Magnetism and MatterThe Bar Magnet

Sprinkle iron filings on a sheet of glass placed over a short bar magnet, tap the glass, and the filings arrange themselves into a striking pattern (Fig. 5.1). Do the same around a current-carrying solenoid and compare.

Pick a source, sprinkle the filings, then tap the board.

The pattern suggests that the magnet has two poles, much like the positive and negative charges of an electric dipole. One pole is called the north pole and the other the south pole; hung freely, they point roughly to the geographic north and south. The same pattern around a solenoid is the first hint of something deep: a bar magnet is a magnetic dipole, and it behaves like a solenoid.

The Magnetic Field Lines

The iron filings let us plot the magnetic field lines, a visual, intuitive picture of the magnetic field. Another way to plot them: place a small compass needle at a point, note the way its north end points, move the compass to that mark, and repeat.

Drag the compass near the magnet, then press the red button again and again to trace a field line.

Some books call these "magnetic lines of force". We avoid that name: unlike electrostatic field lines, magnetic field lines do not show the direction of the force on a (moving) charge. That force, qv × B, is always at right angles to B.

Three Pictures, One Shape

Fig. 5.2 puts the field lines of (a) a bar magnet, (b) a current-carrying finite solenoid and (c) an electric dipole side by side. The dashed circles i and ii are closed surfaces; keep them in mind for Section 5.3.

Tap each picture.

Far from the source, the three patterns are very similarcompletely different. Close up, look inside the bar magnet or the solenoid: there B points from S to NN to S, which closes every loop. The lines are packed more tightly around surface ii than around surface i, so B is larger near iilarger near i.

So a bar magnet is a magnetic dipole: at large distances its field matches the field of a solenoid and has the same shape as the field of an electric dipole. The differences show up close to the source, and the biggest one is that magnetic field lines have no ends.

Speed Round: Field Lines

Get five right to move on.

Bar Magnet as an Equivalent Solenoid

In the last chapter you saw that a current loop acts as a magnetic dipole, and you met Ampère's hypothesis: all magnetic phenomena can be explained in terms of circulating currents. The likeness of their field lines suggests that a bar magnet is a large number of tiny circulating currents, just like the turns of a solenoid. Cutting a bar magnet in half is then like cutting a solenoid in half: you get two smaller solenoids with weaker magnetic properties, and the field lines stay continuous, leaving one face and entering the other. A small compass needle moved around a bar magnet and around a current-carrying finite solenoid is deflected in the same way.

To make the analogy firm, compare the real field on the axis of a finite solenoid, Fig. 5.3(a), with the formula for a magnetic dipole.

Move the point P along the axis. Distance of P from the centre: r = ${r} l, where 2l is the length of the solenoid.

The purple curve is the solenoid's real field on its axis; the dashed teal curve is the dipole formula below.

Far from the solenoid, the size of its field at a point P on the axis is

where m = NIA is the magnetic moment of the solenoid (N turns of area A carrying a current I). This is also the far axial field of a bar magnet, measured in the lab. So a bar magnet and a solenoid produce similar fields, and the magnetic moment of a bar magnet equals the magnetic moment of the equivalent solenoid that produces the same field.

The Dipole in a Uniform Magnetic Field

Now let a small compass needle of known magnetic moment m oscillate in a uniform magnetic field B, as in Fig. 5.3(b). The torque on the needle is

where θ is the angle between m and B. It is a restoring torque: it always turns m back towards B. Here m = 0.5 A m² and B = 0.2 T, so mB = 0.1; both graphs are drawn in units of mB.

Drag the north end of the needle round, watch both graphs, then let go.

The amber curve is the torque, mB sin θ. The green curve is the magnetic potential energy, found just like the potential energy of an electric dipole:

The zero of potential energy can be fixed where we like. Taking the constant of integration as zero puts U = 0 at θ = 90°, with the needle at right angles to the field. Then U is lowest, −mB, at θ = 0° (the most stable position) and highest, +mB, at θ = 180° (the most unstable position). The work an external torque must do to turn the needle from θ₁ to θ₂ is the rise in U:

Example 5.1

Four puzzles about magnets. (a) What happens if a bar magnet is cut into two pieces (i) across its length, (ii) along its length?

(b) A magnetised needle in a uniform field feels a torque but no net forcea net force. An iron nail near a bar magnet, though, is attracted as well as turned, because near a bar magnet the field is non-uniformuniform.

The magnet induces a magnetic moment in the nail. In a non-uniform field there is a net force, and it is attractive: the induced south pole of the nail (say) is closer to the north pole of the magnet than its induced north pole is.

(c) Must every magnetic configuration have a north pole and a south pole? What about the field of a toroid?

Poles appear only when the source has a net non-zero magnetic moment. A toroid's field lines are circles inside it, and an infinite straight wire's field lines circle the wire: neither has a north or a south pole.

(d) You are given two identical-looking iron bars A and B. One of them is definitely magnetised, but you do not know which. Using nothing but the two bars, find out which one is the magnet.

A bar magnet's field is strongest at its two ends (the poles) and weakest in the middle. Lower one end of A onto an end of B and then onto the middle of B. If A feels no force at the middle of B, B is the magnet; if the pull does not change from the end to the middle, A is the magnet. And if you had to check whether both were magnetised: bring different ends together. A repulsion in some position proves both are magnets; if the force is always attractive, only one is.

The Electrostatic Analog

Compare Eqs. (5.1), (5.2) and (5.3) with the equations for an electric dipole in Chapter 1. The field far from a bar magnet of moment m follows from the field of an electric dipole of moment p if you make the replacements

In particular, at a distance r ≫ l, where l is the size of the magnet, the field on its equatorial line (the normal bisector) and on its axis are

Equation (5.5) is just Eq. (5.1) in vector form.

This short magnet has m = 0.5 A m². Drag the compass along the axis or the equatorial line.

The axial curve sits everywhere twice as high as the equatorial curve, and both fall off as 1/r³. On the axis B points along m; on the equatorial line it points opposite to m. Table 5.1 sums up the analogy.

Table 5.1 — the dipole analogy

Example 5.2

Fig. 5.4 shows a small magnetised needle P at a point O; its arrow gives the direction of its magnetic moment. The other needles show different positions (and directions of the moment) of another, identical needle Q. The potential energy of each pair is the energy of one needle, Q, in the field of the other, P. Which pairs are not in equilibrium? Which are in stable and which in unstable equilibrium? Which has the lowest potential energy?

Tap each needle Q to see the field of P where it sits.

Speed Round: The Bar Magnet

Torque, energy, work and field. Get five right to move on.