Electrostatics
Electromagnetics
Electricity & Magnetism
© The scientific sentence. 2010

Mutual inductance
Mutual inductance
Consider two coils 1 and 2 one
in front of the other, of length l1 and l2, crosssectional
areas S1 and S2, with n1, and n2 turns per unit length each.
The two coils carry the currents i1 and i2, . The current i1
produces a magnetic field B1 inside the coil 1 that influences
the nearby coil 2.
When i1 changes (then B1 changes), it produces in the coil 1 a selfinduced
emf = ℰ_{11}, and a induced emf = ℰ_{21} in the
coil 2 through (external field) B1.
Similarly, the current i2 produces a magnetic field B2 inside the
coil 2 that influences the nearby coil 1.
when i2 changes in the coil 2 (then B2 changes), it produces in
the coil 2 a selfinduced emf = ℰ_{22}, and a induced emf =
ℰ_{12} in the coil 1 through (external field) B2.
The effect of inducing an emf by one coil on the
other is called mutual induction.
Selfinduction in the coil 1
The magnetic field inside is B1 = μ_{o} n1 i1
The magnetic flux due to B1 that links this coil is Φ_{11} = B1 S1
Its emf = ℰ_{1} =  n1 dΦ_{B1}/dt
=  μ_{o}n1^{2} S1 l1 di1/dt
Its selfinductance is L1 = μ_{o}n1^{2}S1l1
Selfinduction in the coil 2
The magnetic field inside is B2 = μ_{o} n2 i2
The magnetic flux due to B2 that links this coil is Φ_{22} = B2 S2
Its self = ℰ_{2} =  n2 dΦ_{B2}/dt
=  μ_{o}n2^{2} S2 l2 di2/dt
Its selfinductance is L12 = μ_{o}n2^{2}S2l2
Interaction: Mutualinduction of the coil 1 and the coil 2
The magnetic field inside the two coils is B1 + B2 =
μ_{o} n2 i2 + μ_{o} n1 i1
The magnetic flux due to (B1 + B2) that links this coil 1
is Φ_{112} = (B1 + B2) S1
and the magnetic flux due to (B1 + B2) that links this coil 2
is Φ_{212} = (B1 + B2) S2
The emf of the coil 1 is emf = ℰ_{12}
=  n1 l1 dΦ_{(B1 + B2)}/dt
=  n1 l1 S1 μ_{o} d(n2 i2 + n1 i1)/dt
=  n1 l1 S1 μ_{o} (n2 di2/dt + n1 di1/dt)
The emf of the coil 2 is emf = ℰ_{21}
=  n2 l2 dΦ_{(B1 + B2)}/dt
=  n12 l2 S2 μ_{o} d(n2 i2 + n1 i1)/dt
We define:
The mutualinductance for the coil 1:
ℰ_{12} =  M12 di1/dt, and
The mutualinductance for the coil 2:
ℰ_{21} =  M21 di2/dt, and
M12 and M21 are called coefficient of mutual inductance.
They depend on the geometrical factors of the coils.
We can consider the particular case where M12 = M21 = M.
M is then the inductance of the pair. The SI unit of the
mutual inductance is the Henry.

