Bar magnets and magnetism of matter

A magnet that cannot be cut in two

Cut a bar magnet in half. You do not get a north pole alone and a south pole alone. You get two smaller magnets, each with both poles. Cut again, and the same thing happens. Nobody has found an isolated north pole.

Why? A bar magnet is a bundle of tiny current loops. We saw in the first lesson that each loop is a dipole. A dipole always has two ends.

The field lines of a bar magnet

Sprinkle iron filings round a bar magnet and they show the lines of the field.

The rules of the lines are these.

• The tangent to a line gives the direction of $\vec{B}$ at that point.
• Where lines crowd together, the field is strong. Near the ends, it is strong.
• Outside the magnet, the lines go from the north pole to the south pole. Inside, they go from south to north. Every line is a closed loop.
• Two lines never cross, because the field at a point has one direction.

The closed lines tell us something important. Draw any closed surface, and count the lines that go in and the lines that come out. The numbers are equal. The net flux is zero:

This is Gauss's law for magnetism. It says there are no magnetic monopoles. For electric charges the flux is $q/\varepsilon_0$, which is not zero, because a charge can be alone.

A magnet is a solenoid

Look at the lines of a short solenoid, and then at those of a bar magnet. They look the same. A bar magnet behaves as a solenoid with a magnetic moment $m$ pointing from its south pole to its north pole. For a short magnet, at a distance $r$ from its centre that is large compared with its length,

• on the axis: $B = \dfrac{\mu_0}{4\pi}\dfrac{2m}{r^3}$,
• on the equatorial line: $B = \dfrac{\mu_0}{4\pi}\dfrac{m}{r^3}$, in the direction opposite to $\vec{m}$.

The axial field is twice the equatorial field at the same distance. Here $\dfrac{\mu_0}{4\pi} = 10^{-7}\ \text{T m/A}$. Both fields fall as $1/r^3$, which is fast.

Find the distance where the field on the axis falls to the strength of the Earth's field. Beyond it, a compass needle ignores the magnet.

A magnet in a field

A bar magnet is a dipole, so everything in the first lesson applies. In a uniform field $\vec{B}$ it feels the torque and has the energy

A compass needle is a small magnet that is free to turn. The torque turns it until $\vec{m}$ points along the field.

Push the needle a little and let go. For a small angle $\theta$, $\sin\theta \approx \theta$, so $\tau = -mB\theta$. The torque is in proportion to the angle and opposes it. That is simple harmonic motion. With $J$ as the moment of inertia of the magnet about its middle, the period is

A strong field or a strong magnet makes the swings quicker. Comparing two periods is a good way to compare two fields.

The magnetism of the Earth

The Earth behaves as if a huge bar magnet sat near its centre, with a moment of about $8 \times 10^{22}\ \text{A m}^2$.

The north end of a compass needle points to the geographic north. Unlike poles attract. So the magnetic pole of the Earth near the geographic north is a magnetic south pole.

Three numbers describe the field at a place.

• Declination $D$. The angle between the geographic meridian and the magnetic meridian. It is the error of a compass in finding true north.
• Angle of dip $\delta$. The angle that the whole field makes with the horizontal.
• Horizontal component $B_H$. The part of the field along the ground. A compass needle feels only this part.

The picture below is a vertical slice. The total field $B_E$ has a horizontal part and a vertical part:

At the magnetic equator the dip is $0^\circ$ and the needle lies flat. At a magnetic pole the dip is $90^\circ$ and the needle stands upright. The field of the Earth at the surface is about $3 \times 10^{-5}\ \text{T}$ to $6 \times 10^{-5}\ \text{T}$.

Magnetisation and magnetic intensity

Put a material in a magnetic field. Its atoms act as tiny dipoles, and they respond. The material gets a magnetisation $M$, the net magnetic moment in each unit of volume. Its unit is A/m.

We split the field inside the material into two parts. The first part comes from the currents that we control, such as those in a solenoid. We describe it by the magnetic intensity $H$, also in A/m. For a long solenoid, $H = nI$. The second part comes from the material. The total is

For most materials, $M$ is in direct proportion to $H$:

The susceptibility $\chi$ has no unit. The relative permeability is $\mu_r = 1 + \chi$.

A worked example. A solenoid has $n = 500$ turns in each metre and carries $0.60$ A. An iron rod with $\mu_r = 250$ fills it.

