wires, and he generated a steady (DC) current by rotating a copper disk near the bar magnet with a sliding electrical lead ("Faraday's disk"). Michael Faraday explained electromagnetic induction using a concept he called lines of force. However, scientists at the time widely rejected his theoretical ideas, mainly because they were not formulated mathematically. An exception was James Clerk Maxwell, who in 1861–62 used Faraday's ideas as the basis of his quantitative electromagnetic theory. In Maxwell's papers, the time-varying aspect of electromagnetic induction is expressed as a differential equation which Oliver Heaviside referred to as Faraday's law even though it is different from the original version of Faraday's law, and does not describe motional emf. Heaviside's version (see Maxwell–Faraday equation below) is the form recognized today in the group of equations known as Maxwell's equations. Lenz's law, formulated by Emil Lenz in 1834, describes "flux through the circuit", and gives the direction of the induced emf and current resulting from electromagnetic induction (elaborated upon in the examples below). Faraday's law The most widespread version of Faraday's law states: Mathematical statement For a loop of wire in a magnetic field, the magnetic flux is defined for any surface whose boundary is the given loop. Since the wire loop may be moving, we write for the surface. The magnetic flux is the surface integral: where is an element of surface area of the moving surface , is the magnetic field, and is a vector dot product representing the element of flux through . In more visual terms, the magnetic flux through the wire loop is proportional to the number of magnetic field lines that pass through the loop. When the flux changes—because changes, or because the wire loop is moved or deformed, or both—Faraday's law of induction says that the wire loop acquires an emf, defined as the energy available from a unit charge that has traveled once around the wire loop. (Although some sources state the definition differently, this expression was chosen for compatibility with the equations of special relativity.) Equivalently, it is the voltage that would be measured by cutting the wire to create an open circuit, and attaching a voltmeter to the leads. Faraday's law states that the emf is also given by the rate of change of the magnetic flux: where is the electromotive force (emf) and is the magnetic flux. The direction of the electromotive force is