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

Question

Prepp Team · Accepted Answer

Understanding Electron Velocity in Metal Wires

When a metal wire is connected to a constant potential difference, an electric field is established within the wire. This electric field exerts a force on the free electrons, causing them to drift in a direction opposite to the field. This average velocity with which the electrons drift is called the drift velocity.

Simultaneously, the electrons in a metal are constantly in random motion due to their thermal energy. This random motion velocity is called the thermal velocity. The thermal velocity of electrons is generally much higher than the drift velocity under normal conditions.

Impact of Increasing Temperature on Electron Velocity

Let's analyse how increasing the temperature of the metal wire affects both the drift velocity and the thermal velocity of the electrons.

Effect on Thermal Velocity

Temperature is a measure of the average kinetic energy of the particles in a substance. In the case of a metal, increasing the temperature means increasing the average kinetic energy of the free electrons and the vibrating lattice atoms (ions).

As temperature increases, the random thermal motion of the electrons becomes more vigorous.
The average speed of the electrons due to this random motion increases significantly.
Therefore, the thermal velocity of the electrons increases when the temperature of the metal wire increases.

Effect on Drift Velocity

The drift velocity of electrons in a metal wire is related to the electric field, the charge of the electron, its mass, and the average time between collisions ($\tau$) with the lattice ions or other imperfections. The relationship is often expressed as:

$\vec{v}_d = - \frac{e\vec{E}}{m} \tau$

where $e$ is the electron charge, $\vec{E}$ is the electric field, $m$ is the electron mass, and $\tau$ is the average relaxation time (time between collisions).

When the temperature of the metal wire increases:

The thermal vibrations of the lattice atoms become more intense.
This increases the frequency of collisions between the drifting electrons and the vibrating lattice atoms.
An increased collision frequency means a shorter average time between collisions ($\tau$) decreases.
Since the potential difference (and thus the electric field $\vec{E}$) is kept constant, a decrease in $\tau$ directly leads to a decrease in the magnitude of the drift velocity ($v_d$).

Alternatively, we can consider the effect on resistance. For most metals, resistance increases with temperature. Since the potential difference $V$ is constant, by Ohm's Law ($V = IR$), if resistance $R$ increases, the current $I$ must decrease ($I = V/R$). The current $I$ is also related to the drift velocity $v_d$ by the equation $I = nAe v_d$, where $n$ is the number density of free electrons, $A$ is the cross-sectional area of the wire, and $e$ is the electron charge. Since $n$, $A$, and $e$ are relatively constant with temperature, a decrease in current $I$ implies a decrease in the drift velocity $v_d$.

Conclusion on Velocities and Temperature

Based on the analysis:

When the temperature of a metal wire increases, the thermal velocity of the electrons increases.
When the temperature of a metal wire increases (under constant potential difference), the drift velocity of the electrons decreases.

Comparing this with the given options, the statement "Decreases, thermal velocity of the electrons increases" correctly describes what happens to the drift velocity and the thermal velocity, respectively, when the temperature of the metal wire increases.

Revision Table: Temperature Effects on Metal Wire Properties

Property	Effect of Increasing Temperature	Explanation
Thermal Velocity	Increases	Higher thermal energy leads to faster random electron motion.
Collision Frequency	Increases	More vigorous lattice vibrations cause more frequent collisions.
Relaxation Time ($\tau$)	Decreases	Average time between collisions becomes shorter.
Resistance ($R$)	Increases	Increased collision frequency hinders electron flow.
Current ($I$, for constant $V$)	Decreases	According to Ohm's Law ($I=V/R$).
Drift Velocity ($v_d$)	Decreases	Reduced relaxation time or decreased current density ($v_d = I / (nAe)$).

Additional Information: Drift Velocity vs. Thermal Velocity

It is important to understand the vast difference in magnitude between thermal velocity and drift velocity.

Thermal Velocity: At room temperature, the thermal velocity of electrons in a metal is very high, typically around $10^5$ to $10^6$ m/s. This motion is random in direction, so the average thermal velocity vector is zero, but the average speed is high.
Drift Velocity: Under typical electric fields in a wire carrying current, the drift velocity is very low, on the order of $10^{-4}$ m/s (fractions of a millimeter per second). This is the slow, directed movement superimposed on the rapid, random thermal motion.

Even though the drift velocity is small, the sheer number density ($n$) of free electrons in a metal is extremely high (about $10^{28}$ electrons per cubic meter), which results in a significant electric current ($I = nAe v_d$).

Increasing temperature dramatically increases the thermal velocity but significantly impedes the directed drift motion, thus decreasing the drift velocity and increasing resistance.

Prepp Team · Answer

Increases, thermal velocity of the electrons decreases

Prepp Team · Answer

Decreases, thermal velocity of the electrons decreases

Prepp Team · Answer

Increases, thermal velocity of the electrons increases