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In electromagnetism, current density is the amount of charge per unit time that flows through a unit area of a chosen cross section. [1] The current density vector is defined as a vector whose magnitude is the electric current per cross-sectional area at a given point in space, its direction being that of the motion of the positive charges at this point.
Generically, these equations state that the divergence of the flow of the conserved quantity is equal to the distribution of sources or sinks of that quantity. The divergence theorem states that any such continuity equation can be written in a differential form (in terms of a divergence) and an integral form (in terms of a flux).
For example, if in the mass continuity equation for flowing water, u is the water's velocity at each point, and ρ is the water's density at each point, then j would be the mass flux, also known as the material discharge. In a well-known example, the flux of electric charge is the electric current density.
The equations introduce the electric field, E, a vector field, and the magnetic field, B, a pseudovector field, each generally having a time and location dependence. The sources are the total electric charge density (total charge per unit volume), ρ, and; the total electric current density (total current per unit area), J.
Where no such symmetry exists, Gauss's law can be used in its differential form, which states that the divergence of the electric field is proportional to the local density of charge. The law was first [1] formulated by Joseph-Louis Lagrange in 1773, [2] followed by Carl Friedrich Gauss in 1835, [3] both in the context of the attraction of ...
The term on the right is the divergence of the current density J at the same point. The equation equates these two factors, which says that the only way for the charge density at a point to change is for a current of charge to flow into or out of the point. This statement is equivalent to a conservation of four-current.
The corresponding form of Poynting's theorem is =, where J is the total current density and the energy density u is given by = (| | + | |), where ε 0 is the vacuum permittivity. It can be derived directly from Maxwell's equations in terms of total charge and current and the Lorentz force law only.
In the general case a conservation equation can be also a system of this kind of equations (a vector equation) in the form: [9]: 43 + = where y is called the conserved (vector) quantity, ∇y is its gradient, 0 is the zero vector, and A(y) is called the Jacobian of the current density.