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Concept
Current density
Summary
In electromagnetism, current density is the amount of charge per unit time that flows through a unit area of a chosen cross section. 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. In SI base units, the electric current density is measured in amperes per square metre.
Assume that A (SI unit: m2) is a small surface centred at a given point M and orthogonal to the motion of the charges at M. If I_A (SI unit: A) is the electric current flowing through A, then electric current density j at M is given by the limit:
with surface A remaining centered at M and orthogonal to the motion of the charges during the limit process.
The current density vector j is the vector whose magnitude is the electric current density, and whose direction is the same as the motion of the positive charges at M.
At a given time t, if v is the velocity of the charges at M, and dA is an infinitesimal surface centred at M and orthogonal to v, then during an amount of time dt, only the charge contained in the volume formed by dA and will flow through dA. This charge is equal to where ρ is the charge density at M. The electric current is , it follows that the current density vector is the vector normal (i.e. parallel to v) and of magnitude
The surface integral of j over a surface S, followed by an integral over the time duration t1 to t2, gives the total amount of charge flowing through the surface in that time (t2 − t1):
More concisely, this is the integral of the flux of j across S between t1 and t2.
The area required to calculate the flux is real or imaginary, flat or curved, either as a cross-sectional area or a surface. For example, for charge carriers passing through an electrical conductor, the area is the cross-section of the conductor, at the section considered.
Official source
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