AcademyCurrent and Resistance
Academy
Resistivity
Level 1 - Physics topic page in Current and Resistance.
Principle
Resistivity describes how strongly a material opposes current density for a given electric field.
Notation
\(\rho\)
resistivity
\Omega\,m
\(\sigma\)
conductivity
S m^{-1}
\(\vec E\)
electric field inside the material
\(\mathrm{V\,m^{-1}}\)
\(\vec J\)
current density
\(\mathrm{A\,m^{-2}}\)
\(\rho_0\)
resistivity at reference temperature
\Omega\,m
\(\alpha\)
temperature coefficient of resistivity
\(\mathrm{K^{-1}}\)
\(T\)
temperature
K or ^\circ C
Method
Derivation 1: Relate field to current density
In an ohmic material, a larger internal electric field drives a proportionally larger current density.
Conductivity form
\[\vec J=\sigma\vec E\]
Resistivity definition
\[\rho=\frac{1}{\sigma}\]
Resistivity form
\[\vec E=\rho\vec J\]
Derivation 2: Connect material behavior to a wire
For a uniform wire, the field is potential drop per length and current density is current per area.
Uniform field
\[E=\frac{V}{L}\]
Uniform current density
\[J=\frac{I}{A}\]
Substitute into \(E=\rho J\)
\[\frac{V}{L}=\rho\frac{I}{A}\]
Wire relation
\[\frac{V}{I}=\rho\frac{L}{A}\]
Derivation 3: Temperature dependence
For many metals over modest temperature ranges, resistivity is modeled as changing approximately linearly with temperature.
Linear model
\[\rho(T)=\rho_0\left[1+\alpha(T-T_0)\right]\]
Positive coefficient
\[\alpha>0\Rightarrow\rho\ \text{increases with }T\]
Rules
These are the compact resistivity relations.
Ohmic material
\[\vec E=\rho\vec J\]
Conductivity
\[\sigma=\frac{1}{\rho},\qquad \vec J=\sigma\vec E\]
Uniform wire
\[\frac{V}{I}=\rho\frac{L}{A}\]
Temperature model
\[\rho(T)=\rho_0[1+\alpha(T-T_0)]\]
Examples
Question
A material has
\[\rho=2.0\times10^{-8}\,\Omega\,\mathrm{m}\]
and current density \[3.0\times10^6\,\mathrm{A\,m^{-2}}\]
Find the required electric field.Answer
\[E=\rho J=(2.0\times10^{-8})(3.0\times10^6)=6.0\times10^{-2}\,\mathrm{V\,m^{-1}}\]
Checks
- Resistivity is a material property; resistance also depends on geometry.
- Conductivity is the reciprocal of resistivity.
- Use cross-sectional area, not surface area, in \(J=I/A\).
- The linear temperature model is approximate and material-dependent.