A Candle in the Dark

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Nerd Sniping

From xkcd,

So, how does one find the effective resistance?

The solution to this question, and many other network shapes, is present in Jozsef Cserti’s paper1.

Additionally, this website2 also clearly explains the technique to find the general equivalent resistance between any two nodes. The resistance between the two marked nodes in the xkcd question is,

$\frac{1}{4 \pi^2} \int_{-\pi}^{\pi} \int_{-\pi}^{\pi} \left( \frac{1-\cos(2x+y)}{2-\cos{x} - \cos{y}} \right)$ $dx dy$

which turns out to be $\frac{8-\pi}{2 \pi}$. Check out Appendix A of Cserti’s paper for a technique to evaluate the above integral.

[1] – Application of the lattice Green’s function for calculating the resistance of an infinite network of resistors, Cserti József, American Journal of Physics, Volume 68, Issue 10, pp. 896-906 (2000). arXiv:cond-mat/9909120v4

Written by parseval

January 3, 2008 at 12:02 am

Posted in humour, mathematics

5 Responses

1. Or, just go and ask Dr. Balaji Sampath, genius and teacher extraordinaire. 🙂

January 3, 2008 at 10:05 am

2. er, the truck missed you or what?

Cheers,
Arunn

Arunn

January 3, 2008 at 8:01 pm

3. I’m an engineer, I wait till I cross the road :p

parseval

January 3, 2008 at 9:17 pm

4. Amazing!

N

January 4, 2008 at 9:32 pm

5. I don’t know how I missed this, absolutely lovely find!

Mohan

January 12, 2008 at 3:42 pm