Notifications

Log in to see your notifications.

Electromagnetism

14 min read

Magnetic fields and forces

A magnetic field exerts a force on a moving charge and on a current-carrying conductor. The strength of the field is the magnetic flux density B, measured in tesla (T). The force on a charge q moving with speed v at right angles to the field is F = qvB. It is always at right angles to the velocity, so it changes the direction of motion but does no work: the charge moves in a circle.

The radius of the circular path of a charged particle of mass m in a uniform field. Used in mass spectrometers and cyclotrons.
Try itA charged particle in a magnetic field

Choose the speed of an electron and the magnetic flux density. Find the radius of its circular path. (m = 9.1 × 10⁻³¹ kg, e = 1.6 × 10⁻¹⁹ C)

Magnetic fields of currents

SourceField
Long straight wireB = μ₀I ÷ 2πr (distance r from the wire)
Circular coil of N turns, radius r (at the centre)B = μ₀NI ÷ 2r
Long solenoid with n turns per metreB = μ₀nI (inside)

μ₀ = 4π × 10⁻⁷ T m A⁻¹ is the permeability of free space. Two parallel wires carrying currents in the same direction attract; in opposite directions they repel. This defines the ampere.

Magnetic fields of currentsOn the left, a wire carrying current towards the viewer has circular magnetic field lines around it, in the direction given by the right-hand grip rule. On the right, a solenoid, a long coil of wire, has a field like a bar magnet with a north pole at one end and a south pole at the other.•Straight wire: circular fieldcurrent towards youSolenoid: field like a bar magnetN at one end, S at the other

Electromagnetic induction

The magnetic flux Φ through a coil is BA cos θ (weber, Wb). Faraday’s law: the induced e.m.f. equals the rate of change of flux linkage (NΦ). Lenz’s law (the minus sign) says the induced e.m.f. acts to oppose the change that produced it, which is an example of conservation of energy.

A wire of length L moving at speed v at right angles to a field B has an induced e.m.f. BLv.
Worked exampleE.m.f. induced in a moving rod

A metal rod of length 0.50 m moves at 4.0 m/s at right angles to a magnetic field of 0.20 T. Find the induced e.m.f.

  1. Use E = BLv.

Answer: The induced e.m.f. is 0.40 V.

Generators, motors, transformers and eddy currents

An a.c. generatorA coil turns between the poles of a magnet. As it cuts the magnetic field lines, a voltage is induced. Slip rings and brushes carry it out. The output changes direction twice every turn, so it is an alternating voltage drawn as a sine wave. A faster turn or a stronger magnet gives a bigger voltage.NSslip rings and brushesturnvoltage against time
A coil of N turns and area A rotating at angular speed ω in a field B generates an alternating e.m.f. Mains of 230 V rms has a peak of 325 V.
A transformerTwo coils of wire wound on a soft iron core. An alternating current in the primary coil makes a changing magnetic field in the core, which induces an alternating voltage in the secondary coil. The ratio of the voltages equals the ratio of the numbers of turns. A step-up transformer has more turns on the secondary and a step-down has fewer.primary coilNₚ turns, Vₚsecondary coilNₛ turns, Vₛsoft iron coreVₛ ÷ Vₚ = Nₛ ÷ Nₚ

In a real transformer the efficiency is below 100% because of resistance of the windings (copper loss), eddy currents in the core, hysteresis and flux leakage. Eddy currents are reduced by a laminated iron core. Self-inductance L: a changing current in a coil induces an e.m.f. in the coil itself, opposing the change: E = −L ΔI/Δt. An inductor stores energy ½LI².

Field lines make closed loops.
Check yourself A charged particle moves at right angles to a uniform magnetic field. Its path is…
Check yourself Lenz’s law is an example of conservation of…
Check yourself A coil of 100 turns has a flux change of 0.02 Wb in 0.5 s. The induced e.m.f. is…

Key terms

Magnetic flux density
The force per unit current per unit length on a conductor at right angles to the field, measured in tesla.
Flux linkage
The magnetic flux multiplied by the number of turns.
Eddy current
A circulating current induced in a piece of metal by a changing magnetic field.
Self-inductance
The property of a coil that opposes changes in its own current.

Summary

  • F = BIL = qvB; a charge in a field moves in a circle of radius mv/qB.
  • Faraday: E = −NΔΦ/Δt; Lenz: the induced effect opposes the change.
  • Generators: E = NBAω sin ωt; rms = peak ÷ √2.
  • Real transformers lose energy by resistance, eddy currents and flux leakage.
Current Electricity