Magnetic Dipole Moment

A compass needle in the Thar desert near Jaisalmer spins in its brass housing, settles down, and points north. It has done this for a thousand years, with no external source of energy except the Earth's own magnetic field — a gentle, invisible field roughly 50 microteslas strong, fifty times weaker than a fridge magnet. The needle settles because there is a torque on it that keeps shoving it toward alignment with the field, and once it is aligned, the torque drops to zero. A tiny oscillation remains, damped by friction, and the needle is still.

Inside an AIIMS scanner, something similar — but on a different scale — is happening to four quintillion hydrogen nuclei per cubic millimetre of a patient's brain. Each proton has its own magnetic moment, sixteen orders of magnitude smaller than the compass needle's. In the scanner's 3 T field, each proton is nudged toward alignment with the field. Most stay aligned; a tiny fraction — about ten parts per million at body temperature — are aligned rather than anti-aligned. That ten-parts-per-million excess is what produces the entire MRI image. The whole medical imaging business rests on the quantitative theory of the magnetic dipole moment.

From a current loop to the dipole moment

Start with the simplest system: a single rectangular loop of wire, sides a and b, carrying current I, placed in a uniform external magnetic field \vec{B}. Put the loop flat on a table for now; \vec{B} will come in horizontally. You want to find out what the external field does to the loop.

The force on each of the four sides is given by the current-conductor law from chapter 185: \vec{F} = I\vec{L}\times\vec{B}, where \vec{L} points along the current direction with magnitude equal to the side's length.

• Top side (length a): current along +\hat{x}, so \vec{F}_\text{top} = Ia\hat{x}\times\vec{B}.
• Bottom side (length a): current along -\hat{x}, so \vec{F}_\text{bot} = -Ia\hat{x}\times\vec{B}.
• Left side (length b): current along -\hat{y}, so \vec{F}_\text{left} = -Ib\hat{y}\times\vec{B}.
• Right side (length b): current along +\hat{y}, so \vec{F}_\text{right} = Ib\hat{y}\times\vec{B}.

Top and bottom cancel; left and right cancel. The net force on the loop is zero. This is the first key fact: in a uniform external field, a current loop feels no translational force — it cannot be pushed as a whole; it can only be rotated.

The torque — where the action is

Even though the net force is zero, the lines of action of the four forces are different, so they create a torque. Here is the clean derivation.

Set up the geometry. Place the loop in the xy-plane with its centre at the origin, sides a along \hat{x} and b along \hat{y}. Let the external field be \vec{B} = B\hat{x} — pointing along the x-axis, in the plane of the loop. The current flows counter-clockwise when viewed from the +z side (right-hand rule: fingers along current, thumb along +\hat{z}).

Step 1. Force on the top side. The current in the top goes along +\hat{x}; the field is also along +\hat{x}. The cross product \hat{x}\times\hat{x} = 0.

Similarly \vec{F}_\text{bot} = 0. Why: the current in the top and bottom sides is parallel (or anti-parallel) to \vec{B}, so those sides feel no force at all.

Step 2. Force on the right side. Current along +\hat{y}; field along +\hat{x}.

Why: use the cyclic rule \hat{x}\times\hat{y} = \hat{z}, so \hat{y}\times\hat{x} = -\hat{z}. The force on the right side is -IbB\hat{z} — into the page (in our top-view convention).

Step 3. Force on the left side. Current along -\hat{y}; field along +\hat{x}.

Why: flipping the sign of the current flips the force. Left side points out of the page.

Step 4. The two forces are equal in magnitude (IbB), opposite in direction, and separated by a lever arm of a — the spacing between the left and right sides. That is the definition of a couple: zero net force, nonzero net torque.

Step 5. Compute the torque about the centre of the loop. The right side is at position \vec{r}_\text{right} = (a/2)\hat{x} with force \vec{F}_\text{right} = -IbB\hat{z}. The left side is at \vec{r}_\text{left} = -(a/2)\hat{x} with force \vec{F}_\text{left} = +IbB\hat{z}.

Why: \hat{x}\times\hat{z} = -\hat{y} (going against the cyclic order x\to y\to z). The two contributions add, not cancel — this is the signature of a torque couple.

