Resistivity and Temperature Dependence Reference Cheat Sheet

Key Facts

• Resistance is related to resistivity by $R = \rho \frac{L}{A}$, where $L$ is length and $A$ is cross-sectional area.
• Resistivity can be found from a measured resistance using $\rho = \frac{RA}{L}$.
• Conductivity is the reciprocal of resistivity, so $\sigma = \frac{1}{\rho}$.
• For many metals, resistivity changes with temperature according to $\rho = \rho_0[1 + \alpha(T - T_0)]$.
• If the conductor’s dimensions stay nearly constant, resistance changes with temperature as $R = R_0[1 + \alpha(T - T_0)]$.
• A positive temperature coefficient $\alpha$ means resistance usually increases as temperature increases.
• A negative temperature coefficient $\alpha$ means resistance usually decreases as temperature increases, which is common in many semiconductors.
• The cross-sectional area of a round wire is $A = \pi r^2 = \pi \left(\frac{d}{2}\right)^2$.

Vocabulary

Resistance

Resistance is the opposition to electric current in a specific object, measured in ohms $\Omega$.

Resistivity

Resistivity is a material property that describes how strongly a substance opposes current, measured in $\Omega\cdot\text{m}$.

Conductivity

Conductivity is a material property that describes how easily charge flows, given by $\sigma = \frac{1}{\rho}$.

Temperature Coefficient

The temperature coefficient $\alpha$ tells how much resistivity or resistance changes per degree of temperature change.

Cross-Sectional Area

Cross-sectional area is the area of the cut face of a conductor, such as $A = \pi r^2$ for a round wire.

Reference Temperature

The reference temperature $T_0$ is the starting temperature at which $\rho_0$ or $R_0$ is known.