Ampere's circuital law, one of Maxwell's equations, since absence of this term would not predict electromagnetic waves to propagate, or the time evolution of electric fields in general. Continuity equation Since charge is conserved, current density must satisfy a continuity equation. Here is a derivation from first principles. The net flow out of some volume (which can have an arbitrary shape but fixed for the calculation) must equal the net change in charge held inside the volume: where is the charge density, and is a surface element of the surface enclosing the volume . The surface integral on the left expresses the current outflow from the volume, and the negatively signed volume integral on the right expresses the decrease in the total charge inside the volume. From the divergence theorem: Hence: This relation is valid for any volume, independent of size or location, which implies that: and this relation is called the continuity equation. In practice In electrical wiring, the maximum current density (for a given temperature rating) can vary from 4 A⋅mm−2 for a wire with no air circulation around it, to over 6 A⋅mm−2 for a wire in free air. Regulations for building wiring list the maximum allowed current of each size of cable in differing conditions. For compact designs, such as windings of SMPS transformers, the value might be as low as 2 A⋅mm−2. If the wire is carrying high-frequency alternating currents, the skin effect may affect the distribution of the current across the section by concentrating the current on the surface of the conductor. In transformers designed for high frequencies, loss is reduced if Litz wire is used for the windings. This is made of multiple isolated wires in parallel with a diameter twice the skin depth. The isolated strands are twisted together to increase the total skin area and to reduce the resistance due to skin effects. For the top and bottom layers of printed circuit boards, the maximum current density can be as high as 35 A⋅mm−2 with a copper thickness of 35 μm. Inner layers cannot dissipate as much heat as outer layers; designers of circuit boards avoid putting high-current traces on inner layers. In the semiconductors field, the maximum current densities for different elements are given by the manufacturer. Exceeding those limits raises the following problems: The Joule effect which increases the temperature of the component. The electromigration effect which will erode