A metallic conductor connected to a battery is heated due to the passage of current. The quantity which does not change is:
Answer & explanation
Correct answer: option 3
The correct answer is Option (3) → Number of free electrons per unit volume
The number density \( n \) of free electrons is the number of free electrons per unit volume of the metal. In a metal every atom gives up a fixed number of valence electrons, so \( n \) is a property of the material and does not depend on temperature. Heating only makes the ions vibrate more, so the electrons collide more often and the relaxation time \( \tau \) falls.
\( \displaystyle \rho = \frac{m}{n e^{2} \tau} \)
Option (1) Resistance — Incorrect. The resistance of a metal rises with temperature, because its resistivity rises while the length and area hardly change.
Option (2) Resistivity — Incorrect. A smaller relaxation time \( \tau \) in \( \rho = m/(ne^{2}\tau) \) means a larger resistivity.
Option (3) Number of free electrons per unit volume — Correct. The number of free electrons per unit volume is fixed by the metal itself, so heating does not change it.
Option (4) Drift speed of electrons — Incorrect. The drift speed is \( v_{d} = eE\tau/m \); the battery keeps the field \( E \) the same, but the relaxation time falls on heating, so the drift speed decreases.
NCERT (2026-27):
Temperature Dependence of Resistivity – “… As we increase temperature, average speed of the electrons, which act as the carriers of current, increases resulting in more frequent collisions. The average time of collisions \( \tau \), thus decreases with temperature. In a metal, n is not dependent on temperature to any appreciable extent and thus the decrease in the value of \( \tau \) with rise in temperature causes \( \rho \) to increase as we have observed.”