given by Lenz's law. The laws of induction of electric currents in mathematical form was established by Franz Ernst Neumann in 1845. Faraday's law contains the information about the relationships between both the magnitudes and the directions of its variables. However, the relationships between the directions are not explicit; they are hidden in the mathematical formula. It is possible to find out the direction of the electromotive force (emf) directly from Faraday’s law, without invoking Lenz's law. A left hand rule helps doing that, as follows: Align the curved fingers of the left hand with the loop (yellow line). Stretch your thumb. The stretched thumb indicates the direction of (brown), the normal to the area enclosed by the loop. Find the sign of , the change in flux. Determine the initial and final fluxes (whose difference is ) with respect to the normal , as indicated by the stretched thumb. If the change in flux, , is positive, the curved fingers show the direction of the electromotive force (yellow arrowheads). If is negative, the direction of the electromotive force is opposite to the direction of the curved fingers (opposite to the yellow arrowheads). For a tightly wound coil of wire, composed of identical turns, each with the same , Faraday's law of induction states that where is the number of turns of wire and is the magnetic flux through a single loop. Maxwell–Faraday equation The Maxwell–Faraday equation states that a time-varying magnetic field always accompanies a spatially varying (also possibly time-varying), non-conservative electric field, and vice versa. The Maxwell–Faraday equation is (in SI units) where is the curl operator and again is the electric field and is the magnetic field. These fields can generally be functions of position and time . The Maxwell–Faraday equation is one of the four Maxwell's equations, and therefore plays a fundamental role in the theory of classical electromagnetism. It can also be written in an integral form by the Kelvin–Stokes theorem, thereby reproducing Faraday's law: where, as indicated in the figure, is a surface bounded by the closed contour , is an infinitesimal vector element of the contour , and is an infinitesimal vector element of surface . Its direction is orthogonal to that surface patch, the magnitude is the area of an infinitesimal patch of surface. Both and have a sign ambiguity; to get the correct sign, the right-hand rule is used, as explained