### Maxwell’s equations

 (SI units) (Gaussian units)

 (SI units) Gaussian units) Define the flux and the electromotive force . Then : The electromotive force is the work done per unit charge (W/q = V) 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

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
where    (SI units),   (Gaussian units).

is the coefficient of mutual induction and

is the coefficient of self inductance.  We have

.

For a single filamentary circuit we have

.

To change the current in a circuit we need an external emf, Vext, to overcome the induced emf ε.

.

The energy stored in the circuit is U = (1/2)LI2.  For a system of N circuits we have:

 (SI units) (Gaussian units) or or