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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.
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 ...
where E is the electric field vector with units of volts per meter (analogous to V of Ohm's law which has units of volts), J is the current density vector with units of amperes per unit area (analogous to I of Ohm's law which has units of amperes), and ρ "rho" is the resistivity with units of ohm·meters (analogous to R of Ohm's law which has ...
The members of the algebra may be decomposed by grade (as in the formalism of differential forms) and the (geometric) product of a vector with a k-vector decomposes into a (k − 1)-vector and a (k + 1)-vector. The (k − 1)-vector component can be identified with the inner product and the (k + 1)-vector component with the outer product. It is ...
Then the electric field and current density are constant and parallel, and by the general definition of resistivity, we obtain ρ = E J , {\displaystyle \rho ={\frac {E}{J}},} Since the electric field is constant, it is given by the total voltage V across the conductor divided by the length ℓ of the conductor:
For example, the charge density at a point in a medium, which is a scalar in classical physics, must be combined with the local current density (a 3-vector) to comprise a relativistic 4-vector. Similarly, energy density must be combined with momentum density and pressure into the stress–energy tensor. Examples of scalar quantities in ...
For vector flux, the surface integral of j over a surface S, gives the proper flowing per unit of time through the surface: = ^ =, where A (and its infinitesimal) is the vector area – combination = ^ of the magnitude of the area A through which the property passes and a unit vector ^ normal to the area. Unlike in the second set of equations ...
For spatial density, current, current density and flux, the notations are common from one context to another, differing only by a change in subscripts. For current density, t ^ {\displaystyle \mathbf {\hat {t}} } is a unit vector in the direction of flow, i.e. tangent to a flowline.