Introduction:
When a potential difference (V) is applied across a conductor, the movement of electrons in the conductor is influenced. The thermal speed of electrons also changes due to this potential difference.

Proportional to V:
The thermal speed of electrons in a conductor is proportional to the potential difference (V) applied across it. The higher the potential difference, the greater the thermal speed of electrons in the conductor.

Explanation:
The thermal speed of electrons in a conductor is determined by the temperature of the conductor. However, when a potential difference is applied across the conductor, the electrons experience a force that causes them to move in a particular direction. This force is known as the electric field.

The electric field causes the electrons to accelerate and gain kinetic energy. The kinetic energy of the electrons is proportional to the potential difference applied across the conductor. Therefore, the thermal speed of electrons in the conductor also increases.

The relationship between the thermal speed of electrons and the potential difference applied across the conductor is described by the following equation:

v = (2kT/m)^(1/2)

Where v is the thermal speed of electrons, k is the Boltzmann constant, T is the temperature of the conductor, and m is the mass of the electron.

This equation shows that the thermal speed of electrons is directly proportional to the square root of the temperature of the conductor. However, the temperature of the conductor remains constant when a potential difference is applied across it. Therefore, the thermal speed of electrons is directly proportional to the potential difference applied across the conductor.

Conclusion:
In conclusion, the thermal speed of electrons in a conductor is proportional to the potential difference applied across it. The higher the potential difference, the greater the thermal speed of electrons in the conductor.