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.
$$\rho=\frac{1}{\sigma}=\frac{m}{n e^2 \tau}$$
What are the advantages of the null-point method in a Wheatstone bridge? What additional measurements would be required to calculate $R_{\text {unknown }}$ by any other method?
The advantage of null point method in a Wheatstone bridge is that the resistance of galvanometer does not affect the balance point, there is no need to determine current in resistances and the internal resistance of a galvanometer.
It is easy and convenient method for observer.
The $R_{\text {unknown }}$ can be calculated applying Kirchhoff's rules to the circuit. We would need additional accurate measurement of all the currents in resistances and galvanometer and internal resistance of the galvanometer.