Consider an infinite ladder network shown in the figure. $V$ voltage $V$ is applied between the points $A$ and $B$. This applied voltage is halved after each section.

$A$ battery consisting of $6$ cells,each with an internal resistance of $0.5\, \Omega$ and an $e.m.f.$ of $2\, V$,is being charged by a $220\, V$ $D.C.$ mains using an external resistor of $10\, \Omega$. The charging current is: (in $, A$)

In the circuit shown in the figure,the reading of the voltmeter is $V_1$ when only $S_1$ is closed,and the reading of the voltmeter is $V_2$ when only $S_2$ is closed. The reading of the voltmeter is $V_3$ when both $S_1$ and $S_2$ are closed. Then:

Inside a superconducting ring,six identical resistors each of resistance $R$ are connected as shown in the figure. The equivalent resistance$(s)$:

The thermo $e.m.f.$ of a thermocouple is $25\,\mu V/^{\circ}C$ at room temperature. $A$ galvanometer of $40\,\Omega$ resistance, capable of detecting current as low as $10^{-5}\,A$, is connected with the thermocouple. The smallest temperature difference that can be detected by this system is ................ $^{\circ}C$.

Find the current flowing through the resistance $R_2$ of the circuit shown in the figure,if the resistances are $R_1 = 20 \ \Omega, R_2 = 30 \ \Omega$ and $R_3 = 60 \ \Omega$,and the potentials of points $1, 2$ and $3$ are $V_1 = 20 \ V, V_2 = 30 \ V$ and $V_3 = 60 \ V$ respectively. (in $A$)