Magnetic Fields

A magnetic field is produced by moving charges

A magnetic field is a region in which a magnetic pole, a current-carrying conductor or a moving charge experiences a force, so it is an example of a field of force. It is produced either by moving charges or by permanent magnets, and these are not two separate effects. The magnetism of a permanent magnet comes from the motion of electrons within its atoms, so every magnetic field has moving charge as its origin.

A magnetic field is represented by field lines. The direction of a field line at a point is the direction in which the north pole of a small compass points when the compass is placed there, and the spacing of the lines shows the strength of the field: lines close together mean a strong field, lines far apart mean a weak one. Field lines never cross, because a compass needle placed at a point can only point in one direction.

Where a field is uniform the lines are straight, parallel and equally spaced, so the force on a current is the same everywhere in the region. The field between the poles of a laboratory magnet is nearly uniform over a small region. The Earth's field is nearly uniform but very weak, with a flux density of only about 5.0 x 10-5 T in the United Kingdom, which is why a compass needle is easily pushed aside by the much stronger field of a nearby bar magnet or a current-carrying coil.

• A long straight wire: concentric circles around the wire, in planes at right angles to it.
• A flat circular coil: the field passes through the centre at right angles to the plane of the coil.
• A long solenoid: a uniform field inside, and a field outside like that of a bar magnet.
• A permanent magnet: outside the magnet the lines run from the north pole to the south pole, and inside the material they continue from south to north.

Force on a current-carrying conductor, and the definition of flux density

A wire carrying a current in a magnetic field experiences a force, because each of the moving charges in the wire is pushed sideways by the field. Where the wire is at right angles to a uniform field the force is at right angles to both the wire and the field, and its magnitude is given by F = BIL sin theta. Here B is the magnetic flux density, I is the current, L is the length of the wire that lies in the field, and theta is the angle between the wire and the direction of the field. When theta is 90 degrees, sin theta is 1 and F = BIL; when the wire lies along the field, sin theta is 0 and there is no force at all.

Magnetic flux density B is defined as the force acting per unit current per unit length on a wire placed at right angles to the magnetic field. Rearranging F = BIL gives B = F/(IL), so the unit of flux density is the newton per ampere per metre, N A-1 m-1, which has its own name: the tesla (T). One tesla is also one weber per square metre, so flux density is the flux passing through each square metre at right angles to the field.

The direction of the force is found with Fleming's left-hand rule. Hold the thumb, the first finger and the second finger of the left hand mutually at right angles; point the first finger in the direction of the field and the second finger in the direction of the conventional current, from positive to negative; the thumb then gives the direction of the force, which is also the direction in which the conductor tends to move. This is the principle of every electric motor.

Force on a moving charge: circular motion and the velocity selector

A single charge Q moving with speed v in a magnetic field experiences a force F = BQv sin theta, where theta is the angle between the direction of motion and the field. The force is a maximum when the charge moves at right angles to the field, and it is zero when the charge moves along the field, because a charge at rest or moving parallel to the field feels no magnetic force at all. The direction is given by Fleming's left-hand rule with the second finger along the direction of conventional current, so for an electron, which carries a negative charge, the force is opposite to the direction the rule first suggests.

When the velocity is at right angles to a uniform field, the force is always at right angles to the velocity. A force that is always perpendicular to the velocity changes the direction of motion but not the speed, so it acts as the centripetal force and the charge travels in a circle at constant speed. Putting BQv equal to mv^2/r gives the radius r = mv/(BQ): a faster or heavier particle travels in a larger circle, and a stronger field or a larger charge gives a smaller circle.

This is how a mass spectrometer separates ions of different mass. Ions with the same charge but different mass enter the field with the same speed and follow circular paths of different radius, so they arrive at different points on the detector. The same physics explained why the electron, far lighter than any ion, is deflected so strongly in a magnetic field.

A velocity selector uses an electric field and a magnetic field at right angles to each other and to the direction of the beam. The electric force EQ and the magnetic force BQv act in opposite directions on a charged particle, and when BQv = EQ the resultant force is zero, so the particle passes through undeflected. This rearranges to v = E/B, so only particles with this single speed get through: particles that are slower or faster are deflected out of the beam. A velocity selector is placed before the magnetic field of a mass spectrometer so that every ion entering the field has the same speed.

The Hall effect and the Hall probe

In a slice of conductor or semiconductor carrying a current in a magnetic field, the moving charge carriers are pushed sideways by the magnetic force. Charge therefore builds up on the two opposite faces of the slice until the electric field it creates pushes the carriers back with a force equal to the magnetic force. The potential difference across the slice at that equilibrium is called the Hall voltage.

The equilibrium condition is EQ = BQv, which gives E = Bv. If the slice has thickness t measured along the direction of the magnetic field and is measured across the Hall voltage, then E = VH/d and so VH = Bvd. The current is I = nAvq, where n is the number density of charge carriers and the cross-sectional area is A = td, so v = I/(ntdq). Substituting this into VH = Bvd gives VH = BI/(ntq), where t is the thickness of the slice.

