An infinite ladder network of resistance is constructed with$1 \mathrm{\Omega }$ and $2 \mathrm{\Omega }$ resistance, as shown in the figure. 

$\left(\mathrm{i}\right)$ Show that the effective resistance between $A$ and $B$ is $2 \mathrm{\Omega }.$
$\left(\mathrm{ii}\right)$ What is the current that passes through the $2 \mathrm{\Omega }$ resistance nearest to the battery?

As shown in the figure a variable rheostat of $2 \mathrm{k\Omega }$ is used to control the potential difference across $500 \mathrm{ohm}$ load

(i) If the resistance$\mathit{ }AB$is $500 \mathrm{\Omega },$ what is the potential difference across the load?

(ii) If the load is removed, what should be the resistance at $BC$ to get $40 \mathrm{volt}$ between $B$ and $C$?

$\mathrm{ABCD}$ is a square where each side is uniform wire of resistance $1\mathrm{\Omega }.$ Find a point $\mathrm{E}$ on $\mathrm{CD}$ such that if a uniform wire of resistance $1\mathrm{\Omega }$ is connected across $\mathrm{AE}$ and a potential difference is applied across $\mathrm{A} \mathrm{and} \mathrm{C},$ the points $\mathrm{B} \mathrm{and} \mathrm{E}$ will be equipotential. 

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Three bulbs, each having a resistance of $180 \mathrm{\Omega }$ are connected in parallel to an ideal battery of emf $60 \mathrm{V}.$ Find the current delivered by the battery when

$\left(\mathrm{a}\right)$ all the bulbs are switched on,

$\left(\mathrm{b}\right)$ two of the bulbs are switched on and

$\left(\mathrm{c}\right)$ only one bulb is switched on.

Consider the circuit shown in figure. Find the current through the $10 \mathrm{\Omega }$ resistor when the switch $\mathrm{S}$ is

$\left(\mathrm{a}\right)$opened

$\left(\mathrm{b}\right)$closed.

Six lead-acid type of secondary cells, each of emf $2.0 \mathrm{V}$ and internal resistance $0.015 \mathrm{\Omega }$, are joined in series to provide a supply to a resistance of $8.5 \mathrm{\Omega }$. Determine : (i) the current drawn from the supply and (ii) its terminal voltage.

In the figure each cell has an emf of $1.5 \mathrm{V}$ and internal resistance of $0.40 \mathrm{\Omega }$. Calculate:

(i) current $\mathrm{I}$

(ii) current in the $36 \mathrm{\Omega }$ resistor

(iii) potential difference across $\mathrm{A}$ and $\mathrm{B}$.

In the circuit shown all five resistors have the same value $200 \mathrm{\Omega }$ and each cell has an emf $3 \mathrm{V}$. Find the open circuit voltage and the short circuit current for the terminals $\mathrm{A}$ and $\mathrm{B}$.