Resistivity
For a conductor with resistivity $\rho$. the current density at a point where the electric field is $\vec{E}$:
$$\vec{J} = \frac{1}{\rho} \vec{E}$$
Since $J = \frac{n q^{2} \tau}{m} E$, the expression for resistivity is
$$\rho = \frac{m}{n q^{2} \tau}$$
The SI unit for $\rho$ is $\Omega \, m$.
When Ohm’s law is obeyed,$\rho$ is constant and independent of the magnitude of the electric field $\vec{E}$ – ohmic conductors. Metals, at a given temperature, are examples of Ohmic conductors.
The resistivity of metallic conductor nearly always increases with increasing temperature. Over a small temperature range (up to $100^{\circ}C$ or so):
$$\rho (T) = \rho_{0} [ 1 + \alpha (T – T_{0} )]$$
where $\alpha$ is the temperature coefficient of resistivity
Resistance
Consider a wire with uniform cross-sectional area A and length L. Since $J = \frac{l}{A}$, $J = \frac{E}{\rho}$ and $E = \frac{V}{L}$,
$$\begin{aligned} J &= \frac{E}{\rho} \\ \frac{I}{A} &= \frac{1}{\rho} \frac{V}{L} \\ \rho \frac{L}{A} &= \frac{V}{I} \\ R &= \rho \frac{L}{A} \end{aligned}$$
The ratio of V to I for a particular conductor is called its resistance:
$$R = \frac{V}{I}$$
The SI unit for resistance is ohm $( \Omega )$
Next: Resistance Of A Cylindrical Resistor
