You have seen that a current loop is a magnetic dipole, much like a permanent magnet. We will now look at some important features of how current loops behave in magnetic fields. This discussion will be largely qualitative, but it will highlight some of the important properties of magnets and magnetic fields. We will use these ideas in the next section to make the connection between electromagnets and permanent magnets.

FIGURE 29.47a shows two current loops. Using what we just learned about the forces between parallel and antiparallel currents, you can see that parallel current loops exert attractive magnetic forces on each other if the currents circulate in the same direction; they repel each other when the currents circulate in opposite directions.

FIGURE 29.47 Two alternative but equivalent ways to view magnetic forces.

We can think of these forces in terms of magnetic poles. Recall that the north pole of a current loop is the side from which the magnetic field emerges, which you can determine with the right-hand rule. FIGURE 29.47b shows the north and south magnetic poles of the current loops. When the currents circulate in the same direction, a north and a south pole face each other and exert attractive forces on each other. When the currents circulate in opposite directions, the two like poles repel each other.

Here, at last, we have a real connection to the behavior of magnets that opened our discussion of magnetism—namely, that like poles repel and opposite poles attract. Now we have an explanation for this behavior, at least for electromagnets. Magnetic poles attract or repel because the moving charges in one current exert attractive or repulsive magnetic forces on the moving charges in the other current. Our tour through interacting moving charges is finally starting to show some practical results!

Now let’s consider what happens to a current loop in a magnetic field. FIGURE 29.48 shows a square current loop in a uniform magnetic field along the z-axis. As we’ve learned, the field exerts magnetic forces on the currents in each of the four sides of the loop. Their directions are given by the right-hand rule. Forces $\vec{F}{}_{\text{front}}$ and $\vec{F}{}_{\text{back}}$ are opposite to each other and cancel. Forces $\vec{F}{}_{\text{top}}$ and $\vec{F}{}_{\text{bottom}}$ also add to give no net force, but because $\vec{F}{}_{\text{top}}$ and $\vec{F}{}_{\text{bottom}}$ don’t act along the same line they will rotate the loop by exerting a torque on it.

FIGURE 29.48 A uniform magnetic field exerts a torque on a current loop.

Recall that torque is the magnitude of the force F multiplied by the moment arm d, the distance between the pivot point and the line of action. Both forces have the same moment arm $d=\frac{1}{2}l\text{ sin }\theta$, hence the torque on the loop—a torque exerted by the magnetic field—is

where $m=I{l}^2=IA$ is the loop’s magnetic dipole moment and $\theta$ is the angle between the dipole moment vector $\vec{m}$ and the magnetic field $\vec{B}$.

Although we derived Equation 29.28 for a square loop, the result is valid for a current loop of any shape. Notice that Equation 29.28 looks like another example of a cross product. We earlier defined the magnetic dipole moment vector $\vec{m}$ to be a vector perpendicular to the current loop in a direction given by the right-hand rule. Figure 29.48 shows that $\theta$ is the angle between $\vec{B}$ and $\vec{m}$, hence the torque on a magnetic dipole is

The torque is zero when the magnetic dipole moment $\vec{m}$ is aligned parallel or antiparallel to the magnetic field, and is maximum when $\vec{m}$ is perpendicular to the field. It is this magnetic torque that causes a compass needle—a magnetic moment—to rotate until it is aligned with the magnetic field.