Ohm's law

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 &sigma; 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 &sigma;0  times a 3&times;3 identity matrix. The scalar &sigma;0 is the conductivity coefficient and is the inverse of the resistivity  &rho; of the (isotropic) material,
 * $$ \rho = \frac{E}{J} = \frac{1}{\sigma_0}. $$

Reference

 * H.D. Young & R.A. Freedman (2004). University Physics 11th Edition. International Edition. Addison Wesley, ISBN 0-321-20469-7