Question

An infinite ladder is constructed with 1 $\Omega$ and 2 $\Omega$ resistors as shown in figure. Find the total current provided by the battery of EMF 12 V?

• A. $2\Omega$
• B. $-1\Omega$
• C. Infinite
• D. None of these

Answer 1

The correct option is A

$2\Omega$

Its an infinte circuit. The first thing we need to identify the repeating unit. There has to be a repeating unit or the infinte circuit will become impossible to solve. Let’s see…

So we see the unit encircled is the repeating unit here...it comes again and again I can take out one of those units and the total number of units still remains infinte…Like this.

All I did was take out just one repeating unit and express the rest as Req
The 2 $\Omega$ resistor is in parallel with Req so they together become $\frac{2Req}{2+Req}$
And this would be in series with 1 $\Omega$ resistor so the resistance across 12 V battery becomes $1+\frac{2Req}{(2+Req)}$
But wait!!! Isnt the equivalent resistance across 12 V supposed to be just Req?
Yeah right! So lets equate them
$\frac{2Req}{(2+Req)}$= $R_{eq}$
This will be a quadratic equation...
$R_{eq}$= 2 $\Omega$
We ignore the negative answer!