Magnetic fields and induction

Electricity and magnetism are two faces of one phenomenon. Moving charges create magnetic fields, magnetic fields push on moving charges, and a changing magnetic field creates an electric field. These three facts run the modern world: every electric motor, generator, transformer, loudspeaker, MRI scanner and induction cooktop is built on them.

This chapter covers the magnetic force on moving charges and currents, the torque that drives motors, the magnetic fields made by wires and coils, and Faraday's law of induction, which explains generators and transformers. It finishes with why power is transmitted at high voltage.

Magnetic fields and the force on a moving charge

A magnetic field \(\mathbf{B}\) is measured in teslas (T). The Earth's field is about 50 μT; a fridge magnet about 5 mT; an MRI scanner 1.5 to 3 T. Unlike the electric field, the magnetic field pushes only on charges that are moving, and the force is perpendicular to both the velocity and the field:

This is the cross product from Linear Algebra, with its right-hand rule: point your fingers along \(\mathbf{v}\), curl them toward \(\mathbf{B}\), and your thumb gives the force on a positive charge. For a negative charge, reverse it.

Because the force is always perpendicular to the velocity, a magnetic field does no work. It changes a charge's direction but never its speed or kinetic energy.

In diagrams, a field pointing out of the page is drawn as dots (⊙, the tip of an arrow coming toward you) and into the page as crosses (⊗, the tail feathers going away).

Circular motion

A charge moving perpendicular to a uniform field feels a constant-size force toward the centre of a circle. Setting \(qvB = mv^2/r\) gives the radius and the period:

Remarkably, the period does not depend on the speed: faster particles move in bigger circles and take the same time. That is the principle of the cyclotron, which accelerates protons to make medical isotopes. Mass spectrometers use the radius to measure the mass of ions.

Force on a current-carrying wire

A current is a stream of moving charges, so a wire carrying current in a magnetic field feels a force. For a straight length \(L\) of wire at angle \(\theta\) to the field:

directed by the right-hand rule with the fingers along the current.

Torque on a coil: the electric motor

A rectangular loop of current in a field feels equal and opposite forces on its two sides, which form a couple. For a coil of \(N\) turns and area \(A\), the torque is

where \(\theta\) is the angle between the field and the coil's axis (its normal). An electric motor uses this torque, with a commutator or electronic switching to reverse the current every half turn so the torque always acts in the same rotational direction.

Magnetic fields from currents

Currents produce magnetic fields. Around a long, straight wire, the field lines are circles centred on the wire, with direction given by a second right-hand rule: thumb along the current, fingers curl in the direction of \(\mathbf{B}\). The size of the field falls off as \(1/r\):

Two parallel wires carrying current exert forces on each other: each sits in the other's field. The force per unit length is \(F/L = \mu_0 I_1I_2/(2\pi d)\), attractive for currents in the same direction and repulsive for opposite currents. In a short circuit, currents of tens of thousands of amperes make these forces large enough to bend busbars, so switchgear is braced to withstand them.

Solenoids

A long coil of wire, a solenoid, produces a nearly uniform field inside, \(B = \mu_0 nI\), where \(n\) is the number of turns per metre. A 0.50 m solenoid with 1000 turns carrying 2.0 A has \(n = 2000\) per metre and \(B = (4\pi \times 10^{-7})(2000)(2.0) = 5.0\) mT. An iron core multiplies the field by hundreds or thousands, which is how electromagnets, relays and solenoid valves get their strength.

Electromagnetic induction

In 1831 Michael Faraday discovered the reverse effect: a changing magnetic field produces a voltage. To state it precisely, define the magnetic flux through a loop of area \(A\) as \(\Phi = BA\cos\theta\) (in webers, Wb), where \(\theta\) is the angle between the field and the loop's normal. Faraday's law says the emf induced in a coil of \(N\) turns is

Any change in flux induces an emf: a changing field strength, a changing area, or a changing angle. The minus sign expresses Lenz's law: the induced current flows in the direction that opposes the change in flux that produced it. Lenz's law is conservation of energy in disguise; if the induced current helped the change, you would get energy for nothing.

Motional emf

A conductor moving through a field has its free charges pushed along it by the magnetic force, producing an emf \(\mathcal{E} = BLv\) for a rod of length \(L\) moving at speed \(v\) perpendicular to the field. A 0.50 m rod moving at 4.0 m/s through a 0.80 T field generates \(0.80 \times 0.50 \times 4.0 = 1.6\) V. This is the simplest generator, and it is also the principle of electromagnetic flowmeters, which measure the speed of a conducting fluid in a pipe from the voltage across it.

Lenz's law also explains eddy currents: changing fields induce swirling currents in solid metal, which oppose the motion. They are the basis of magnetic braking in trains and roller coasters, of induction cooktops (which heat the pan directly), and of the metal detectors at airports. In transformers and motors they waste energy, so cores are made of thin insulated laminations to block them.

Generators and alternating current

A coil of \(N\) turns and area \(A\) rotating at angular speed \(\omega\) in a field \(B\) has flux \(\Phi = BA\cos\omega t\). By Faraday's law, the emf is

This is alternating current (AC): a sinusoidal voltage that reverses direction twice each cycle. The emf is largest when the coil's plane is parallel to the field, where the flux is zero but changing fastest, and zero when the coil faces the field, where the flux is greatest but momentarily not changing.

Power-station generators work exactly this way, usually with the magnet rotating inside stationary windings, driven by steam, gas, water or wind turbines. Household voltage of 120 V is an rms value; its peak is about 170 V.

Transformers and power transmission

A transformer has two coils wound on a shared iron core. Alternating current in the primary coil creates a changing flux in the core, which induces an emf in the secondary. Since the same flux passes through every turn, the voltages are in the ratio of the turns:

An ideal transformer conserves power, so stepping the voltage up steps the current down by the same factor. Real transformers are 95% to over 99% efficient. Transformers need changing flux, which is the main reason the grid uses AC rather than DC.

This is why electricity leaves power stations at hundreds of kilovolts, is stepped down at substations to distribution voltages (25 kV is common in Alberta), and is stepped down again by pole-top or pad-mounted transformers to the 120/240 V supplied to homes. A 25 kV to 240 V transformer has a turns ratio of about 104 to 1.