Ohm's law

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Ohm's law is the name of the relationship between an electric current (denoted by I) flowing through a conductor and the voltage difference V between the ends of the conductor causing the current,

 V = I R,\,

where R is the resistance of the conductor. The law was discovered by Georg Simon Ohm in 1826. Ohm's equation implies that R is constant, i.e., independent of V. While a resistor is an ohmic conductor, a semiconductor diode is not, as its resistance varies with the voltage applied.

Ohm's law was generalized to the proportionality of current density \vec J and electric field \vec E that is observed in many materials (especially metals),

 J_\alpha = \sum_{\beta=x,y,z} \sigma_{\alpha \beta}\, E_\beta, \qquad\alpha=x,y,z .

The symmetric tensor σ is the conductivity tensor, which in general depends on temperature and is specific for the material. For homogeneous and isotropic materials the tensor is a real number σ0 times a 3×3 identity matrix. The scalar σ0 is the conductivity coefficient and is the inverse of the resistivity ρ of the (isotropic) material,

 \rho = \frac{E}{J} = \frac{1}{\sigma_0}.

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