Concept:Metals have a positive temperature coefficient of resistance (\( \alpha > 0 \)), while intrinsic semiconductors have a negative temperature coefficient (\( \alpha < 0 \)).
Formula:$$R_{\text{metal}}(T) = R_0 (1 + \alpha T) \quad \text{and} \quad R_{\text{semiconductor}}(T) \propto e^{\frac{E_g}{2 k T}}$$
Solution:- Copper (Metal): Cooling reduces lattice vibrations (phonon scattering), allowing free electrons to move with higher mobility, which decreases its resistance.
- Germanium (Semiconductor): Cooling reduces thermal excitation, causing electrons to fall back from the conduction band into the valence band. This drastically reduces the carrier density (\( n \) and \( p \)), which increases its resistance.
Why other options are incorrect:- Option A: Metals do not increase in resistance when cooled.
- Option C: This reverses the temperature dependencies of metals and semiconductors.
- Option D: Semiconductors do not decrease in resistance when cooled.
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