# Electric current | current density and ohm's law

## Current density and Ohm's law

Current density is a measure of the density of an electric current. It is defined as a vector whose magnitude is the electric current per cross-sectional area. In SI units, the current density is measured in amperes per square metre.

${\displaystyle I=\int {\vec {J}}\cdot d{\vec {A}}}$

where ${\displaystyle I}$ is current in the conductor, ${\displaystyle {\vec {J}}}$ is the current density, and ${\displaystyle d{\vec {A}}}$ is the differential cross-sectional area vector.

The current density (current per unit area) ${\displaystyle {\vec {J}}}$ in materials with finite resistance is directly proportional to the electric field ${\displaystyle {\vec {E}}}$ in the medium. The proportionality constant is called the conductivity ${\displaystyle \sigma }$ of the material, whose value depends on the material concerned and, in general, is dependent on the temperature of the material:

${\displaystyle {\vec {J}}=\sigma {\vec {E}}\,}$

The reciprocal of the conductivity ${\displaystyle \sigma }$ of the material is called the resistivity ${\displaystyle \rho }$ of the material and the above equation, when written in terms of resistivity becomes:

${\displaystyle {\vec {J}}={\frac {\vec {E}}{\rho }}}$ or
${\displaystyle {\vec {E}}=\rho {\vec {J}}}$

Conduction in semiconductor devices may occur by a combination of drift and diffusion, which is proportional to diffusion constant ${\displaystyle D}$ and charge density ${\displaystyle \alpha _{q}}$. The current density is then:

${\displaystyle J=\sigma E+Dq\nabla n,}$

with ${\displaystyle q}$ being the elementary charge and ${\displaystyle n}$ the electron density. The carriers move in the direction of decreasing concentration, so for electrons a positive current results for a positive density gradient. If the carriers are holes, replace electron density ${\displaystyle n}$ by the negative of the hole density ${\displaystyle p}$.

In linear anisotropic materials, σ, ρ and D are tensors.

In linear materials such as metals, and under low frequencies, the current density across the conductor surface is uniform. In such conditions, Ohm's law states that the current is directly proportional to the potential difference between two ends (across) of that metal (ideal) resistor (or other ohmic device):

${\displaystyle I={V \over R}\,,}$

where ${\displaystyle I}$ is the current, measured in amperes; ${\displaystyle V}$ is the potential difference, measured in volts; and ${\displaystyle R}$ is the resistance, measured in ohms. For alternating currents, especially at higher frequencies, skin effect causes the current to spread unevenly across the conductor cross-section, with higher density near the surface, thus increasing the apparent resistance.

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