NEET 2020 Physics Question 144 Solution

According to the Drude-Lorentz free electron model, the electrical resistivity of a solid is given by: $$\rho = \frac{m}{n e^2 \tau}$$ where $$m$$ is electron mass, $$e$$ is electron charge, $$n$$ is charge carrier number density, and $$\tau$$ is the average relaxation time between collisions.

In metals, the carrier concentration $$n$$ is independent of temperature ($$\sim 10^{28}\text{ m}^{-3}$$). As temperature increases, lattice thermal vibrations amplify, reducing relaxation time $$\tau$$. Consequently, $$\rho$$ increases with temperature, yielding a POSITIVE temperature coefficient of resistance ($$\alpha > 0$$).

In semiconductors and insulators, valence electrons are separated from the conduction band by an energy band gap ($$E_g$$). As temperature rises, thermal energy excites electrons across the band gap into the conduction band, causing the carrier density $$n$$ to increase exponentially according to Boltzmann distribution: $$n(T) \propto e^{-E_g / (2 k_B T)}$$ This immense increase in $$n$$ overwhelmingly compensates for any modest decrease in relaxation time $$\tau$$. Consequently, resistivity and resistance decrease exponentially as temperature increases.

Because $$\frac{dR}{dT} < 0$$, both insulators and semiconductors possess a NEGATIVE temperature coefficient of resistance ($$\alpha < 0$$).