Question

A piece of copper and another of germanium are cooled from room temperature to $100K$. What will happen to their conductivities? Explain.

Answer 1

Hint: The behavior of conductor and semiconductor varies to the change in temperature. Here copper is a conductor and germanium is a semiconductor. To know about the conductivity, we want to know how the resistance changes according to temperature.

Complete solution:
We are given a piece of copper which is a conductor and a piece of germanium which is a semiconductor. These two pieces are cooled to $100K$ from the room temperature. The temperature decreases.
The resistance of the conductor is proportional to the temperature.
$R \propto T$
When the temperature decreases the resistance also decreases. This leads to the conductivity to increase. So, the rise in temperature can decrease the electrical conductivity of the conductor. When the temperature is raised this will cause the electrons in the conduction band to collide each other. This will gradually decrease the conductivity.
In the case of a semiconductor, the resistance is inversely proportional to the temperature.
$R \propto \dfrac{1}{T}$
Therefore, when the temperature decreases the resistance of the semiconductor will increase. And this leads to conductivity to decrease. So, the rise in temperature can increase the electrical conductivity of the semiconductor. When the temperature is raised in the semiconductor more electrons will occupy the conduction band thus increasing the conductivity.
Thus, when the temperature is decreased to $100K$ from temperature the conductivity of copper increases and conductivity of germanium decreases.

Note: We have to note that the resistance and the conductivity are inversely proportional. The metal has resistance means it resists the movement of electrons. This will decrease the conductivity hence the conductance is through electrons.