A Hall probe uses this effect to measure magnetic flux density. A steady current is passed through a slice of semiconductor and the Hall voltage is measured with a voltmeter; because VH is proportional to B, the probe can be calibrated so that flux density is read off directly. A semiconductor is used rather than a metal because it has a much smaller number density of charge carriers, and a small n makes the Hall voltage much larger and therefore much easier to measure.

Hall probes are used to measure the flux density inside a solenoid or near a magnet, as non-contact switches in keyboards and door sensors, and as wheel-speed sensors in a car's anti-lock braking system.

• VH = BI/(ntq), where t is the thickness of the slice measured along the magnetic field.
• The Hall voltage is proportional to the flux density B, which is what makes a Hall probe a flux-density meter.
• A small number density of charge carriers gives a large Hall voltage, so Hall probes are made from semiconductors.
• The sign of the Hall voltage shows whether the charge carriers are positive or negative.

Magnetic fields due to currents, and the forces between currents

Every current produces a magnetic field, and you must be able to sketch the pattern for three arrangements. For a long straight wire the field lines are concentric circles in planes at right angles to the wire, centred on the wire, and the spacing of the circles increases with distance from the wire, showing that the field is strongest close to it. The direction is given by the right-hand grip rule: point the thumb of the right hand along the conventional current and the fingers curl in the direction of the field lines.

For a flat circular coil, the field lines pass through the middle of the coil at right angles to its plane and are closest together at the centre, where the field is strongest, so the pattern is like the field of a short bar magnet. For a long solenoid, the field inside is uniform, with straight, parallel and equally spaced lines along the axis, while the field outside is like that of a bar magnet, with one end behaving as a north pole and the other as a south pole.

Inserting a ferrous core into a solenoid increases the magnetic field inside it many times over. The iron is magnetised by the field of the coil, and its aligned magnetic domains produce an extra field in the same direction, so the two fields add. This is why electromagnets, transformers and the field coils of motors contain iron.

Two long parallel wires carrying currents exert forces on each other. Each wire produces a magnetic field at the position of the other, and the current in that second wire experiences a force in the field. Applying Fleming's left-hand rule shows that the force on each wire is towards the other wire when the currents are in the same direction, so the wires attract; when the currents are in opposite directions the force is outwards and the wires repel. This force is measured directly in a current balance, in which the force on a horizontal wire is balanced against the weight of small masses so that flux density can be found from B = F/(IL).

Magnetic flux and magnetic flux linkage

Magnetic flux is defined as the product of the magnetic flux density and the cross-sectional area perpendicular to the direction of the flux density: flux = B x A. The unit of flux is the weber (Wb). Since B = flux/A, one weber per square metre is one tesla, so the flux density is the flux passing through each square metre at right angles to the field.

The word perpendicular in that definition matters. If the plane of a coil is not at right angles to the field, use the component of the flux density perpendicular to the plane, which is the same as the area projected onto a plane at right angles to the field. If the field makes an angle theta with the normal to the plane of the coil, then flux = B x A x cos theta.

A coil has many turns, and the same flux passes through each turn, so what matters in electromagnetic induction is the total flux through the whole coil. This is the magnetic flux linkage, the product of the number of turns and the flux through one turn: flux linkage = N x flux = N x B x A. Flux linkage is measured in webers. A coil of 150 turns, each carrying a flux of 2.0 x 10-4 Wb, therefore has a flux linkage of 150 x 2.0 x 10-4 = 3.0 x 10-2 Wb.

Electromagnetic induction: Faraday's law and Lenz's law

An e.m.f. is induced in a circuit whenever the magnetic flux linkage through the circuit changes. Faraday's law states that the magnitude of the induced e.m.f. is proportional to the rate of change of magnetic flux linkage, written as e.m.f. = N x (change in flux)/(change in time). Notice what is missing from that statement: a large but steady flux linkage induces no e.m.f. at all, because it is the change that matters.

Lenz's law states that the induced e.m.f. is in such a direction as to oppose the change producing it. This law is a statement of the conservation of energy. If the induced current helped the change along instead of opposing it, electrical energy would be produced from nothing. The opposition is real and can be felt: push a magnet quickly into a coil and it resists, because you have to do work to generate the electrical energy.

Several standard experiments show this. Take a coil connected to a sensitive galvanometer and push a bar magnet into it: the galvanometer deflects while the magnet is moving, reads zero when the magnet is held still, and gives a larger deflection when the magnet is pushed in more quickly. Pull the magnet out and the deflection reverses; push the opposite pole in and the deflection reverses as well. In every case the size of the deflection follows Faraday's law, because it depends on the rate of change of flux linkage, and the direction follows Lenz's law, because the induced current opposes the change that produced it.

Two further experiments use the same physics. A magnet dropped down a vertical copper tube falls slowly, because the changing flux linkage induces currents in the tube and the force on those currents opposes the motion. In an a.c. generator a coil rotates in a uniform magnetic field, so its flux linkage varies continuously; the induced e.m.f. therefore reverses direction twice in every revolution.