Question

The resistivity $\rho$ of a metal increases with rise in temperature because :

• A. only relaxation time ‘$\tau$’ of electrons decreases with temperature.
• B. only number of electrons per unit volume ‘$n$’ increases appreciably.
• C. ‘$\tau$’ decreases with temperature but ‘$n$’ does not change appreciably.
• D. ‘$\tau$’ decreases with temperature and ‘$n$’ increases.

Hint:

In metals: $n \approx \text{constant}$, $\tau \downarrow \implies \rho \uparrow$.
In semiconductors: $n \uparrow\uparrow$ exponentially with temperature, dominating over $\tau \downarrow \implies \rho \downarrow$.

Answer 1

The Correct Option is C

Concept:
• Resistivity of a metallic conductor is given by $\rho = \frac{m}{n e^2 \tau}$, where $m$ is electron mass, $e$ is electron charge, $n$ is free electron concentration, and $\tau$ is relaxation time.

Step 1: Behavior of free electron density $n$ in metals
In metals, free electron density $n$ is extremely high ($\approx 10^{28}\text{ m}^{-3}$) and determined by metallic bonding.
Raising the temperature does not appreciably alter $n$ because thermal energy is insufficient to liberate significant additional free electrons.

Step 2: Behavior of relaxation time $\tau$ with temperature
As temperature rises, thermal energy increases the amplitude of vibration of lattice ions.
Free electrons collide much more frequently with vibrating ions.
The average time interval between consecutive collisions (relaxation time $\tau$) decreases significantly ($\tau \downarrow$).

Step 3: Effect on resistivity
From $\rho = \frac{m}{n e^2 \tau}$, since $n$ remains almost constant while $\tau$ decreases, resistivity $\rho$ increases proportionally ($\rho \uparrow$).

Step 4: Conclusion
Resistivity increases because $\tau$ decreases with temperature while $n$ does not change appreciably, corresponding to option (C).