Question

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

Answer 1

(a) Let the effective resistance of the combination be R. The circuit can be redrawn as shown below.


From the figure,

$$\begin{gathered} \frac{2R}{R+2}+1=R \\ \Rightarrow 3R+2=R^2+2R \\ \Rightarrow R^2-R-2=0 \\ \Rightarrow R=\frac{+1+\sqrt{1+4\times 1\times 2}}{2\times 1} \\ \Rightarrow R=\frac{+1+\sqrt{9}}{2.1}=2\mathrm{\Omega } \end{gathered}$$

(b) Total current sent by the battery $=\frac{6}{R}=\frac{6}{2}=3\mathrm{A}$

Applying Kirchoff's Law in loop 1, we get:

$$\begin{gathered} 3\times 1+2i=6 \\ \Rightarrow 2i=3 \\ \Rightarrow i=\frac{3}{2}=1.5\mathrm{A} \end{gathered}$$