In the study of the induction of electromotive force in circuits by the flux fields arising from magnetic materials, the usual starting point is Faraday's law of electromagnetic induction. It is shown that, for systems limited to coil and magnet, or coil and coil, there is an alternative expression of Faraday's law which forms a useful basis for mathematical development. A basic general equation is derived, some theorems are noted, and equations for special cases are derived. Relationships are found between the electromotive forces in the coil-magnet situation and the mechanical forces that arise on the magnet if a current is passed through the coil.