# Current density

Current density
Common symbols
J
In SI base unitsA m−2
DimensionI L−2

In electromagnetism, current density is the electric current per unit area of 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 charges at this point. In SI units, the electric current density is measured in amperes per square metre.[1]

## Definition

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 IA (SI unit: A) is the electric current flowing through A, then electric current density J at M is given by the limit:[2]

${\displaystyle J=\lim \limits _{A\rightarrow 0}{\frac {I_{A}}{A}},}$

surface A remaining centred 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 charges at M.

At a given time t, if v is the speed 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 l = v dt will flow through dA. This charge is equal to ρ ||v|| dt dA, where ρ is the charge density at M, and the electric current at M is I = ρ ||v|| dA. It follows that the current density vector can be expressed as:

${\displaystyle J=\rho {\mathbf {v} }.}$

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 (t2t1):

${\displaystyle q=\int _{t_{1}}^{t_{2}}\iint _{S}{\mathbf {J} }\cdot {\mathbf {\hat {n}} }\,{\rm {d}}A{\rm {d}}t.}$

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.

The vector area is a combination of the magnitude of the area through which the charge carriers pass, A, and a unit vector normal to the area, ${\displaystyle {\mathbf {\hat {n}} }}$. The relation is ${\displaystyle {\mathbf {A} }=A{\mathbf {\hat {n}} }}$.

If the current density J passes through the area at an angle θ to the area normal ${\displaystyle {\mathbf {\hat {n}} }}$, then

${\displaystyle {\mathbf {J} }\cdot {\mathbf {\hat {n}} }=J\cos \theta }$

where is the dot product of the unit vectors. That is, the component of current density passing through the surface (i.e. normal to it) is J cos θ, while the component of current density passing tangential to the area is J sin θ, but there is no current density actually passing through the area in the tangential direction. The only component of current density passing normal to the area is the cosine component.

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