Question

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.

• A. Increases, thermal velocity of the electrons decreases
• B. Decreases, thermal velocity of the electrons decreases
• C. Increases, thermal velocity of the electrons increases
• D. Decreases, thermal velocity of the electrons increases

Answer 1

The Correct Option is D

The drift velocity of an electron in a metal wire placed under a constant potential difference is influenced by changes in temperature. The drift velocity \( v_d \) is determined by the equation:

\[ v_d = \frac{eE\tau}{m} \]

where:

• \( e \) is the electron charge
• \( E \) is the electric field
• \( \tau \) is the mean free time between collisions
• \( m \) is the electron mass

As temperature increases, the lattice ions in the metal vibrate more vigorously. This increased vibration causes a rise in the collision rate of electrons within the lattice, effectively reducing the mean free time \(\tau\) between collisions. Consequently, since \(\tau\) appears in the numerator, a decrease in \(\tau\) leads to a decrease in drift velocity \(v_d\).

Furthermore, the thermal velocity of electrons, which is the speed component caused by thermal agitation, increases with temperature as electrons gain more kinetic energy. Therefore, the correct statement is: the drift velocity of the electron decreases, and the thermal velocity increases.

Thus, the correct answer is: Decreases, thermal velocity of the electrons increases

Answer 2

Let's analyze the effect of temperature on drift velocity and thermal velocity of electrons in a metal wire.

1. Drift Velocity (v_d):

The drift velocity is given by:

v_d = (eEτ) / m

Where:

• e is the charge of an electron
• E is the electric field
• τ is the relaxation time (average time between collisions)
• m is the mass of an electron

When the temperature increases:

• The number of collisions between electrons and the lattice ions increases.
• The relaxation time (τ) decreases.
• Since v_d is directly proportional to τ, the drift velocity (v_d) decreases.

2. Thermal Velocity (v_th):

The thermal velocity is the random velocity of electrons due to their thermal energy. It is given by:

v_th ∝ √T

Where T is the temperature.

When the temperature increases:

• The thermal energy of the electrons increases.
• The thermal velocity (v_th) increases.

Therefore, when the temperature of the metal wire increases, the drift velocity of the electron decreases, and the thermal velocity of the electrons increases.

The correct answer is:

Option 4: Decreases, thermal velocity of the electrons increases