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∫_AB·n dA.
Define the flux
F = ∫_AB·n 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∫_AB·n 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  F_i = ∑_j=1^N F_ij,
where  F_ij = M_ijI_j  (SI units),  F_ij = cM_ijI_j  (Gaussian units).

M_ij = M_ji is the coefficient of mutual induction and  
M_ii = L_i is the coefficient of self inductance.  We have
ε_i = -∑_jM_ij∂I_j/∂t.
For a single filamentary circuit we have ε = -L∂I/∂t.
To change the current in a circuit we need an external emf, V_ext, to overcome the induced emf ε. 
V_ext = L∂I/∂t.

The energy stored in the circuit is U = ½LI².  For a system of N circuits we have
U = ½∑_m=1^N F_mI_m,  or
U = ½∫_{all_space} j·A dV,  or
U = (1/(2μ₀))∫_{all_space} B·B dV.