Question

An infinite ladder is constructed with 1 $\Omega$ and 2$\Omega$ resistors as shown in figure (32-E22). (a) Find the effective resistance between the points A and B. (b) Find the current that passes through the 2$\Omega$ resistor nearest to the battery. Figure 32 - E22

Answer 1

Let the equivalent resistance of the combination be R

So,$\frac{2R}{R+2}+1=R$

$3R+2=R^2+2R$

$\Rightarrow R^2-R-2=02$

$\Rightarrow R=\frac{+1+\sqrt{1+\mathrm{4.1.2}}}{2.1}$

$=\frac{+1+\sqrt{9}}{2.1}=2\Omega$

(b) Total current send by battery

$=-\frac{6}{R_{eff}}=\frac{6}{2}=3$

Potential difference between A and B

$3.1+2i=6$

$\Rightarrow 2i=3$

$\Rightarrow i=\frac{3}{2}=1.5A$