Temperature dependence of resistivity $\rho(T)$ of semiconductors, insulators and metals is significantly based on the following factors
The measurement of an unknown resistance $R$ is to be carried out using Wheatstones bridge as given in the figure below. Two students perform an experiment in two ways. The first students takes $R_2=10 \Omega$ and $R_1=5 \Omega$. The other student takes $R_2=1000 \Omega$ and $R_1=500 \Omega$. In the standard arm, both take $R_3=5 \Omega$. Both find $R=\frac{R_2}{R_1}, R_3=10 \Omega$ within errors.
In a meter bridge, the point $D$ is a neutral point (figure).
Is the motion of a charge across junction momentum conserving? Why or why not?
When an electron approaches a junction, in addition to the uniform electric field $\mathbf{E}$ facing it normally. It keep the drift velocity fixed as drift velocity depend on $E$ by the relation drift velocity
$$v_d=\frac{e E \tau}{m}$$
This result into accumulation of charges on the surface of wires at the junction. These produce additional electric field. These fields change the direction of momentum. Thus, the motion of a charge across junction is not momentum conserving.
The relaxation time $\tau$ is nearly independent of applied $E$ field whereas it changes significantly with temperature $T$. First fact is (in part) responsible for Ohm's law whereas the second fact leads to variation of $p$ with temperature. Elaborate why?
Relaxation time is inversely proportional to the velocities of electrons and ions. The applied electric field produces the insignificant change in velocities of electrons at the order of $1 \mathrm{~mm} / \mathrm{s}$, whereas the change in temperature $(T)$, affects velocities at the order of $10^2 \mathrm{~m} / \mathrm{s}$.
This decreases the relaxation time considerably in metals and consequently resistivity of metal or conductor increases as.
$$\rho=\frac{1}{\sigma}=\frac{m}{n e^2 \tau}$$