Question

An infinite ladder network of resistances is constructed with $1\Omega$ and $2\Omega$ resistance as shown in figure. The $6$V battery between A and B has negligible internal resistance. The equivalent resistance between A and B is?

• A. $1\Omega$
• B. $2\Omega$
• C. $3\Omega$
• D. $4\Omega$

Answer 1

The correct option is A $1\Omega$

$\text{Step 1: Find Smallest Repeating unit [Refer Fig. 1]}$

$\text{Step 2: Replace with equivalent resistance R. [Refer Fig. 2]}$

After the first smallest repeating unit, the remaining circuit is also same as the initial circuit.

Therefore, if we assume the equivalent resistance between A & B as $R$, then the resistance of the remaining circuit will also be $R$ as shown in the figure.

$\text{Step 3: Find equivalent resistance between A and B and equate with R. [Refer Fig. 3]}$

From figure, the equivalent resistance between A & B will be equal to the equivalent resistance of the remaining circuit. So,

$R_{AB}=1+\frac{2R}{2+R}$

$\Rightarrow R=1+\frac{2R}{2+R}$

$\Rightarrow 3R+2=R^2+2R$

$\Rightarrow R^2-R-2=0$

$\therefore R=2\Omega$ (Here, Negative value can is rejected)

Hence Equivalent resistance between points A and B is 2$\Omega$.