Quasi-static situations

Maxwell's equations (SI units)

∇∙E = ρ/ε0×E = -∂B/∂t,
∇∙B = 0,  ×B = μ0j + (1/c2)∂E/∂t.

Faraday's law

×E = -∂B/∂t,
Γ E∙dr = -∂/∂t∫ABn dA.
Define the flux
F = ∫ABn dA
and the electromotive force
ε = ∮Γ E∙dr.
Then ε = -∂F/∂t.
The electromotive force is the work done per unit charge (W/q = ε) if it is moved once around  the loop Γ.

Any induced emf tries to oppose the flux changes that produce it.  This is Lenz's rule.
In the above integral formulas the "loop" Γ can be any fixed curve in space, i.e. a loop that does not change its shape.

Motional emf

Consider a well-defined filamentary circuit which can change its shape.  For such circuit we may write 
ε = -d/dt∫ABn dA,
i.e. we can combine the emf due to flux changes and the emf due to shape changes into one equation.  (The partial derivative changes to a total derivative.)

Quasi-static situations

Quasi-static situations refer to non-static situations in which electromagnetic radiation can be neglected.
Consider N filamentary circuits.  Then the flux through the ith circuit is  Fi = ∑j=1N Fij,
where  Fij = MijIj  (SI units),  Fij = cMijIj  (Gaussian units).

Mij = Mji is the coefficient of mutual induction and  
Mii = Li is the coefficient of self inductance.  We have
εi = ∑jMij∂Ij/∂t.
For a single filamentary circuit we have ε = -L∂I/∂t.
To change the current in a circuit we need an external emf, Vext, to overcome the induced emf ε. 
Vext = L∂I/∂t.

The energy stored in the circuit is U = ½LI2.  For a system of N circuits we have
U = ½∑m=1N FmIm,  or
U = ½∫all_space jA dV,  or
U = (1/(2μ0))∫all_space BB dV.