11.1.Electromagnetic Induction
Let us first introduce the concept of electromotive force.
We define the electromotive force, or emf, around a circuit by
With static E-and B-field, this emf was always zero. We now take up cases where it is not zero. Since this E-field cannot be defined from Coulomb’s law, it is legitimate to ask what does define it. It is defined so that the Lorenz force
F = q(E + v x B)
Is always the electromagnetic force on a test charge q
This result is the differential form of Faraday’s law
Lenz’s law. In case of a change in a magnetic system, that thing happens which tends to oppose the change.
11.2.Self-Inductance
In case, the self-inductance, L, is defined as
L = 
11.3.Mutual Inductance
The coeficients 
are constants, independent of the current, if the magnetic media in the problem are linear
are constants, independent of the current, if the magnetic media in the problem are linearIn either case, linear or nonlinear,
Mij = 
Is defined as the mutual inductance between circuit i and circuit j,
11.4.The Neumann Formula
For two rigid stationary circuits in a linear medium (vacuum for the present), the mutual inductance is just
M21 = 
The equation is valid simply because 
is proportional to 
, making and 
equal.
is proportional to , making and equal. M21 = = 
= Using Stoke’s theorem to transform the surface integral gives
M21 = 
Which is known as Nowmann’s formula for the mutual inductance. Neumann’s formula is equally applicable to self-inductance, in which case it is written as
L = = 
11.5.Inductances in Series and in Parallel
Thus the effective inductance of two iductors in parallel is
Leff = 
Where again the sign of M depends on the way in which the inductors are connected
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