Magnet-to-magnet force, repulsion and stiffness

Why the catalogue number does not apply

Every catalogue pull-force figure, including ours, is measured the same way: the magnet is pulled directly off a thick, flat, ground low-carbon steel plate, in contact, along the magnetization axis. That is a useful, repeatable benchmark. It is also a specific test that most designs do not reproduce.

Put a second magnet in place of the steel plate and three things change at once.

There is a clean way to relate the two cases. An infinitely permeable steel plate acts as a mirror: a magnet at gap d from ideal steel behaves as though an identical magnet sat at distance 2d, oriented to attract. That gives a result worth carrying around:

At contact the two cases converge, which is why a magnet on steel and two magnets stuck together feel similar. At any real gap they diverge quickly, and always in the same direction: two magnets attract each other more strongly at a given gap than one magnet is pulled to steel across that same gap. If you have sized a gapped design from a catalogue pull figure, you have underestimated a magnet pair, sometimes by a lot.

The far-field formula

Once the separation is comfortably larger than the magnets themselves, each magnet looks like a point dipole and the force between two coaxial, axially magnetized magnets has a simple closed form.

The exponent is the part to internalise. Force between two dipoles falls off as the fourth power of separation. Double the gap and the force drops to one sixteenth. That is far more aggressive than most intuitions allow for, and it is why designs that work on the bench at 5 mm fail at 10 mm.

A worked pair

Two N42 discs, 10 mm diameter × 5 mm thick, B_r = 1.30 T, coaxial and aligned.

Ten millimetres of extra gap took 94% of the force away. When somebody says a magnetic latch "barely works", the gap has usually grown by a couple of millimetres somewhere in the tolerance stack.

Why contact estimates from remanence are wrong

Close in, the dipole model fails and people reach for the Maxwell stress expression instead, usually in this form:

The formula is correct. The trouble is what gets substituted for B. Remanence is the flux density the material holds in a closed magnetic circuit — a magnet whose flux has an uninterrupted path back around through iron. An open magnet in air is nowhere near that state, and the flux density at its pole face is substantially lower.

Take the 10 × 5 mm N42 disc from the previous section. Substituting B_r = 1.30 T over the 78.5 mm² pole face gives roughly 52 N. The measured pull of that part against a steel plate is closer to 15–20 N. The formula overstates it by around a factor of three, and it does so consistently.

The usable version of this is to find the working point first — the intersection of the load line with the demagnetization curve — and use that B. The permeance coefficient calculator does that, and Magnets 201 covers the derivation. For a first pass, a stocked part's published pull figure is a better starting point than any hand calculation, precisely because it was measured rather than derived.

Repulsion, and the coercivity it costs

For two identical magnets, the magnitude of the repulsive force at a given separation equals the magnitude of the attractive force at that separation. Flip one magnet and the sign changes, not the size. So a repulsion design can be sized with the same arithmetic.

What differs is what the field is doing to the magnets themselves. In attraction, each magnet sits in a field that reinforces its own magnetization. In repulsion, each sits in a field that opposes it — and an opposing external field moves the operating point down and to the left along the demagnetization curve, exactly the direction that risks irreversible loss.

What makes it worse

The risk concentrates where you would least like it to. Force is highest when the magnets are closest, which is also when the opposing field is strongest — so the demagnetizing condition peaks at the same moment the design is working hardest. Short, wide magnets are worst affected, because they already sit low on the curve before any external field arrives. And on NdFeB the knee moves upward as temperature rises, so a stack that is comfortable at 20 °C can cross the knee at 80 °C and lose output permanently.

Stiffness, and why it matters more than force

For anything that moves — a suspension, a coupling, a compliant mount — the useful quantity is not force but its rate of change with position. That is stiffness, and it sets the resonant frequency and the stability of the whole arrangement.

The relation k = 4F/z is worth remembering. It says stiffness rises much faster than force as the gap closes, so a magnetic suspension that feels soft at a wide gap becomes abruptly harsh as it approaches — and any system with a nonlinear spring of this shape will have a resonance that shifts with load.

Earnshaw's theorem, and what it forbids

You cannot stably levitate an object using static permanent magnets alone. This is not an engineering limitation to be designed around; it follows from the field equations. In a region free of the sources themselves the magnetic potential satisfies Laplace's equation, so it has no local minimum — and a stable equilibrium requires one.

The practical statement is that the stiffnesses in the three axes must sum to zero. Achieve positive stiffness vertically and you have necessarily created negative stiffness in at least one lateral direction. Every magnetic suspension that works does one of the following:

If a proposal claims passive static levitation with permanent magnets only, look for which of these six it is actually using. Magnets for magnetic levitation covers the workable architectures in more detail.

Stacks, arrays and what does not add up

Multiple magnets do not combine the way a parts list suggests, and the errors run in both directions depending on how they are arranged.

Stacking in the magnetization direction

Two identical magnets stacked pole to tail behave much like one magnet of double the length. Pull force against steel rises, but well short of double, because what has changed is the permeance coefficient rather than the pole area. A tall stack approaches a limit and further magnets add very little. As a rough guide, two stacked gives perhaps 1.3–1.5× the single-magnet pull, and beyond four the return is marginal.

Placing magnets side by side

Same polarity facing the same way, side by side, is a repulsion arrangement between neighbours. Each magnet's flux is pushed away from its neighbours, the working point falls, and total force is less than the sum of the parts. Spacing them out recovers some of it, at the cost of area.

Alternating polarity

Reversing every other magnet gives each one a short return path through its neighbour. Flux is concentrated close to the surface and falls away rapidly with distance. Excellent for a thin-gap holding application, poor for reaching across a gap.

That last row is usually the highest-value change available. Adding a steel cup or backing plate is cheaper than adding magnet material and generally does more — backing plates, pot cups and flux return works through the sizing. The Halbach arrangement is the specialised case where one-sidedness is the objective.

Working practice

A short sequence that avoids most of the trouble described above.

On step six: the gap between calculated and measured force in a real assembly is routinely 30% or more, almost always in the unfavourable direction, and the causes are mundane — surface flatness, a coating thicker than nominal, a fixture that is not quite square, a steel part thinner than the flux wants. None of those appear in any formula. Build the assembly and pull it.