Why a Magnet Grabs Iron Nails but Ignores Aluminium Foil

A magnet grabs iron nails, barely notices aluminium foil, and very slightly pushes water away. These differences come from the tiny magnetic moments of electrons inside atoms and how they line up.

This part covers the bar magnet, the magnetic dipole's field and torque, magnetisation and Gauss's law for magnetism, and magnetic materials.

A bar magnet's field lines leave its north pole, curve round outside, enter its south pole and continue through the magnet as closed loops; this pattern matches that of a current-carrying solenoid, because both behave like collections of tiny current loops.

Properties of magnetic field lines:

- They form continuous closed loops, unlike electric field lines
- Outside the magnet they run from N to S; inside, from S to N
- They never intersect, and their crowding shows field strength

Bar magnet as a solenoid. A current-carrying solenoid produces field lines of exactly the same shape as a bar magnet's, with one end acting as a north pole and the other as a south pole. Cutting either in two gives two smaller magnets, each with both poles.

Magnetic moment. A bar magnet's moment points from S to N; a solenoid of $N$ turns and area $A$ carrying current $I$ has $m=NIA$.

Worked example. A solenoid of $200$ turns and area $5\times 10^{-4}$ m${}^2$ carries $2$ A:

$$m=NIA=200\times 2\times 5\times 10^{-4}=0.2{\text{ A m}}^2$$

An everyday example. Iron filings sprinkled on paper over a bar magnet trace out its looping field lines, just as they would around a solenoid.

The substance. Isolated magnetic poles do not exist — cutting a magnet always creates a new N and S pole at the cut.

**Far from a magnetic dipole of moment $m$, the field on its axis is $\frac{{\mu }_0}{4\pi }\frac{2m}{r^3}$ and on its perpendicular bisector is $\frac{{\mu }_0}{4\pi }\frac{m}{r^3}$, and in a uniform field $B$ it feels a torque $\tau =mB\sin \theta$ that turns it into line with the field.

Fields, with the same form as for the electric dipole:

- Axial line**: $B=\frac{{\mu }_0}{4\pi }\frac{2m}{r^3}$, along $\vec{m}$
- Equatorial line: $B=\frac{{\mu }_0}{4\pi }\frac{m}{r^3}$, opposite to $\vec{m}$

Worked example. For $m=0.2$ A m${}^2$ at $r=0.1$ m on the axis, using $\frac{{\mu }_0}{4\pi }=10^{-7}$ T m A${}^{-1}$:

$$B=10^{-7}\times \frac{2\times 0.2}{(0.1{)}^3}=4\times 10^{-5}\text{ T}$$

On the equatorial line at the same distance, $B=2\times 10^{-5}$ T.

Torque and energy:

$$\vec{\tau }=\vec{m}\times \vec{B},U=-mB\cos \theta$$

An everyday example. A compass needle settles along the north-south line because Earth's magnetic field exerts a torque on the needle's magnetic moment.

The substance. Electric and magnetic dipole results have exactly the same form — replace $\frac{1}{4\pi {\epsilon }_0}$ by $\frac{{\mu }_0}{4\pi }$ and $p$ by $m$.

**Magnetisation $M$ is the net magnetic moment per unit volume, magnetic intensity $H$ is the part of the field due to external currents, they combine as $B={\mu }_0(H+M)$ with $M=\chi H$, and Gauss's law for magnetism says the net magnetic flux through any closed surface is zero.

Definitions:

- Magnetisation** — $M=\frac{m_{net}}{V}$, in A m${}^{-1}$
- Magnetic intensity — $H=\frac{B_0}{{\mu }_0}$, such as $H=nI$ inside a solenoid
- Susceptibility — $\chi =\frac{M}{H}$, a pure number

Worked example. A solenoid with $1000$ turns per metre carries $1$ A and is filled with a material of $\chi =499$:

$$H=nI=1000{\text{ A m}}^{-1},M=\chi H=4.99\times 10^5{\text{ A m}}^{-1}$$

$$B={\mu }_0(1+\chi )H=4\pi \times 10^{-7}\times 500\times 1000\approx 0.63\text{ T}$$

Gauss's law for magnetism:

$$\oint \vec{B}\cdot d\vec{S}=0$$

Every field line entering a closed surface also leaves it, because there are no magnetic monopoles.

An everyday example. The iron core of a transformer has a very large susceptibility, so a modest current in its coil produces a strong magnetic field.

The substance. The electric version of Gauss's law depends on enclosed charge, but the magnetic version always gives zero, because magnetic poles come in pairs.

Diamagnetic substances are weakly repelled by a magnet and have small negative susceptibility, paramagnetic substances are weakly attracted with small positive susceptibility, and ferromagnetic substances are strongly attracted with very large susceptibility; paramagnetism weakens with rising temperature, and a ferromagnet turns paramagnetic above its Curie temperature.

Diamagnetic:

- Weakly repelled, moving from stronger to weaker field; $\chi$ small and negative
- Examples: bismuth, copper, lead, water and nitrogen gas

Paramagnetic:

- Weakly attracted, moving towards stronger field; $\chi$ small and positive
- Examples: aluminium, sodium, calcium and oxygen gas
- Curie's law: $\chi =\frac{C}{T}$, so susceptibility falls as temperature rises

Ferromagnetic:

- Strongly attracted, because atoms form domains that align easily; $\chi$ very large
- Examples: iron, cobalt, nickel and gadolinium
- Above the Curie temperature, a ferromagnet becomes paramagnetic

Worked example — Curie's law. A paramagnetic salt has $\chi =3\times 10^{-4}$ at $300$ K. At $200$ K:

$$\chi =3\times 10^{-4}\times \frac{300}{200}=4.5\times 10^{-4}$$

An everyday example. A fridge magnet sticks to a steel door but not to an aluminium kadhai, because steel is ferromagnetic while aluminium is only weakly paramagnetic.

The substance. Heating a permanent magnet strongly can destroy its magnetism, as thermal motion scrambles its aligned domains.