Resistant Infinite ladder

Electricity and Magnetism Level 1

In the figure above, the effective resistance between the points $A$ and $D$ of the infinite ladder can be represented as $\frac{a+b\sqrt{c}}{2} R$ , where $c$ is square free.

Find the value of $a+b+c+2$ .

The answer is 9.

2 solutions

Nihar Mahajan
Jun 25, 2015

Let the resistance between $A \ and \ D$ be $r$ .As the ladder is infinite , $r$ is also equivalent resistance of the ladder to the right of points $B \ and \ C$ .So we can redraw the circuit like this:

Thus we can form an equation:

$r=R+\dfrac{rR}{r+R} \\ \Rightarrow rR+r^2=R^2+2rR \\ \Rightarrow r^2-rR-R^2=0 \\ \Rightarrow r=\dfrac{R+\sqrt{R^2+4R^2}}{2} \\ \Rightarrow r = \dfrac{1+\sqrt{5}}{2} R \\ \Rightarrow a+b+c+2=1+1+5+2=\boxed{9}$

Probably (1+✓5)/2 is called the golden ratio....am I right?

Istiak Reza - 5 years, 11 months ago

Yes you are right. :)

Nihar Mahajan - 5 years, 11 months ago

Perfect solution!cheers!

Rohit Ner - 5 years, 11 months ago

Thanks!!! :)

Nihar Mahajan - 5 years, 11 months ago

@Nihar Mahajan This is a nice way of doing. But What I did was something like this. I started at the rightmost resistance, assuming there existed one. And from there I go on evaluating the equivalent resistances. Fun thing is if you do that you will see, that the terms 2 ohm, $3/2$ ohm, $5/3$ ohm, $8/5$ ohm, ... and so on. But as we already know and have probably observed, this is the ratio of $\frac{F_{n+1}}{F_n}$ , where F are the Fibonacci numbers, and we need to find the limit of the ratio at n->infinity. But as we already know, that is, $\frac{1+\sqrt{5}}{2}$ .

When I did this problem first, it seemed to be a very surprising occurrence of Fibonacci in resistances. XD

Soumava Pal - 5 years, 3 months ago

That is really a nice way, I solved it using the way Nihar did in his solution. However, this solution is really amazing!!

Raushan Sharma - 5 years, 2 months ago

Yes , I also found out the relation $\dfrac{F_{n+1}}{F_n}$ , but at that time I didn't know how to evaluate limits , so the problem remained unsolved by me by this approach :P

Nihar Mahajan - 5 years, 3 months ago

I see, ... lol..

Soumava Pal - 5 years, 3 months ago

Nicely done! Just like solving infinite series sum.

P.S; I got mad and input $-1$ as the answer and then understood my stupid mistake and hit the correct number!

Sravanth C. - 5 years, 11 months ago

Thanks! Yes , it is like the technique of solving infinite series sum.

Nihar Mahajan - 5 years, 11 months ago

Nice Generalization !!

Ninad Mhalgi - 5 years, 11 months ago

Thank you Pjan :P

Nihar Mahajan - 5 years, 11 months ago

Jake Ok
May 23, 2016

Classic golden ratio example

Yeah, you come across the golden ratio very frequently in math and science.

Pranshu Gaba - 5 years ago

Resistant Infinite ladder

The answer is 9.

2 solutions

0 pending reports