Talha's Physics Academy
Resistivity and Temperature Coefficient
Video Lecture
Watch the complete video lecture below to understand resistivity, temperature coefficient of resistance, and resistivity derivations.
Resistivity
Derivation of Resistivity Expression
The electrical resistance of a conductor depends upon two primary geometrical factors: its length ($L$) and its cross-sectional area ($A$).
- The resistance $R$ of a conductor is directly proportional to its length $L$:
$$R \propto L \quad \text{--- (i)}$$
- The resistance $R$ of a conductor is inversely proportional to its cross-sectional area $A$:
$$R \propto \frac{1}{A} \quad \text{--- (ii)}$$
Combining relations (i) and (ii), we get:
$$R \propto \frac{L}{A}$$
$$R = \rho \frac{L}{A}$$
Where $\rho$ (rho) is a constant of proportionality known as resistivity or specific resistance. Rearranging the formula to solve for resistivity:
$$\rho = R \frac{A}{L}$$
Unit of Resistivity: Its SI unit is ohm-meter ($\Omega\cdot\text{m}$).
Temperature Coefficients
When an electric current flows through a conductor, moving electrons collide with vibrating metal ions. As temperature increases, metal ions vibrate with greater amplitude, causing more frequent collisions with electrons and thereby increasing electrical resistance.
1. Temperature Coefficient of Resistance
It is defined as "the fractional change in resistance per degree change in temperature."
Consider a wire having resistance $R_0$ at $0\,^\circ\text{C}$ and resistance $R_t$ at a higher temperature $t\,^\circ\text{C}$. The change in resistance is $\Delta R = R_t - R_0$ for a temperature change $\Delta t = t - 0 = t$.
The change in resistance is directly proportional to the initial resistance $R_0$ and the temperature change $\Delta t$:
$$\Delta R \propto R_0 \Delta t$$
$$\Delta R = \alpha R_0 \Delta t$$
Where $\alpha$ is the temperature coefficient of resistance:
$$\alpha = \frac{\Delta R}{R_0 \Delta t} = \frac{R_t - R_0}{R_0 t}$$
Rearranging for $R_t$:
$$R_t = R_0 (1 + \alpha t)$$
2. Temperature Coefficient of Resistivity
It is defined as "the fractional change in resistivity per degree change in temperature."
Since resistivity ($\rho$) is directly proportional to the resistance of the metal, the temperature dependence follows a similar relation:
$$\rho_t = \rho_0 (1 + \alpha t)$$
Where $\rho_0$ is the resistivity at $0\,^\circ\text{C}$, $\rho_t$ is the resistivity at $t\,^\circ\text{C}$, and $\alpha$ is the temperature coefficient of resistivity.
The temperature coefficients help quantify how sensitive a material's electrical properties are to thermal variations.
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