Question

An infinite ladder network of resistances is constructed with $1 \, \Omega$ and $2 \, \Omega$ resistances. The $6 \, V$ battery has negligible internal resistance. The current that passes through the $2 \, \Omega$ resistance nearest to the battery is

• A. $1 \, A$
• B. $1.5 \, A$
• C. $2 \, A$
• D. $2.5 \, A$

Answer 1

Answer: (B)

Let $x$ be the equivalent resistance of the infinite ladder then,
the equivalent circuit is

also, the equivalent resistance is
$x=1+\frac{2 x}{2 + x}$
$x^{2}-x-2=0$
$x=\frac{1 + \sqrt{1 + 8}}{2}=2 \, \Omega$
current through the battery is $3 \, A$ .

Current through $2 \, \Omega$ resistor is
$i=3\left(\frac{x}{2 + x}\right)=3\left(\frac{2}{2 + 2}\right)=1.5 \, A$