A metallic resistor R is connected across a battery of emf E . In the…

The correct answer is Option (4) → increase

In a metal the resistivity depends on the relaxation time \( \tau \), the average time between collisions of the free electrons with the lattice.

    \( \displaystyle \rho = \frac{m}{ne^{2}\tau} \)

Cooling reduces the lattice vibrations, so the electrons collide less often and \( \tau \) increases. Hence \( \rho \) and the resistance R decrease, while the emf E is unchanged.

    \( \displaystyle I = \frac{E}{R}\ \text{increases} \)

Note: A similar question was also asked in CUET (UG) 2026 on 31-05-2026 (Slot 2).

NCERT (2026-27):
Temperature Dependence of Resistivity – “… We can qualitatively understand the temperature dependence of resistivity, in the light of our derivation of Eq. (3.23). From this equation, resistivity of a material is given by

    \( \displaystyle \rho = \frac{1}{\sigma} = \frac{m}{n e^{2} \tau} \qquad (3.27) \)

\( \rho \) thus depends inversely both on the number \( n \) of free electrons per unit volume and on the average time \( \tau \) between collisions. 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.”