Step 6. Notice that ab = A, the area of the loop. So

The direction \hat{y} is perpendicular to \vec{B} and to the loop's normal \hat{n} = \hat{z}. This torque rotates the loop about the y-axis — turning it so that \hat{n} swings toward \vec{B}.

Defining the dipole moment and writing the torque as \vec{m}\times\vec{B}

Define

where \hat{n} is the normal to the loop (right-hand rule: fingers curl along I, thumb points along \hat{n}). For our setup, \vec{m} = IA\hat{z}.

Rewrite the torque from step 6:

That is:

Why: we pulled the expression IAB\hat{y} apart and recognised it as a cross product using \hat{z}\times\hat{x} = \hat{y}. This repackages the torque in a form that makes the physics jump out — the torque is the dipole moment crossed with the external field.

For a loop with N turns carrying the same current, each turn contributes its own moment IA\hat{n}, all adding in the same direction:

And the torque

where \theta is the angle between \vec{m} and \vec{B}. Although we derived this for a rectangle, the result holds for any planar loop of any shape — only the enclosed area A matters.

A quick sanity check: at \theta = 0 (moment aligned with field), the torque is zero. At \theta = 90^\circ (moment perpendicular to field), the torque is maximum. At \theta = 180^\circ (anti-aligned), the torque is again zero — but unstably so, as you will see in a moment.

Potential energy — U = -\vec{m}\cdot\vec{B}

The torque \vec{\tau} = \vec{m}\times\vec{B} is conservative: the work done by the torque depends only on the initial and final orientations, not on the path. So there is a potential energy U(\theta).

Step 1. The work done by the torque when the loop rotates by a small angle d\theta (in the direction of decreasing \theta, since the torque pulls the moment toward the field) is

Why: \tau = mB\sin\theta is the magnitude of the torque at angle \theta; the negative sign is because rotating in the direction that increases \theta does negative work (the torque pulls the other way). When you rotate from alignment (\theta = 0) toward \theta, you do positive work against the torque.

Step 2. The change in potential energy is minus the work done by the torque:

Step 3. Integrate from some reference orientation \theta_0 to the current \theta:

Step 4. Choose the standard convention: set U = 0 at \theta_0 = 90^\circ (where \cos\theta_0 = 0). Then

Step 5. Recognise that mB\cos\theta = \vec{m}\cdot\vec{B} by the definition of the dot product. So

Why: the dot product bundles the magnitudes and the angle into a single scalar. The minus sign encodes the physical fact that alignment is the low-energy configuration — aligned moments "want to be there."

Reading the energy landscape

The function U(\theta) = -mB\cos\theta has a simple shape on [0, 180^\circ].

• \theta = 0: U = -mB. Stable equilibrium. \vec{m}\parallel\vec{B}. Small disturbances produce a restoring torque that pulls \vec{m} back to alignment.
• \theta = 90^\circ: U = 0. The reference level. Maximum torque here; but this is not an equilibrium because any tilt sends the moment further away.
• \theta = 180^\circ: U = +mB. Unstable equilibrium. \vec{m} anti-parallel to \vec{B}. Zero torque (since \sin 180^\circ = 0), but any tiny disturbance grows — the moment flips all the way around to alignment.

This energy landscape is what compass needles minimise, what proton spins in an MRI scanner settle toward (partially, against thermal agitation), and what makes ferromagnetic domains align with an applied field.

Interactive — explore the torque and energy as you rotate the dipole

Notice the interplay: where the torque is largest (\theta = 90^\circ), the energy is steepest (the slope dU/d\theta = mB\sin\theta is maximum). Where the torque is zero, the energy is flat (an extremum — either a minimum or maximum). The two curves tell the same story from two angles.

Worked examples

The bar magnet — a dipole from atoms all the way up

You know a bar magnet from the physics lab — a rectangular block of magnetised steel with a north end and a south end. The surprising claim of the dipole theory is that a bar magnet, seen from sufficiently far away, is indistinguishable from a small current loop of the appropriate area and current. You cannot tell, from the external field, whether you are looking at a loop or a magnet.

