This notes sheet looks at magnetic fields & electromagnetism, including induction, the magnetomotive force, permeability, reluctance and electromagnets.

Magnetic Fields

A magnetic field is a vector field in which a magnetic material experiences a force. There are two
causes of magnetic fields:

  1. Permanent Magnets
bar magnet field, magnetic field lines, bar magnet magnetic field
  1. Moving Charge
    • This could be around a straight, current-carrying wire:
Magnetic Field around a straight current carrying wire, wire electric field, electric magnetic field
  • A flat coil:
magnetic field around a flat coil
  • or a solenoid:
solenoid magnetic field


Flux Density, B, is a measure of field strength, the density of field lines.

The units of flux are Tesla, T (N/Am).

Magnetic flux, φ, is the flow of the field, represented as field lines.

  • For fixed magnets, flux always goes from north to south
  • For a current-carrying wire, the right-hand grip rule shows the direction of flux
  • Like poles repulse, opposite poles attract
  • Magnetic flux lines never cross

    \[\phi=BA\cos \theta\]

The units of flux are Weber, Wb.

Flux Linkage, Nφ , applies when a wire is coiled, for example a solenoid. In this instance, the flux
linkage is the flux multiplied by the number of turns:

    \[N\phi=BAN \cos \theta\]

The units of flux linkage are Weber turns.


When a current carrying wire is in a magnetic field, it experiences a force. This is known as the
Lorentz Force, and is given as:

    \[F=BIL\sin \theta\]

The direction of the force is found using Fleming’s left-hand rule:

Fleming's left hand rule, lorentz force, induction hand rule, induction force direction

Alternatively, if a charged particle moved through the magnetic field with velocity v, it expereinces a force:


Faraday’s Law

According to Faraday’s Law, an emf is induced in the circuit whenever it experiences a change in
magnetic flux linkage:


    \[E=\frac{d}{dt}(BAN\cos \theta)\]

In a closed circuit, this emf causes a current to flow.

Generally, the change of flux occurs because the wire/circuit is moving. In this case, the direction
of the induced current is found using the right-hand rule:

Fleming's right hand rule, the right hand rule, right hand rule for induced force,how to find the direction of an induced force, electromagnetism

Lenz’s Law

According to Faraday’s Law, when a wire moves through a magnetic field, a current is induced.
This current opposes the current that caused the initial Lorentz force that moved the wire.

This means all generated power has resistance and is known as Lenz’s Law.

Electromagnetic Quantities

Magnetism is very similar to electricity – for every electrical property (emf, current, resistance etc.),
there is an equivalent electromagnetic quantity that performs in exactly the same way. Electric
equations and relations, like Ohm’s law, have their direct equivalents too.

Magnetomotive Force, mmf

Just like electrical current is the flow of electromotive force (emf) around a circuit, magnetic flux is
the flow of magnetomotive force around a circuit.

A solenoid, or any coil of current-carrying wire is an mmf source – equivalent to an emf source in
an electrical circuit. The magnitude of the mmf delivered is given as:


  • N is the number of turns
  • I is the current in the wire

Reluctance, S

This is the magnetic equivalent to electrical resistance. Therefore, magnetic Ohm’s Law is:

    \[mmf=\phi S\]

When connected in series, reluctances add up like resistors:


Similarly in parallel:


A key difference in magnetic circuits, however, is that reluctance is not given in a ‘reluctor’ in the
way that a resistance comes from a resistor.

Instead, reluctance rises from cores made of magnetic materials. The reluctance depends on the
material, cross-section, and length:


  • L is the length of the core
  • A is the cross-sectional area


Magnetic Permeability is a material property. It is defined relative to the permeability of free space:

    \[\mu = \mu_r\mu_0\]

  • \mu_r is the relative permeability of the material
  • \mu_0 is the permeability of free space

Ferromagnetic materials have a high relative permeability, making them excellent magnetic
conductors. They are easy to magnetise and magnetise and are often called soft magnets. They
are called ferromagnetic, as they generally have high iron content.

Other magnetic materials are hard, however: it is significantly harder to magnetise and
demagnetise these.

Comparing Electrical and Magnetic Quantities

Electrical QuantityUnitsMagnetic QuantityUnits
Current, IAFlux, \phiWb
Resistance, R\OmegaReluctance, SA.turns Wb^{-1}
Resistivity, \rho1/Permeability, \frac{1}{\mu}

Know these equivalents, and magnetic circuits will be a breeze.

Energy in Magnetic Circuits

Energy in a magnetic circuit is given as:


This can be written in many forms:

    \[energy=\frac{1}{2} \frac{N^2I^2}{S}\]

    \[energy=\frac{1}{2}\phi^2 S\]

Air Gaps

Many magnetic circuits have air gaps. The relative permeability of these is 1, so the reluctance in an air gap is just:


Where x is the thickness of the gap.

This reluctance is significantly larger than that of the core, and the two reluctances are in series:


where S_{core}\<< S_{gap}

Therefore, the vast majority of the energy stored in the circuit is in the air gap:

    \[energy = \frac{1}{2} \phi^2(S_{core}+S_{gap})\]

    \[energy\approx \frac{1}{2} \phi^2 S_{gap}\approx\frac{AB^2x}{2\mu_0}\]

Since work done is force times distance moved:



This is how electromagnets work: an electric winding on a core produces a vast force in a small
air gap between the core and a magnetic object (the armature). If this force is greater than the
frictional and gravitational forces holding the armature stationary, it will move towards the
electromagnet. There are two currents:

  1. Pull-in current
    This is the current required to move the armature over the air gap
  2. Holding current
    This is the current required to hold the armature once it has made contact with the core
    (there is no air gap)

The holding current is significantly lower than the pull-in current.

  • A magnetic field is a vector field in which a magnetic object experiences a force
    • They are either caused by permanent magnets, or by moving charges
  • Flux density, B, is a measure of field strength. The units are Tesla, T.
  • Flux, \phi, is the flow of the field
  • The Lorentz Force is experienced by
    • a current carrying wire in a magnetic field: F=BIL \sin \theta
    • a charged particle moving in a magnetic field: F=BQv
  • Electric quantities all have their magnetic counterpart – see above
  • The energy in a magnetic circuit is given as E=\frac{1}{2}\phi^2 S
  • The energy stored in an air gap is far higher than that stored in the core, as the reluctance is so much higher