When the current in a coil is changing an e.m.f will be induced
in a nearby circuit due to some of the magnetic flux produced by the first circuit linking the second.
The phenomenon is known as mutual induction. It is important to realise that the induced e.m.f. lasts
only as long as the current in the first circuit is changing.

The mutual inductance M is defined by
the equation

where E is the e.m.f induced in
the secondary coil and dI/dt the rate of change of current in the primary.

Two coils are said to
have a mutual inductance of 1 H if an e.m.f. of 1 V is induced in the secondary when the current in the
primary changes at the rate of 1 A s^{-1}.

Induction coils such as this are used in car ignition circuits,
and used to be a source of high voltage for research.

Consider the mutual inductance of a long solenoid and a coil as
shown in the diagram.

Suppose that a short coil of N_{2} turns is wound round a
solenoid of N_{1} turns, with a cross-sectional area A, length x and carrying a current I.

The flux at the centre of the solenoid is: B = m_{o}N_{1}I/x

The flux linking the short coil is
f = BA and therefore the flux linkage of the short coil
is

If the current in the primary changes by dI in time dt, giving a change in flux linkage of d(Nf) in the secondary, then the e.m.f. induced in the secondary will be

Writing M as the mutual inductance, we have that E = - MdI/dt and therefore:

Calculate the mutual inductance of a pair of coils if the primary has 1000 turns of radius 2 cm and is 1 m long while the secondary has 1200 turns and is wound round the centre of the primary.

Mutual inductance = 4px 10