Question

The effective resistance of the following infinite ladder network between the points $A$ and $B$ is

• A. $3.23\, \Omega$
• B. $8.6 \,\Omega$
• C. $5.8\, \Omega$
• D. $1.23\, \Omega$

Answer 1

Answer: (A)

Since the sequence is infinite; it can be replaced by
$R_{e f f}=1 \,\Omega+\frac{1}{\frac{1}{2}+\frac{1}{R_{e f f}}}+1$
$R_{e f f}=2+\frac{2 R_{e f f}}{R_{e f f}+2}=\frac{4+4 R_{e f f}}{R_{e f f}+2}$
Or $R_{e f f}^{2}+2 R_{e f f}=4+4 R_{e f f}$
$R_{e f f}^{2}-2 R_{e f f}-4=0 $
Or $ R_{e f f}=\frac{+2 \pm \sqrt{4+16}}{2}=\frac{2 \pm 2 \sqrt{5}}{2}$
taking positive value $R_{e f f}=1+\sqrt{5}=(1+2.236)=3.23\, \Omega$