1. The intensity. $H = nI = 500\ \text{m}^{-1} \times 0.60\ \text{A} = 300\ \text{A/m}$.
2. The field. $B = \mu_0\mu_rH = 4\pi \times 10^{-7}\ \text{T m/A} \times 250 \times 300\ \text{A/m} = 0.094\ \text{T}$.
3. Without the rod. $B = \mu_0H = 3.8 \times 10^{-4}\ \text{T}$. The rod multiplies the field by $\mu_r$.

Three kinds of material

• Diamagnetic. Atoms have no net moment of their own. A field induces tiny moments that point against it. So $\chi$ is small and negative, about $-10^{-5}$, and $\mu_r$ is a little below $1$. The material is weakly pushed away from a strong field. Examples: bismuth, copper, gold, silicon, water. Every material has this effect. A superconductor is a perfect case, with $\chi = -1$: it pushes the field out of itself completely.
• Paramagnetic. Atoms have permanent moments, pointing in random directions. A field lines some of them up. So $\chi$ is small and positive, from about $10^{-5}$ to $10^{-3}$. The material is weakly drawn into a strong field. Examples: aluminium, sodium, calcium, oxygen. Heat shakes the moments out of line, so $\chi = \dfrac{C}{T}$. This is Curie's law. $C$ is the Curie constant, and $T$ is the temperature in kelvin.
• Ferromagnetic. Strong forces between neighbouring atoms line their moments up in small regions called domains, even with no field. A field makes the domains that point along it grow, and turns the others. So $\chi$ is large and positive, hundreds or thousands. Examples: iron, cobalt, nickel. Above the Curie temperature, 1043 K for iron, the order breaks and the metal turns paramagnetic.

Hysteresis

Magnetise a piece of iron, and then reduce the field to zero. The iron stays magnetised. Reverse the field and bring it back, and the graph of $B$ against $H$ makes a closed loop.

• Retentivity. The value of $B$ left when $H$ falls to zero.
• Coercivity. The reverse $H$ needed to bring $B$ to zero.
• The area of the loop is the energy turned into heat in each cycle for each unit of volume.

The loop decides the use. Soft iron has a narrow loop: it magnetises and demagnetises easily, and it wastes little energy. It is used for electromagnets and the cores of transformers. Steel and the alloy alnico have a wide loop. They stay magnetised and resist change. They make permanent magnets.

Practise with new numbers

Try it in Python

The program finds the horizontal part of the Earth's field at a place. It then finds how fast three compass needles swing when pushed a little.

Common mistakes

• Saying a magnet can have one pole. Every line is closed, and every magnet has two poles.
• Mixing up axis and equator. The field on the axis is twice that on the equatorial line.
• Forgetting that the geographic north is a magnetic south pole. That is why the north end of the needle points there.
• Using the sine for the horizontal part. It is $B_H = B_E\cos\delta$.
• Giving a diamagnetic material a positive $\chi$. It is small and negative.
• Using Celsius in Curie's law. Use kelvin.
• Using a hard material for an electromagnet. It needs a narrow loop, soft iron.

Remember these

• Field lines are closed loops, and $\oint \vec{B}\cdot d\vec{A} = 0$.
• A short magnet: axis $\dfrac{\mu_0}{4\pi}\dfrac{2m}{r^3}$; equatorial line $\dfrac{\mu_0}{4\pi}\dfrac{m}{r^3}$.
• In a field: $\tau = mB\sin\theta$, $U = -mB\cos\theta$, $T = 2\pi\sqrt{J/(mB)}$.
• The Earth: $B_H = B_E\cos\delta$, $B_V = B_E\sin\delta$, $\tan\delta = B_V/B_H$.
• $B = \mu_0(H + M)$, $M = \chi H$, $\mu_r = 1 + \chi$.
• Dia: $\chi$ small, negative. Para: $\chi$ small, positive, $\chi = C/T$. Ferro: $\chi$ large, positive, domains, hysteresis.

Check yourself

Try each one on your own first. If you are stuck, open a hint. Then check with Show the answer.

1. A bar magnet is cut in two across its middle. What do you get, and what happens to the magnetic moment of each piece?

2. On the equatorial line of a short magnet, $0.60$ m from its centre, the field is $2.4\ \mu\text{T}$. What is the field at the same distance on the axis?

3. A compass needle is placed at the magnetic equator. What is the angle of dip, and what is the vertical part of the field there?

4. A substance is drawn weakly into a strong magnet, and its susceptibility gets smaller when it is heated. What kind of substance is it?

5. Why is soft iron used for the core of an electromagnet, but steel for a compass needle?