Magnetism and Matter

A bar magnet behaves, from the outside, just like a current-carrying solenoid, the equivalent solenoid picture. A solenoid of $N$ turns, each carrying current $I$ and enclosing area $A$, has a magnetic dipole moment:

A bar magnet is assigned the same quantity $M$, pointing from its south pole to its north pole, precisely because it produces the same external field pattern and obeys the same torque law as that equivalent solenoid.

Placed in a uniform external field $B$ at angle $\theta$ to $M$, the two forces on the (equal and opposite) poles form a couple. Each force has magnitude $qB$ (pole strength times field), separated by a perpendicular lever arm $2a\sin\theta$, giving:

Potential energy is defined by the work needed to rotate the dipole from a reference angle (conventionally $90°$, where $U=0$) to angle $\theta$, which integrates the torque above:

With this sign convention, $U$ is lowest (most negative) when $M$ and $B$ point the same way, exactly where the simulation above reads Stable.

Both $\theta=0°$ and $\theta=180°$ give $\sin\theta=0$, so $\tau=MB\sin\theta=0$ at both. It’s tempting to read “zero torque” as “nothing is happening here, so it must just sit still”, and treat the two positions as interchangeable. They are not. Zero torque only describes this exact instant; it says nothing about what happens the moment something disturbs the angle even slightly, which, in the real world, something always does.

To tell the two apart, look at potential energy, not torque, just either side of each point. Near $\theta=0°$, $U=-MB\cos\theta$ is at a minimum, nudge $\theta$ a little either way and $U$ rises, meaning the system has to do work to move away from there. A restoring torque appears the instant you leave $\theta=0°$, and it always points back toward it. That is exactly what a stable equilibrium means.

Near $\theta=180°$, $U$ is at a maximum instead. Nudge $\theta$ slightly away from 180° and $U$ falls, the dipole can lower its energy by continuing to rotate away, not by returning. The torque that appears doesn’t restore $\theta=180°$, it accelerates the dipole away from it, all the way around to $\theta=0°$, never back to $180°$. That is an unstable equilibrium: a true zero-torque point that no real disturbance leaves alone.

This is precisely why a compass needle always settles pointing to magnetic north and never to magnetic south, even though “pointing south” is also, momentarily, a position of zero torque. The rule to remember: zero torque is necessary for equilibrium, but you must check whether $U$ sits at a minimum (stable) or a maximum (unstable) before calling it one.