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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.
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.
More technically, the divergence represents the volume density of the outward flux of a vector field from an infinitesimal volume around a given point. As an example, consider air as it is heated or cooled. The velocity of the air at each point defines a vector field. While air is heated in a region, it expands in all directions, and thus the ...
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 ...
where this time is the charge density, is the current density vector, and is the current source-sink term. The current source and current sinks are where the current density emerges σ > 0 {\displaystyle \sigma >0} or vanishes σ < 0 {\displaystyle \sigma <0} , respectively (for example, the source and sink can represent the two poles of an ...
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 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.
An example of a solenoidal vector field, (,) = (,) In vector calculus a solenoidal vector field (also known as an incompressible vector field , a divergence-free vector field , or a transverse vector field ) is a vector field v with divergence zero at all points in the field: ∇ ⋅ v = 0. {\displaystyle \nabla \cdot \mathbf {v} =0.}