Electrical Resistivity and Conductivity

1.0What is Electrical Resistivity(Specific Resistance)?

Electrical resistivity (ρ) is an intrinsic property of a material that quantifies its opposition to the flow of electric current. It's a measure of how strongly a material resists the movement of charge carriers. A high resistivity means the material is a poor conductor (an insulator), while a low resistivity means it's a good conductor

The resistivity (specific resistance) of a material is equal to the resistance of a wire of that material with unit cross sectional area and unit length.

$\rho =\frac{RA}{\ell }.If\ell =1unitandA=1unit,then\rho =R$

The SI unit of resistivity is the ohm-meter (Ω⋅m).

From Ohm’s law:

$R=\frac{V}{I}=\frac{E\ell }{neA{V}_d}=\frac{E\ell }{neA\left(\frac{eE\tau }{m_e}\right)}=\frac{m_e\ell }{n{e}^{2}A\tau }$

$R=\left(\frac{m_e}{n{e}^2\tau }\right)\times \frac{\ell }{A}=\frac{\rho \ell }{A}$

Where $\rho =\left(\frac{m_e}{n{e}^2\tau }\right)$, Resistivity

Resistivity Formula

Resistivity is defined from the relationship between resistance (R), length (L), and cross-sectional area (A) of a conductor.

$R=\rho \frac{L}{A}$

From this, the resistivity can be expressed as:

$\rho =\frac{RA}{L}$

This formula is crucial for calculating resistivity from a given resistance and dimensions of a wire.

2.0What is Electrical Conductivity?

Electrical conductivity (σ) is the reciprocal of resistivity. It measures a material's ability to conduct electric current. A high conductivity means the material is a good conductor.

The SI unit of conductivity is the siemens per meter $(S/m)oh{m}^{-1}\cdot m^{-1}$

Conductivity Formula

The relationship between conductivity and resistivity is simply:

$\sigma =\frac{1}{\rho }$

This shows that a material with high resistivity will have low conductivity, and vice versa.

Where $\sigma =\frac{n{e}^2\tau }{m}$ is Conductivity of a material

Conductivity is degree to which a material conducts electricity.

Note:

Resistivity ($\rho$) and conductivity (\sigma) depends on:

1. Nature of material
2. Temperature of material

Resistivity and conductivity do not depend on the size and shape of the material because they are characteristic properties of the conducting material.

3.0Resistivity and Temperature Dependence

For most materials, resistivity changes with temperature.

• For Metals (Conductors): As temperature increases, the positive ions in the metallic lattice vibrate more vigorously. This increases the chances of collision with the free electrons, hindering their motion. Consequently, the resistivity of metals increases with temperature. The relationship can be approximated as:
  ${\rho }_T={\rho }_0[1+\alpha (T-T_0)]$
  where ${\rho }_T$​ is the resistivity at temperature ​ T and ${\rho }_0$ is the resistivity at a reference temperature $T_0$​, and α is the temperature coefficient of resistivity.
• For Semiconductors: The number of charge carriers (electrons and holes) in a semiconductor increases exponentially with temperature. This effect dominates over the increased lattice vibrations. Therefore, the resistivity of semiconductors decreases with temperature.
• For Insulators: The resistivity of insulators is very high and decreases with increasing temperature, similar to semiconductors, but the effect is less pronounced.

4.0Factors Affecting Resistivity

Resistivity is an intrinsic property, meaning it primarily depends on the nature of the material. However, for a given material, it is significantly influenced by temperature.

Important Note: Resistivity does not depend on the dimensions (length or area) of the material. Resistance (R) depends on these factors, but resistivity (ρ) is a fundamental property of the material itself.

5.0Solved Problems

Problem 1: A wire of resistance 10Ω has a length of 50 cm and a cross-sectional area of $2\times 10^{-4}{\text{ m}}^2$. Find the resistivity of the material.

Solution:

$R=10\Omega$

$L=50\text{ cm}=0.5\text{ m}$

$A=2\times 10^{-4}{\text{ m}}^2$

Using the formula $\rho =\frac{RA}{L}$:

$\rho =\frac{(10)(2\times 10^{-4})}{0.5}=4\times 10^{-3}\Omega \cdot m$

Problem 2: The resistance of a metal wire is 20Ω at 20^∘C. If its temperature coefficient of resistivity is $2\times 10^{-3}{/}^{\circ }C$ find its resistance at 100^o C.

Solution:

$R_0=20\Omega$

$T_0=20^{\circ }C$

$T=100^{\circ }C$

$\alpha =2\times 10^{-3}{/}^{\circ }C$

The formula for resistance change with temperature is $R_T=R_0[1+\alpha (T-T_0)]$

$R_{100}=20[1+(2\times 10^{-3})(100-20)]$

$R_{100}=20[1+(2\times 10^{-3})(80)]$

$R_{100}=20[1+0.16]=20(1.16)=23.2\Omega$

Problem 3:

The current-voltage graphs for a given metallic wire at two different temperatures $T_{1}and{T}_2$ are shown in the figure. Which one is higher, $T_{1}or{T}_2$?

Solution:

Slope of I - V Curve is $\frac{1}{R}$

${\left(\frac{1}{R}\right)}_1>{\left(\frac{1}{R}\right)}_2\Rightarrow R_2>R_1\Rightarrow T_2>T_1$