Question
Answers

If temperature is decreased, then relaxation time of electrons in metals will
A. Increase
B. Decrease
C. Fluctuate
D. Remain constant

Answer Verified Verified
Hint: As we know that the relaxation time is inversely proportional to the resistance of the metals. The variation of resistance with temperature in metals is directly proportional so when the temperature increases the resistance increases and vice versa.

Formula used:
$R\propto \dfrac{1}{\tau }$
$R\propto T$
Where $R$ - resistance of metals
$T$ - temperature
$\tau $ - relaxation time

Complete step by step answer:
As we know that the resistivity of metal is connected with relaxation time by:
$\rho =\dfrac{m}{n{{e}^{2}}\tau }$
Where $\rho $ - resistivity of metals
$m$ - mass of electron
$e$ - electronic charge
$n$ - electron density
And the relation between resistance and resistivity is:
$\begin{align}
  & R=\dfrac{\rho l}{A} \\
 & \Rightarrow R=\dfrac{ml}{n{{e}^{2}}\tau A} \\
 & \\
\end{align}$
Or we can write as $R\propto T\propto \dfrac{1}{\tau }$
So, when we increase temperature then resistance of the metals will also increase and the relaxation time decreases with increase in temperature.

So, the correct answer is “Option A”.

Additional Information:
A time constant appears within the only expression for the transport property of electrical conductivity, which states that the electrical conductivity equals the merchandise of the relief time, the density of conduction electrons, and the square of the electron charge, divided by the electron effective mass in the solid. See Band theory of solids.

Note:
Although mostly collision times in metals are quite short (on the order of 10$^{-14}$ s at room temperature), mean free paths range from about 100 atomic distances at room temperature to 106 atomic distances in pure metals near temperature. So the relaxation time will depend upon the metal we choose and then the variation of resistance and temperature.