Question

A piece of copper and another of germanium are cooled from room temperature to 80 K. Then, which of the following is true regarding their resistances?
${\text{A}}{\text{.}}$ Each of them increase
${\text{B}}{\text{.}}$ Each of them decreases
${\text{C}}{\text{.}}$ Copper increases and germanium decreases
${\text{D}}{\text{.}}$ Copper decreases and germanium increases

Answer 1

Hint: Here, we will proceed by giving the relation between resistance and resistivity. Then, we will discuss the variation of resistivity with respect to temperature for metals and semiconductors.
Formulas Used: ${\text{R}} = \dfrac{{\rho {\text{L}}}}{{\text{A}}}$ and $\rho = {\rho _0}\left[ {1 + \alpha t\left( {{\text{T}} - {{\text{T}}_0}} \right)} \right]$.

Complete Step-by-Step solution:
Resistivity is a material 's fundamental property, and it shows how strongly the material opposes the present.
The relation between resistance R and resistivity $\rho $ is given as under.
${\text{R}} = \dfrac{{\rho {\text{L}}}}{{\text{A}}}$ where L denotes the length of the conductor and A denotes the area of cross-section of the material
The resistivity of any material depends on its temperature and thus, the resistance of that material also depends on the material 's temperature.
The difference in the resistivity of material with temperature in various materials is significant.
Metals: The number density n of free electrons in most metals does not change with temperature but an increase in temperature raises the vibration rate of the metal's lattice ions. Consequently, the collision of free electrons with ions or atoms when traveling towards the conductor's positive end is more common, leading to a decrease in relaxation time. Thus conductor resistivity increases with temperature rise. Resistivity increases at a higher T-power at low temperature.
The temperature dependence of a metal's resistivity is determined by the relation
$\rho = {\rho _0}\left[ {1 + \alpha t\left( {{\text{T}} - {{\text{T}}_0}} \right)} \right]$
where $\rho $ and ${\rho _0}$ denotes the resistivity at temperature T and ${{\text{T}}_0}$ respectively and $\alpha $ is called the temperature coefficient of resistivity.
For Conductors, the value of the metal temperature coefficient is positive , indicating that their resistivity increases with temperature rises and a metal 's resistance often rises with temperature increases and vice versa. The resistivity increases linearly over a temperature range of around 500 K above the room temperature for most metals.
Semiconductors: The value of the temperature coefficient is negative in semiconductors. It means that semiconductor resistivity decreases as temperature increases and semiconductor resistance decreases with temperature increases and vice versa.
When a piece of copper is cooled from room temperature (298 K) to 80 K, the temperature of the copper metal is decreased. With the decrease in temperature of a metal, the resistivity and hence resistance also decreases. So, the resistance of copper pieces will decrease when cooled to 80 K.
When a piece of germanium is cooled from room temperature (298 K) to 80 K, the temperature of the germanium semiconductor is decreased. With the decrease in temperature of a semiconducting material, the resistivity and hence resistance increases. So, the resistance of germanium pieces will increase when cooled to 80 K.
Therefore, the resistance of copper decreases and that of germanium increases when both are cooled to 80 K.
Hence, option D is correct.

Note- The calculation of the tendency of a substance to impede the movement of electrical current through it is called resistance, and is denoted by R. The resistance units are ohm. Resistivity is defined as the resistance of a unit length material and cross section area of a device. It is denoted by $\rho $. The resistivity unit SI is the ohm-metre.