What if magnets became stronger?

A magnet can pull a paperclip. Zip! The paperclip jumps to it.

But move the paperclip away, and the magnet stops pulling it. What if the magnet were stronger?

Hold a magnet near a paperclip and the clip slides over and snaps onto it. Move the clip a little farther away and nothing happens at all. The magnet is still pulling, but not hard enough.

So what if magnets became much stronger? Would a magnet three times stronger pull from three times as far away? In this lab you can make the magnet stronger and measure how far it really reaches.

Magnets attract iron and steel through their magnetic field, which weakens rapidly with distance. A paperclip on a table only moves when the magnetic pull beats the table's grip.

This lab asks how a magnet's reach, the farthest distance from which it can pull a clip in, depends on its strength (its magnetic dipole moment, in A·m²). Intuition suggests reach should grow in step with strength. The model shows otherwise.

Olly has an ordinary magnet. The paperclip is a little way from the magnet.

The magnet pulls a little. But the table holds the clip still. It does not move.

In the model, Olly uses a 2.5-centimeter ceramic magnet. A paperclip lies 5 centimeters from its middle. At that distance, the magnetic field is still about 160 times stronger than Earth's own field.

But the pull on the clip is only about a third as strong as the table's grip, so the clip stays put. The model says this magnet can only pull clips in from about 4 centimeters away.

The ceramic block is modeled as a point dipole with m = 5 A·m². At r = 5 cm on its axis, B = (μ₀/4π)(2m/r³) ≈ 8 mT, about 160 times the 50 µT taken for Earth's field.

The pull on the 0.5 g steel clip is about 0.58 mN, while the table's grip (μ = 0.4) is μ·m·g ≈ 1.96 mN. The pull is only 30% of the grip, so the clip does not move. The model's reach for this magnet is 4.2 cm.

Now Olly uses a magnet three times stronger. The clip is in the same place.

Zip! This time the clip slides right over to the magnet.

Now switch to a neodymium magnet of the same size, about three times stronger. At the same 5 centimeters, the pull is now about three times the table's grip, so the clip slides over and sticks.

But look at the reach: it grows only from about 4 to about 6 centimeters. Even an imaginary super magnet, ten times stronger than the ceramic one, reaches only about 8 centimeters. Try moving the clip farther away and see where each magnet gives up.

With m = 16 A·m², B at 5 cm rises to 25.6 mT, and because the pull scales as m², it jumps about tenfold to 5.96 mN: about three times the grip. The clip moves.

The reach, however, follows r* ∝ m^(2/7): tripling the moment (×3.2) raises it by only 3.2^(2/7) ≈ 1.39, from 4.2 to 5.9 cm. A tenfold increase (the imaginary 50 A·m² magnet) gives 10^(2/7) ≈ 1.93, a reach of 8.1 cm.

A magnet's pull gets weak very fast as you move away.

So a much stronger magnet still only reaches a little bit farther.

A magnet's field fades astonishingly fast. Move twice as far away and the field is only one eighth as strong. And a paperclip is pulled by how quickly the field changes, so its pull fades even faster: twice as far means less than one hundredth of the pull.

With a pull that drops off that steeply, making the magnet stronger only pushes the reach out a little. Our model is a simplified picture: it treats the clip as a tiny ball of steel right in line with the magnet.

On a dipole's axis B ∝ m/r³. A soft-iron object gains an induced moment proportional to B and feels F ∝ B·|dB/dr| ∝ m²/r⁷. Setting F equal to the friction force gives r* ∝ m^(2/7).

The model's stated limits matter: the dipole formula is poor within a few magnet lengths, so near contact the true pull is smaller than modeled; a real, long, thin paperclip magnetizes more easily than a sphere, so real reaches can be noticeably larger, though they scale the same way. Off the axis the field is weaker, and saturation and temperature are ignored.

Earth has its own gentle magnetic field. A compass needle lines up with it.

Near a magnet, the magnet's pull is much stronger. Far away, Earth's field is all a compass feels.

Earth itself has a magnetic field, and a compass needle swings to line up with it. The model takes Earth's field as 50 millionths of a tesla, the unit scientists use for magnetic fields.

Close to a magnet, the magnet wins easily: 5 centimeters from the ceramic magnet, its field is 160 times Earth's. But that field fades so fast that about 27 centimeters away it has dropped to the same strength as Earth's. That is why even strong magnets only grab steel from close by.

The model compares fields with Earth's, taken as 50 µT (NOAA gives Earth's field strength as roughly 25 to 65 µT); NOAA's geomagnetism FAQ also explains that a compass aligns with the horizontal component of that field wherever it is. In the model, the 5 A·m² ceramic magnet's axial field is 8 mT (160 times Earth's) at 5 cm, but the 1/r³ fall-off brings it down to 50 µT at r = (μ₀m/(2π·B))^(1/3) ≈ 27 cm.

The clip's pull falls off faster still, as 1/r⁷, which is why the reach is only a few centimeters even when the field is far stronger than Earth's: what moves a clip is the field's gradient, not its strength alone.