Animated Solution for Physics - Current Electricity: A piece of copper and another of germanium are cooled from room temperature to $80\text{ K}$. The resistance of

The problem asks us to determine the effect of cooling on the electrical resistance of two distinct materials: Copper and Germanium. To solve this, we must dive into the microscopic world of metals and semiconductors.

We are given a piece of Copper (Cu) and a piece of Germanium (Ge). Copper is a classic example of a metal (or conductor), while Germanium is a well-known semiconductor.

The problem states that both materials are cooled from room temperature (approximately $300\text{ K}$) down to $80\text{ K}$. To predict how their resistance changes, we need to understand what causes resistance in each type of material.

Copper

In a metal like Copper, there is an abundance of free electrons that act as charge carriers. The resistance in a metal arises primarily from the collisions between these free electrons and the vibrating positive ions in the crystal lattice.

The resistivity $\rho$ is given by the formula: where $n$ is the number density of free electrons and $\tau$ is the relaxation time (the average time between collisions).

When the temperature is high, the lattice ions vibrate with greater amplitude, increasing the frequency of collisions. This decreases the relaxation time $\tau$, which in turn increases the resistance.

Conversely, when we cool the Copper from $300\text{ K}$ to $80\text{ K}$, the thermal agitation of the lattice ions decreases. The electrons can travel further before colliding, meaning the relaxation time $\tau$ increases. As a result, the resistance of Copper decreases.

Germanium

Semiconductors like Germanium behave entirely differently. At absolute zero ($0\text{ K}$), a pure semiconductor acts as a perfect insulator because all its electrons are tightly bound in covalent bonds.

As the temperature rises, thermal energy breaks some of these bonds, exciting electrons from the valence band into the conduction band and leaving behind positively charged holes. Both these electrons and holes act as charge carriers. The number density of charge carriers $n$ increases exponentially with temperature according to: where $E_g$ is the bandgap energy and $k$ is the Boltzmann constant.

When we cool the Germanium, we are removing thermal energy. Fewer electrons have the energy required to jump across the bandgap. This causes a drastic reduction in the number of available charge carriers $n$. Although the relaxation time $\tau$ also increases slightly due to reduced lattice vibrations, the exponential drop in carrier concentration $n$ completely dominates.

Therefore, with fewer charge carriers available to conduct electricity, the resistance of Germanium increases.

By understanding the fundamental microscopic differences between metals and semiconductors, the answer becomes clear: - Cooling a metal (Copper) reduces lattice vibrations, allowing electrons to flow more easily, so its resistance decreases. - Cooling a semiconductor (Germanium) deprives it of the thermal energy needed to generate charge carriers, so its resistance increases.

Thus, the correct option is (d): copper decreases and germanium increases.