----- Original Message ---- From: Fred lunnon <fred.lunnon@gmail.com> To: math-fun <math-fun@mailman.xmission.com> Sent: Sunday, January 6, 2008 12:50:21 PM Subject: Re: [math-fun] xkcd points out dangers of math fun On 1/6/08, Eugene Salamin <gene_salamin@yahoo.com> wrote:
... On a square grid of 1 ohm resistors, the effective resistance between the node at (0,0) and the node at (p,q) is given by the following expression.
R[p,q] = (1/(4 pi^2)) int (F(x,y), x=-pi..pi, y=-pi..pi),
F(x,y) = [1 - cos(p x + q y)] / [2 - (cos x + cos y)].
This is pretty much identical to formula (18) in Cserti, for the 2-D case. No doubt one could be convinced of its correctness simply by checking that it satisfied the Kirchhoff-inspired equations I quoted earlier [ my u^{kl} being your R[p,q] ] : [Lat1] u^{00} = 0; [Lat2] u^{01} = 1/2; [Lat3] \del^2 u^{kl} = 0 for (k,l) <> (0,0); where \del^2 u^{kl} == u^{k,l-1} + u^{k,l+1} + u^{k-1,l} + u^{k+1,l} - 4 u_{kl} . But how did you arrive at it; and exactly what assumptions did you make in order to do so? Fred Lunnon _____________________________________________ We have the Kirchhoff equations (1/4)(v[j+1,k] + v[j-1,k] + v[j,k+1] + v[j,k-1]) - v[j,k] = 0, except at the nodes where current enters or exits. If 1 A of current exits at a node, the current is divided equally among the 4 resistors, and the 4 adjacent nodes will be (1/4) V higher. So at that node, the RHS should be (1/4). If 1 A exits at (p,q) and enters at (r,s), the RHS is (1/4)(delta(j,p) delta(k,q) - delta(j,r) delta(k,s)). Now define the Fourier series V(x,y) = sum(v[j,k] exp(i j x + i k y)) summed over all integers j and k. Applying this to the Kirchhoff equations, we can solve for V(x,y). It is a fraction whose numerator is exp(i r x + i s y) - exp(i p x + i q y), and whose denominator is 4 - 2(cos x + cos y). To calculate v[j,k], multiply V(x,y) by exp(- i j x - i k y)/(2 pi)^2 and integrate over (x,y) in [-pi, pi]^2. The effective resistance between (r,s) and (p,q) is equal to the voltage difference v[r,s] - v[p,q]. You can set (p,q) = (0,0), and get my solution, where I replaced (r,s) by (p,q). Gene ____________________________________________________________________________________ Be a better friend, newshound, and know-it-all with Yahoo! Mobile. Try it now. http://mobile.yahoo.com/;_ylt=Ahu06i62sR8HDtDypao8Wcj9tAcJ