A metal wire is subjected to a constant potential difference. When the temperature of the metal wire increases, the drift velocity of the electron in it_________.

Fill in the blank with the correct answer from the options given below.

The correct option is: decreases, thermal velocity of the electrons increases.

Let us understand why this is the correct choice step-by-step.

1. Effect of Temperature on Thermal Velocity:
The thermal motion of free electrons in a metal is due to their thermal energy. The root-mean-square (rms) thermal velocity of electrons is directly related to the absolute temperature ($T$) by the formula:
$v_{\mathrm{rms}}=\sqrt{\frac{3k_{B}T}{m}}$
where $k_B$ is the Boltzmann constant and $m$ is the mass of an electron. As the temperature of the metal wire increases, the thermal energy of the electrons increases, which directly causes the thermal velocity of the electrons to increase.

2. Effect of Temperature on Drift Velocity:
The drift velocity ($v_d$) of electrons in a conductor under a constant potential difference is given by the relation:
$v_d=\frac{eV\tau }{mL}$
where $e$ is the charge of an electron, $V$ is the constant potential difference across the wire of length $L$, $m$ is the mass of the electron, and $\tau$ is the relaxation time (the average time between successive collisions of electrons with the metal ions).

When the temperature of the metal increases, the metal ions vibrate more vigorously. As a result, the electrons collide with these vibrating ions much more frequently. This increased frequency of collisions causes the relaxation time ($\tau$) to decrease.

Since the drift velocity ($v_d$) is directly proportional to the relaxation time ($\tau$), a decrease in $\tau$ causes the drift velocity of the electrons to decrease.

Conclusion:
Therefore, when the temperature of the metal wire increases, the drift velocity of the electrons in it decreases, while the thermal velocity of the electrons increases.