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# 7.5: Magnetic Moment of a Plane, Current-carrying Coil

• • Contributed by Jeremy Tatum
• Emeritus Professor (Physics & Astronomy) at University of Victoria

A plane, current carrying coil also experiences a torque in an external magnetic field, and its behaviour in a magnetic field is quite similar to that of a bar magnet or a compass needle. The torque is maximum when the normal to the coil is perpendicular to the 3 magnetic field, and the magnetic moment is defined in exactly the same way, namely the maximum torque experienced in unit magnetic field.

Let us examine the behavior of a rectangular coil of sides a and b, which is free to rotate about the dashed line shown in Figure $$\text{VII.2}$$. $$\text{FIGURE VII.2}$$

In Figure $$\text{VII.3}$$, I am looking down the axis represented by the dashed line in Figure $$\text{VII.2}$$, and we see the coil sideways on. A current $$I$$ is flowing around the coil in the directions indicated by the symbols $$\bigodot$$ and $$\bigotimes$$. The normal to the coil makes an angle $$\theta$$ with respect to an external field $$B$$. $$\text{FIGURE VII.3}$$

According to the Biot-Savart law there is a force $$F$$ on each of the b-length arms of magnitude $$bIB$$, or, if there are $$N$$ turns in the coil, $$F = NbIB$$. These two forces are opposite in direction and constitute a couple. The perpendicular distance between the two forces is $$a \sin θ$$, so the torque $$\tau$$ on the coil is $$NabIB \sin θ$$, or $$\tau = NAIB \sin θ$$, where $$A$$ is the area of the coil. This has its greatest value when $$\theta = 90^\circ$$ , and so the magnetic moment of the coil is $$NIA$$. This shows that, in SI units, magnetic moment can equally well be expressed in units of $$\text{A m}^2$$ , or ampere metre squared, which is dimensionally entirely equivalent to $$\text{N m T}^{−1}$$. Thus we have

$\tau = p_mB\sin \theta,$

where, for a plane current-carrying coil, the magnetic moment is

$p_m = NIA.$

This can conveniently be written in vector form:

$\tau=\textbf{p}_m \times \textbf{B},$

where, for a plane current-carrying coil,

$\textbf{p}_m = NI\textbf{A}.$

Here $$\textbf{A}$$ is a vector normal to the plane of the coil, with the current flowing clockwise around it. The vector $$\tau$$ is directed into the plane of the paper in Figure $$\text{VII.3}$$

The formula $$NIA$$ for the magnetic moment of a plane current-carrying coil is not restricted to rectangular coils, but holds equally for plane coils of any shape; for (see Figure $$\text{VII.4}$$) any curve can be described in terms of an infinite number of infinitesimally small vertical and horizontal segments. $$\text{FIGURE VII.4}$$

We understand that a magnet, or anything that has a magnetic moment, experiences no net force in a uniform magnetic field, although it does experience a torque. Furthermore, as in the case of an electric dipole in an electric field, a magnetic dipole situated in an inhomogeneous magnetic field does experience a net force. If the magnetic moment and the gradient of the magnetic field are in the same direction, the net force on the dipole is

$\nonumber p_m\frac{dB}{dx}.$

[$$\text{N m T}^{-1} \times \text{T m}^{-1} = \text{N}.$$]

See Section 3.5 for further details relating to a dipole in an inhomogeneous field.

An important historical experiment that readers may come across, using the force on a magnetic dipole in an inhomogeneous magnetic field, is the 1922 experiment of Stern and Gerlach, demonstrating the spatial quantization of the magnetic moment associated with electron spin.