Question

Define resistivity of a material. Does it depend on temperature?

Answer 1

Hint: Here, we will proceed by defining the term resistivity of a material. Then, we will derive the equation for resistivity in terms of resistance. Finally, we will discuss the variation of resistivity of conductors, semiconductors and insulators with temperature.

Complete Step-by-Step solution:
Electrical resistivity is the reciprocal of electrical conductivity. It is the measure of the ability of a material to oppose the flow of current. Materials resist the flow of electric current. Some of the materials are better at conducting electric current than the others.
Materials that conduct electrical current easily are called conductors and have a low resistivity. Those that do not conduct electricity easily are called insulators and these materials have a high resistivity.
Resistivity is defined as the electrical resistance of a conductor of unit cross-sectional area and unit length.
The resistance of any conductor is directly proportional to the length of the conductor and inversely proportional to the area of cross-section of the conductor.
If L is the length of the conductor and A is the area of cross-section of the conductor. Then, the resistance R of the conductor is given by
$$\begin{gathered} \text{R}\propto {\displaystyle \frac{\text{L}}{\text{A}}} \\ \Rightarrow \text{R}=\rho {\displaystyle \frac{\text{L}}{\text{A}}} \end{gathered}$$
Where $\rho$ is known as the resistivity of the conductor
The general formula to determine the resistivity of any conductor will be given by
$\Rightarrow \rho ={\displaystyle \frac{\text{RA}}{\text{L}}}$
The SI unit of resistance (R) is ohm ($\mathrm{\Omega }$) , SI unit of area of cross-section (A) is square meters (${\text{m}}^2$), SI unit of length is meters (m). Then, the SI unit of resistivity will be given by
SI unit of resistivity = ${\displaystyle \frac{\left(\mathrm{\Omega }\right)\left({\text{m}}^2\right)}{\left(\text{m}\right)}}=\mathrm{\Omega }\text{m}$
SI unit of resistivity is ohm-meters.
The resistivity of materials depends on the temperature according to the below mentioned equation.
$\rho ={\rho }_0\left[1+\alpha \left(\text{T}-{\text{T}}_0\right)\right]$
where $\rho$ denotes the resistivity of the material at the temperature T${}^0\text{C}$, ${\rho }_0$ denotes the resistivity of the material at the reference temperature ${\text{T}}_0{}^0\text{C}$ and $\alpha$ is the temperature coefficient of resistivity.
Variation of Resistivity in Conductors
When the temperature increases the vibrations of the metal ions in the lattice structure increases. The atoms start to vibrate with higher amplitude. These vibrations in turn cause frequent collisions between the free electrons and the other electrons. Each collision drains out some energy of the free electrons and causes them unable to move. Thus it restricts the movement of the delocalized electrons. When the collision happens the drift velocity of the electrons decreases. This means that the resistivity of the metal increases and thus current flow in the metal is decreased. The resistivity increases mean that the conductivity of the material decreases.

Variation of Resistivity in Semiconductors
When the temperature is increased the forbidden gap between the two bands becomes very less and the electrons move from the valence band to the conduction band. Thus some electrons from the covalent bonds between the atoms are free to move within the structure. This increases the conductivity of the material. The conductivity increases mean the resistivity decreases. Thus when the temperature is increased in a semiconductor, the density of the charge carriers also increases and the resistivity decreases.

Variation of Resistivity in Insulators
When the temperature is increased, the atoms of the material vibrate and it makes the valence electrons present in the valence band to shift to the conduction band. This in turn increases the conductivity of the material. When the conductivity of the material increases, it means that the resistivity decreases and so the current flow increases. Thus some insulators at room temperatures change to conductors at high temperature.

Note- Resistivity is a measure of the resistance of a given size of a specific material to electrical conduction. Resistivity may also be referred to as the specific electrical resistance, or volume resistivity, although these terms are less widely used. The resistivity is a figure that enables comparisons of the way in which different materials allow or resist current flow.