Question

# If temperature is decreased, then relaxation time of electrons in metals willA. IncreaseB. DecreaseC. FluctuateD. Remain constant

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

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”.

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.