Question

On increasing the temperature of a conductor. Its resistance increases because
A. Relaxation time decreases
B. Mass of the electrons increase
C. Electron density decreases
D. None of the above

Answer 1

Hint:To answer the given question, if we consider Ohm’s law then we cannot get any relation between resistance and temperature. So we will take a different approach. We know that resistance also depends on resistivity, length and area of cross section. Therefore, we use this dependence to know the reason for an increase in resistance with an increase in temperature of a conductor.

Formula Used:
According to to Ohm’s law,
$R = \dfrac{V}{I}$
Where, $I$ is the current and $V$ is the potential difference.
The formula of resistance is,
$R = \dfrac{{\rho l}}{A}$
where $\rho $ is the specific resistance (or resistivity), $l$ is the length of each side of the given manganin cube and $A$ is the area of the cross section of the conductor.

Complete step by step solution:
Let us first see if Ohm’s law gives any relationship between resistance and temperature. According to Ohm’s law, the voltage across two points of a conductor is directly proportional to the current flowing through it. That is, $R = \dfrac{V}{I}$ .

From this we get that Ohm’s law just gives dependence of resistance on Voltage across the two points of conductor and the current flowing through it. Therefore, we need a different approach. We know that, $R = \dfrac{{\rho l}}{A}$ which tells the dependence of resistance on resistivity, length and area of cross section of the conductor.

Now, for a given value of resistivity, length and area of cross section of a conductor, if the temperature is increased then the free electron's thermal velocity also rises. As a result of this increase in velocity, there are also more collisions between free electrons. Additionally, when the number of collisions rises, the free flow of current and the relaxation time decreases as well, which causes resistance to rise.

Hence, option A is the correct answer.

Note: The time elapsed between two subsequent electron collisions in a conductor when current is flowing through it is known as the relaxation time. It is denoted by $\tau $ . As the temperature of a conductor rises, its resistivity also rises and the conductor's relaxation period shortens.