A battery is connected across a resistor. The resistor is heated due to the current drawn from the battery. Choose the quantity from the following which remains unchanged:
Answer & explanation
Correct answer: option 1
The correct answer is Option (1) → Number of free electrons
As the resistor (a metallic conductor) heats up, its ions vibrate more strongly and the free electrons collide with them more often, so the relaxation time \( \tau \) falls. The number of free electrons in a metal, however, is fixed by the material itself (roughly one or two per atom) and does not change with temperature.
\( \displaystyle \rho = \frac{m}{ne^{2}\tau} \)
\( \displaystyle v_d = \frac{eE}{m}\,\tau \)
With n unchanged and \( \tau \) smaller, the resistivity rises and the drift velocity falls.
Why the other options are incorrect
Option (2) Resistance — Incorrect. Resistance \( R = \rho l/A \) increases because the resistivity increases on heating.
Option (3) Resistivity — Incorrect. Resistivity increases with temperature because the relaxation time decreases.
Option (4) Drift velocity — Incorrect. Drift velocity decreases, since \( v_d \) is proportional to the relaxation time, which becomes shorter.
Note: A similar question was also asked in CUET (UG) 2025 on 21-05-2025 (Slot 2).