adopted sign convention and loop orientations.) Per Ampère's law, the excitation is the product of the current and the number of complete loops made and is measured in ampere-turns. Stated more generally: By Stokes's theorem, the closed line integral of around a contour is equal to the open surface integral of curl across the surface bounded by the closed contour. Since, from Maxwell's equations, , the closed line integral of evaluates to the total current passing through the surface. This is equal to the excitation, , which also measures current passing through the surface, thereby verifying that the net current flow through a surface is zero ampere-turns in a closed system that conserves energy. More complex magnetic systems, where the flux is not confined to a simple loop, must be analysed from first principles by using Maxwell's equations. Applications Air gaps can be created in the cores of certain transformers to reduce the effects of saturation. This increases the reluctance of the magnetic circuit, and enables it to store more energy before core saturation. This effect is used in the flyback transformers of cathode-ray tube video displays and in some types of switch-mode power supply. Variation of reluctance is the principle behind the reluctance motor (or the variable reluctance generator) and the Alexanderson alternator. Multimedia loudspeakers are typically shielded magnetically, in order to reduce magnetic interference caused to televisions and other CRTs. The speaker magnet is covered with a material such as soft iron to minimize the stray magnetic field. Reluctance can also be applied to variable reluctance (magnetic) pickups. See also Magnetic capacitance Magnetic complex reluctance Tokamak References External links Magnetic–Electric Analogs by Dennis L. Feucht, Innovatia Laboratories (PDF) Interactive Java Tutorial on Magnetic Shunts National High Magnetic Field Laboratory Electromagnetism Electric and magnetic fields in matter Electrical analogies