∇∙E = ρ/ε0, ∇×E = -∂B/∂t,
∇∙B = 0, ∇×B = μ0j + (1/c2)∂E/∂t.
∇×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.
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 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 j∙A dV, or
U = (1/(2μ0))∫all_space B∙B dV.