Using ρ = m/(ne²τ), explain how the resistivity of (i) a metal and (ii) a semiconductor changes with temperature. Draw the graph of ρ against T for each.

1. 1 · Start from the formula. ρ = m/(ne²τ): ρ depends on the carrier density n and the relaxation time τ.
2. 2 · Metal. n is nearly constant. Heating increases ion vibrations, so τ falls and ρ rises. Over a limited range ρ_T = ρ₀[1 + α(T − T₀)] with α positive.
3. 3 · Semiconductor. Heating frees many more carriers. The rise in n outweighs the fall in τ, so ρ falls and α is negative.
4. 4 · Draw the graphs. [Two graphs of ρ against T: copper curving up then nearly straight; silicon falling steeply then levelling off; axes labelled with units]
5. 5 · Real life. Manganin, with a tiny α, is used for standard resistors.
6. 6 · Close in one line. So, ρ increases with temperature for metals and decreases for semiconductors.

Scoring tip: Name both factors, n and τ, and say which one wins in each case.