Why? Because the bar magnet's magnetisation is the sum of microscopic current loops — each iron atom has unpaired electron spins whose intrinsic magnetic moments add up. All the interior loops cancel (adjacent atoms' moments point the same way, so their currents cancel on shared boundaries). What survives is an effective current running around the outside surface of the magnet — exactly what a macroscopic single-turn solenoid would look like. Seen from outside, this surface current is the source of the bar magnet's field.

The equivalence. A bar magnet of length L and cross-section A, with pole strength p (defined so that the magnetic moment m = pL — a legacy convention), is equivalent to a current loop of the same area A with current I and N turns satisfying

At large distances, both produce the same dipole field:

These are the exact analogues of the electric dipole field with \vec{m}\leftrightarrow\vec{p} and \mu_0/(4\pi)\leftrightarrow 1/(4\pi\varepsilon_0). The 1/r^3 fall-off is the hallmark of every dipole field — electric or magnetic — at large distances. Closer in (within a few magnet-lengths), the field deviates from the ideal dipole form and you need the full Biot-Savart integral, but that is a finer correction.

Common confusions

• "A magnetic dipole is a pair of north and south poles." Historically yes, physically no. There are no isolated "magnetic monopoles" in nature — no one has ever found a north pole without a south pole attached. The "pole" picture is a bookkeeping device that matches the current-loop picture at large distances but fails at small ones. The dipole moment vector \vec{m} is the right modern concept; it is what nature provides.

• "The torque on a loop is zero in a non-uniform field." No — the torque formula \vec{\tau} = \vec{m}\times\vec{B} holds just as well in a non-uniform field (evaluated at the loop's location). What changes in a non-uniform field is that the net force is no longer zero. A current loop in a field that varies from one side to the other gets pushed: \vec{F} = \nabla(\vec{m}\cdot\vec{B}). That is why a fridge magnet sticks to the fridge (the magnet's field has a gradient at the steel surface, so each iron domain in the fridge feels a non-zero force).

• "Aligned is the same as anti-aligned — both are equilibria." Both are equilibria, yes, but only one is stable. At \theta = 0 (aligned), any small tilt produces a restoring torque pulling back to alignment — like a pendulum hanging down. At \theta = 180^\circ (anti-aligned), any small tilt produces a torque that increases the tilt, sending the moment into a full flip — like a pendulum balanced upside down. Only the aligned state is where a real dipole settles.

• "A bar magnet has magnetic charge at its poles." It has effective magnetic charge for bookkeeping — so-called "pole strength" p = m/L. But there is no real charge there; the ends of the magnet simply represent the location where the internal currents can no longer cancel. Try cutting a bar magnet in half — each half becomes a complete dipole with its own north and south. You cannot isolate a pole.

• "The dipole moment is just a number, the current times the area." It is a vector: magnitude NIA, direction perpendicular to the loop (right-hand rule). The vector nature is essential — it is what allows \vec{\tau} = \vec{m}\times\vec{B} and U = -\vec{m}\cdot\vec{B} to be meaningful. A scalar "moment" would tell you nothing about direction and could not describe the torque.

• "Non-planar loops don't have a dipole moment." They do, as long as the loop is small compared to the distances at which you are observing. For a non-planar current distribution, define the dipole moment vector by \vec{m} = \tfrac{1}{2}\oint\vec{r}\times I\,d\vec{l} — a line integral over the current path. This reduces to NIA for planar loops but is defined for any closed-current distribution. See the going-deeper section for the derivation.

Where this leads next

• Force on a Current-Carrying Conductor — the microscopic origin of the torque on a current loop in a field.
• Force on a Moving Charge — The Lorentz Force — the deeper single-charge law from which dipole torques are built.
• Magnetic Field of Common Current Configurations — the dipole field of a single loop, derived from the Biot-Savart law.
• Electric Dipole — the electrical analogue: \vec{\tau} = \vec{p}\times\vec{E}, U = -\vec{p}\cdot\vec{E}, with the same structure.
• The Cyclotron — the circulating current in a cyclotron is itself a macroscopic current loop, with its own dipole moment.