What if a magnet were farther away?

Hold a magnet near a paperclip. Zip! The clip jumps to the magnet.

Now move the clip a little farther away. Will the magnet still pull it in?

Bring a magnet toward a paperclip and, at first, nothing happens. Then, suddenly, the clip leaps across the last little gap. Why does a magnet seem to do nothing, then everything?

This lab answers that with a model of a 2.5-centimeter neodymium magnet and a paperclip lying on a table. The clip starts 5 centimeters from the middle of the magnet. Then we move it to 10 centimeters, twice as far, and see what happens to the field and to the pull.

How quickly does a magnet's influence fade with distance? Far from a small magnet its field is a dipole field, which OpenStax derives along the axis of a current loop: B = μ₀μ/(2πr³) for distances much larger than the loop.

This lab uses the magnet model, a point dipole of moment m = 16 A·m² (a 2.5 cm neodymium block) pulling a 0.5 g steel clip across a table with friction coefficient 0.4. It compares the clip at r = 5 cm and r = 10 cm, and asks why the pull falls so much faster than the field.

The clip is close to the strong magnet.

The magnet pulls hard, harder than the table holds the clip. Zip! In it goes.

In the model, at 5 centimeters the magnetic field at the clip is 25.6 millitesla, about 512 times the 50 microtesla the model uses for Earth's own magnetic field at the ground.

The magnet pulls on the clip with about 6 millinewtons, roughly three times the grip of the table, which is about 2 millinewtons. The pull wins, so the clip slides in and snaps onto the magnet.

At r = 5 cm the model gives B = 25.6 mT (512 times its 50 µT reference for Earth's field) and a pull of 5.96 mN. Table friction on the 0.5 g clip is μ m g = 0.4 × 0.0005 kg × 9.81 m/s² = 1.96 mN, so the pull is 304% of the grip and the clip moves.

The model's reach, where pull equals grip, is 5.9 cm. Check it: at 5.8 cm the pull is 108% of the grip and the clip slides in; at 5.9 cm it is 95.4% and the clip stays put.

Now the clip is farther away. The magnet tries to pull.

But the pull is too weak. The table holds the clip still.

Now move the clip to 10 centimeters, twice as far. The field there is 3.20 millitesla, one eighth of what it was. You might expect the pull to drop to one eighth too. It drops much more: to about 0.05 millinewtons, 128 times weaker. That is only about 2 percent of the table's grip, so the clip does not move.

The field is still 64 times the model's value for Earth's field, yet it cannot budge the clip. Even the model's imaginary super magnet, about three times stronger, manages only about a quarter of the grip at 10 centimeters.

At r = 10 cm: B = 3.20 mT (×1/8) and F = 0.0466 mN (×1/128), 2.37% of the 1.96 mN grip, so the clip stays put. Doubling r divides B by 2³ = 8 and F by 2⁷ = 128.

Strength helps only slowly. With the imaginary 50 A·m² magnet, the pull at 10 cm is 23.2% of the grip; the reach grows only from 5.9 to 8.1 cm, because r* ∝ m^(2/7). Lowering the table grip to μ = 0.1 extends the reach to 7.1 cm but still leaves the clip at 10 cm at 9.50% of the grip.

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

The clip is not a magnet by itself. The magnet has to make it a little magnet first. Far away, it can't do much.

Two things fade at once. First, the magnet's field gets much weaker with distance: twice as far, one eighth as strong.

Second, an ordinary steel paperclip is not a magnet on its own. The magnet's field turns it into a weak magnet, and the weaker the field, the weaker that magnet. On top of that, the clip is pulled by how quickly the field changes from one spot to the next. Put all of this together and the pull falls off far faster than the field does.

OpenStax shows that far from a current loop the axial field reduces to the dipole form B = μ₀μ/(2πr³). In the model the clip, treated as a small soft-iron sphere, gains a moment proportional to B, and the force on that induced dipole is proportional to B·dB/dr. Since B ∝ r⁻³ and dB/dr ∝ r⁻⁴, F ∝ r⁻⁷.

The model's limits: the dipole formula is poor within a few magnet-lengths, so near contact real fields and pulls are smaller than modeled; a real long, thin clip magnetizes more easily than a sphere, so real reaches can be noticeably larger, though they scale the same way; and the field is only computed along the magnet's axis.

Earth is a giant magnet. Its pull is gentle, but a compass can feel it.

Hospitals have very strong magnets for taking pictures inside the body. People check that nothing made of iron goes near them.

Earth itself is a magnet. NOAA says its field at the surface is roughly 25,000 to 65,000 nanotesla, far weaker than the field near a small magnet. A compass needle lines up with the field where it sits.

MRI scanners in hospitals use powerful magnets. The U.S. Food and Drug Administration explains that their strong field attracts magnetic objects, from keys to much bigger things, so people and objects are carefully screened before they go into the scanner room.

NOAA's geomagnetism FAQ gives Earth's field intensity as roughly 25,000 to 65,000 nT, and notes that a compass aligns with the horizontal component of the local field, not with a single point. The model's 50 µT reference sits inside that range.

MRI shows distance and strength at hospital scale. The NIH's imaging institute (NIBIB) notes that the scanner's field extends beyond the machine and exerts very powerful forces on iron and other magnetizable objects; the FDA says careful screening is critical so nothing enters the magnet area that could become a projectile.