Definition 0.1.6 The resistance denoted $R$ of a prism of material with cross sectional area $A$, length $L$ and resistivity $\rho$ is given by the following relation.
\[R = \rho \frac{\ell}{A}\]
Definition 0.1.1 A current density denoted $\vec{j}$ is a vector field describing the average density of charge flowing through a particular point in space per second.
Definition 0.1.2 The electric conductivity denoted $\sigma$ of a material is the coefficient or tensor that relates the electric field $\vec{E}$ to the current density $\vec{j}$.
\[\vec{j} = \sigma \vec{E}\]
Definition 0.1.3 The resistivity denoted $\rho$ of a material is the coefficient or tensor that relates the current density $\vec{\rho}$ flowing through a material with the electric field $\vec{E}$ required to drive that current.
\[\vec{E} = \rho\vec{j}\]
Corollary 0.1.4 The conductivity $\sigma$ and resistivity $\rho$ of a material are inverses of each other.
\[\sigma = \frac{1}{\rho},\quad \rho = \frac{1}{\sigma}\]
Definition 0.1.5 The Drude conducitivity denoted $\sigma_D$ is the electric conductivity predicted by the Drude model for a pure electric field $\vec{E}$ ($\vec{B}=\vec{0}$) where $\tau$ is the mean scattering time, $n_e$ is number of electrons, $e$ is the elemental charge and $m_e$ is the mass of charge carriers.
\[\vec{j}_D = \frac{e^2n_e\tau}{m_e}\vec{E} = \sigma_D\vec{E}\]
Definition 0.1.6 The resistance denoted $R$ of a prism of material with cross sectional area $A$, length $L$ and resistivity $\rho$ is given by the following relation.
\[R = \rho \frac{\ell}{A}\]
Law 0.1.7 Ohm's law states that the total current $I$ flowing through a material is equal to the resistance $R$ times the bias voltage across the material $V$.
\[V = IR\]