Temperature Dependence of Resistivity

The relation

• Over a limited range of temperature that is not too large, the resistivity of a metallic conductor is approximately

$$\rho_T = \rho_0\left[1 + \alpha (T - T_0)\right]$$

• $\rho_T$ is the resistivity at temperature $T$, $\rho_0$ the resistivity at a reference temperature $T_0$, and $\alpha$ the temperature coefficient of resistivity.
• Since $R = \rho l/A$ and the dimensions barely change, the same relation may be used for resistance:

$$R_T = R_0\left[1 + \alpha(T - T_0)\right] \qquad \Rightarrow \qquad \alpha = \dfrac{R_T - R_0}{R_0(T - T_0)}$$

• $\alpha$ is the fractional increase in resistivity per unit rise in temperature. Its dimension is (temperature)$^{-1}$, so the unit is K$^{-1}$ or $^{\circ}$C$^{-1}$. For metals $\alpha$ is positive.
• The relation implies that a graph of $\rho_T$ against $T$ is a straight line. At temperatures much below $0\ ^\circ$C the real graph deviates considerably from a straight line, so the formula may only be used over a limited range around $T_0$.

Explaining the sign of $\alpha$ — the marks are here

• Start always from the microscopic expression

$$\rho = \dfrac{1}{\sigma} = \dfrac{m}{ne^{2}\tau}$$

• Resistivity depends inversely on both the free-carrier density $n$ and the average time $\tau$ between collisions.
• Raising the temperature increases the average speed of the electrons and makes the lattice ions vibrate more, so collisions become more frequent and $\tau$ decreases. This happens in every material.
• In a metal $n$ is essentially independent of temperature. With $n$ fixed and $\tau$ falling, $\rho$ increases: hence $\alpha$ is positive for a metal.
• In a semiconductor or insulator, $n$ increases with temperature, and this increase more than compensates the decrease in $\tau$. Hence $\rho$ decreases with temperature and $\alpha$ is negative.

The three graphs to be able to sketch

• Copper (a metal): $\rho$ against $T$ is a curve that starts from a very small value near $T = 0\ \text{K}$, rises, and becomes very nearly a straight line of small positive slope over the ordinary range of temperatures. Label the axes $\rho_T$ and $T$ and mark the near-linear region.
• Nichrome (an alloy): $\rho$ against absolute temperature is a straight line with a very small positive slope — nearly flat over the working range.
• A typical semiconductor: $\rho$ against $T$ is a curve falling steeply as $T$ increases, flattening at high temperature.
• Mercury near 4 K: the resistance decreases smoothly on cooling and then drops abruptly to zero at a critical (transition) temperature $T_c \approx 4.2\ \text{K}$. Below $T_c$, mercury offers no resistance at all to the flow of current — it becomes a superconductor.

Reading $I$–$V$ graphs for resistance and temperature

• On a plot of $I$ (vertical) against $V$ (horizontal), $R = \Delta V / \Delta I$, so $R$ is the reciprocal of the slope: the steeper line is the smaller resistance.
• Two wires of equal length and diameter at constant temperature, lines X (less steep) and Y (steeper): X carries less current for the same $V$, so $R_X > R_Y$; since $R = \rho l/A$ with $l$ and $A$ equal, X is made of the material with the larger resistivity.
• The same metallic wire at two temperatures, $T_1$ steeper than $T_2$: $R_1 < R_2$; the resistance of a metal increases with temperature, so $T_1$ is the lower temperature.

Worked numericals in NCERT's own style

• Nichrome toaster. At $27.0\ ^\circ$C, $R_1 = 75.3\ \Omega$. On 230 V the steady current is 2.68 A, so $R_2 = 230/2.68 = 85.8\ \Omega$. With $\alpha = 1.70\times10^{-4}\ ^\circ\text{C}^{-1}$,

$$T_2 - T_1 = \dfrac{85.8 - 75.3}{75.3 \times 1.70\times10^{-4}} = 820\ ^\circ\text{C} \qquad \Rightarrow \qquad T_2 = 847\ ^\circ\text{C}$$

• Platinum resistance thermometer. With $R_0 = 5\ \Omega$ at the ice point, $R_{100} = 5.23\ \Omega$ at the steam point and $R_t = 5.795\ \Omega$ in the bath,

$$t = \dfrac{R_t - R_0}{R_{100} - R_0}\times 100 = \dfrac{5.795 - 5}{5.23 - 5}\times 100 = \dfrac{0.795}{0.23}\times 100 = 345.65\ ^\circ\text{C}$$

Standard results

• $\rho_T = \rho_0[1+\alpha(T-T_0)]$, $R_T = R_0[1+\alpha(T-T_0)]$, $\alpha$ in K$^{-1}$.
• Metal: $\tau$ falls, $n$ fixed, $\rho$ rises, $\alpha > 0$. Semiconductor: $n$ rises faster than $\tau$ falls, $\rho$ falls, $\alpha < 0$. Alloy: $\alpha \approx 0$.
• On an $I$–$V$ graph, $R \propto 1/\text{slope}$.
• Mercury becomes superconducting below $T_c \approx 4.2\ \text{K}$.
• Platinum thermometer: $t = 100(R_t-R_0)/(R_{100}-R_0)$.