Relationship between thermal speed of electrons and potential difference:
When a potential difference (V) is applied across a conductor, the thermal speed of electrons is directly proportional to the magnitude of the potential difference. This relationship can be explained by the following factors:

Electric Field:
- The potential difference creates an electric field within the conductor.
- This electric field exerts a force on the free electrons, causing them to accelerate.

Acceleration of Electrons:
- As the electrons accelerate under the influence of the electric field, their speed increases.
- The thermal speed of electrons is a measure of the average speed of electrons within the conductor.

Direct Proportionality:
- The thermal speed of electrons is directly proportional to the magnitude of the potential difference applied.
- This means that as the potential difference increases, the thermal speed of electrons also increases.

Effect on Current:
- The increased thermal speed of electrons results in a higher drift velocity and thus a higher current flow through the conductor.
- This is the basis for the relationship between potential difference and the thermal speed of electrons in a conductor.
In conclusion, the thermal speed of electrons in a conductor is directly proportional to the potential difference applied across it. This relationship is fundamental in understanding the behavior of electrons in the presence of an electric field.