Electromechanics

The relationship between force and magnetic field:
f = quB sin theta
The flux is given by
phi = int{B dA}
This can be approximated by
phi = B . A

Faraday's law states that a time-varying flux causes an induced electromotive force,
e = - d phi/dt

The polarity of the induced voltage can usually be determined from physical considerations; therefore the minus sign in the above equation is usually left out.

For an N-turn coil with cross-sectional area A, we have the emf
e = N d phi/dt

The flux linkage lambda is given by
lambda = N phi

So that,
e = d lambda/dt

In addition, flux linkage and current is a linear relationship,
lambda = L i

Therefore,
v = L d i /dt where L is the ideal self-inductance

In addition to self-inductance, it is important to consider the magnetic coupling.

The magnetic coupling between the coils established by virtue of the proximity is described by a quantity called mutual inductance and defined by the symbol M. The mutual inductance is defined by the equation
v2 = M d i1 / dt

In practical electromagnetic circuits, the self-inductance of a circuit is not necessarily constant; the inductance L is not constant, in general. It will not be possible to use such a simple relationship as v = L di/dt, with L constant.

If we revisit the definition of the transformer voltage e = N d phi/dt we see that in an inductor coil, the inductance is given by
L = N phi/i = lambda/i