Question

Three different circuits(I, II and III) are constructed using identical batteries and resistors of R and $2R$ ohm. What can be said about current I in arm AB of each circuit?

• A. $I_{II}<I_I<I_{III}$
• B. $I_I<I_{II}<I_{III}$
• C. $I_I=I_{II}=I_{III}$
• D. $I_I>I_{II}=I_{III}$

Answer 1

Answer: (C)

$I_I=I_{II}=I_{III}$

Here the whole network in all the three cases , except of cell 1 is replaced by a resistance R_CD, representing an input resistance of a ladder consisting of cells 2,3, and the rest. Terminals denoted as C, D can be considered “input” terminals of this “depleted” ladder. Also this depleted ladder has infinite length and therefore it holds true that R_AB = R_{CD
$R_1=R$, $R_2=2R$
$$R_{AB}=R+2R||R_{AB}=R+\frac{2R\times R_{AB}}{2R+R_{AB}}$$
$$R_{AB}=\frac{R}{2}(1+\sqrt{1+\frac{4\times 2R}{R}})=2R$$
So current I in arm AB of each circuit no matter how many loops are there in the ladder circuits like those shown here will be the same. Hence}

$I_I=I_{II}=I_{III}$