The temperature dependence of the resistivity \( \rho(T) \) of semiconductors, insulators and metals is significantly based on the following factors:
(a) The number of charge carriers can change with the temperature \( T \).
(b) The time interval between two successive collisions can depend on \( T \).
(c) The length of the material can be a function of \( T \).
(d) The mass of the carriers is a function of \( T \).
Which of the above are correct?
Answer & explanation
Correct answer: option 3
The correct answer is Option (3) → (a) and (b)
From the drift of electrons, the resistivity of a material is \( \rho=m/(ne^{2}\tau) \). The charge \( e \) and mass \( m \) of the carriers are fixed. So the temperature can change \( \rho \) only through the number density of carriers \( n \) and the average time between collisions \( \tau \).
(a) The number of charge carriers can change with the temperature \( T \). — True. In insulators and semiconductors, \( n \) increases with temperature. So their resistivity falls as they warm up.
(b) The time interval between two successive collisions can depend on \( T \). — True. At a higher temperature the electrons move faster and collide more often, so \( \tau \) decreases. In a metal this makes the resistivity rise.
(c) The length of the material can be a function of \( T \). — False. Length is a dimension of the sample, and it enters the resistance \( R=\rho l/A \). The resistivity depends on the material of the conductor. It does not depend on the dimensions of the sample.
(d) The mass of the carriers is a function of \( T \). — False. The mass of an electron is a constant. It does not change with temperature, so it plays no part in \( \rho(T) \).
NCERT (2026-27):
Temperature Dependence of Resistivity – “… From this equation, resistivity of a material is given by
\( \displaystyle \rho=\frac{1}{\sigma}=\frac{m}{ne^{2}\tau} \qquad (3.27) \)
ρ thus depends inversely both on the number n of free electrons per unit volume and on the average time τ between collisions.”
Temperature Dependence of Resistivity – “… As we increase temperature, average speed of the electrons, which act as the carriers of current, increases resulting in more frequent collisions. The average time of collisions τ, thus decreases with temperature.”
Temperature Dependence of Resistivity – “For insulators and semiconductors, however, n increases with temperature.”
Ohm’s Law – “… where the constant of proportionality ρ depends on the material of the conductor but not on its dimensions. ρ is called resistivity.”