{m}$ is the magnetic dipole moment of the rotating current carrying coil and $\vec{B}$ is the magnetic field.

This is exactly analogous to the torque on an electric dipole of moment $\vec{p}$ in an electric field $\vec{E}$, learnt in Std XI:

$$\vec{\tau} = \vec{p}\times\vec{E}$$

10.9 Magnetic potential energy of a dipole

A magnetic dipole freely suspended in a magnetic field possesses magnetic potential energy because of its orientation in the field. The electric analogue is the potential energy of an electric dipole $\vec{p}$ in a field $\vec{E}$:

$$U = -\vec{p}\cdot\vec{E}$$

Analogously, the magnetic potential energy of a magnetic dipole $\vec{\mu}$ in a magnetic field $\vec{B}$ is

$$U = -\vec{\mu}\cdot\vec{B} = -\mu B\cos\theta$$

where $\theta$ is the angle between $\vec{\mu}$ and $\vec{B}$.

Case (i): $\theta = 0^\circ$ gives $U = -\mu B\cos(0^\circ) = -\mu B$. This is the minimum potential energy of a magnetic dipole in a magnetic field, reached when $\vec{\mu}$ and $\vec{B}$ are parallel. It is the stable orientation.

Case (ii): $\theta = 180^\circ$ gives $U = -\mu B\cos(180^\circ) = +\mu B$. This is the maximum potential energy, reached when $\vec{\mu}$ and $\vec{B}$ are antiparallel.

Diagram (Fig. 10.14): two sketches side by side, both showing a uniform field $\vec{B}$ drawn as parallel upward arrows. In case (i) the dipole arrow $\vec{\mu}$ points along $\vec{B}$ and is labelled minimum energy $-\mu B$; in case (ii) it points opposite to $\vec{B}$ and is labelled maximum energy $+\mu B$.

Standard results

• $\mu = m = NIA$; unit A m$^2$ or J/T; direction along $\hat{n}$ by the right hand thumb rule
• $\vec{m} = I\vec{A}$ for a single loop; $A = \pi r^2$ for a circular coil of radius $r$
• $\tau = mB\sin\theta$, $\vec{\tau} = \vec{m}\times\vec{B}$
• $U = -\vec{m}\cdot\vec{B} = -mB\cos\theta$
• $U_{\min} = -mB$ at $\theta = 0^\circ$ (stable); $U_{\max} = +mB$ at $\theta = 180^\circ$
• Electrostatic analogues: $\vec{\tau} = \vec{p}\times\vec{E}$ and $U = -\vec{p}\cdot\vec{E}$
• Worked type (Example 10.4): a coil of 500 turns, area $1.26\times10^{-4}$ m$^2$, carrying 100 $\mu$A has $\mu = 500\times100\times10^{-6}\times1.26\times10^{-4} = 6.3\times10^{-6}$ A m$^2$.