Electromagnetic induction
Selfinductance, mutual inductance
Définition : Selfinductance
A closed wire loop is travelled by an intensity .
Its own magnetic field , which is given by BiotSavart law , is proportional to .
The flux of the proper magnetic field through an outline which is oriented in the positive direction of the chosen current, is also called “own flux”.
It is proportional to :
The coefficient depends on the geometric characteristics of the circuit only.
It is called the selfinductance of the circuit .
Sign of :

If , the magnetic field has the direction represented on the figure.
Its flux is positive so .

Si , direction of the magnetic field changes and the flux is negative.
Hence, is a positive coefficient.
The unit of the flux is Weber and the unit of the inductance is Henry.
Attention : Selfinduction of a wire circuit
Exemple : Selfinductance of a solenoid
The magnetic field inside an infinite solenoid is :
The proper flux through wire loops occupying a given length is : ( )
Hence the selfinductance :
Order of magnitude :
For a coil of length , formed of whorls and which has a diameter of , the selfinductance is around : Henry is a pretty big unit.
Bigger Selfinductances can be obtained by ironcore coils.
But the equation that gives is more complicated ( not only depends on the geometry of the circuit but also on the intensity).
Exemple : Selfinductance of a torus coil of rectangular section
A torus coil has a rectangular section of height and radii and .
It contains joined whorls travelled by intensity .
Any meridian plane is a plane of symmetry.
In any point of this plane, in cylindrical coordinates, the proper field is orthoradial and depends on and :
The field lines are circles around axis.
Ampere's circuital law applied to a field line of radius :
The field actually does not depend on .
Its own field through the whorls is :
Hence the inductance :
Définition : Mutual Inductance
Two wire circuits and are travelled by intensities and .
The flux of the magnetic field through the closed outline oriented by the positive direction of current is proportional to :
Likewise, the flux of the magnetic field through the closed outline oriented by the positive direction of is proportional to :
is the mutual inductance of these two circuits.
Unlike the inductance, which is always positive, can be positive or negative, depending on the orientation of the circuits.
Remark :
If can be calculated, and can be deduced.
Sometimes, one of the two fluxes is hard to calculate whereas the other one is easier.