Is it possible to have resistance less than 1
for example 0.5, 0.25.. or any fraction number
Dear Peter
what I mean is the following
If you go to my works We consider for example he equivalent resistance between the origin and the lattice site (n,m) in an infinite square network. Here n, and m are integers
My question is it possible to have n, and m fraction numbers ?????
Although the question is, perhaps, not as clearly posed or directed as it might, I think the following may be what is sought. The values of the coordinates, (m,n), of any node on the grid may be any real numbers but the point is that, since the grid is square, both m and n must be integer multiples of the distance between adjacent nodes, say d, so it is really a question of scaling. By making d the unit length, the coordinates become integer and the arithmetic concomitantly simpler and the likely reason that you always see integer coordinates. (Of course, electrical resistance, if under discussion, between arbitrary pairs of nodes will, in general, be non-integral using Ohm's law, even if the resistance of the material is 1ohm/d.)
What I meant by my question is the following:
Consider for example an infinite square network of identical resistors each of resistance R. It has been showed that the equivalent resistance between the origin (0,0) and any lattice site (m,n) can be expressed in terms of Lattice Green's Function (LGF) as:
R(0,0;m,n)=R[G(0,0)-G(m,n)]
where G(0,0) is the LGF at the origin, and G(m,n) is the LGF at the site (m,n)
Note that in the network m, and n are integers
Now, my question is:
Is it possible to find R(0,0;0.5,0.5)
m, and n half integers
Dear Prof. Peter
Thanks for your answer
I am asking if we can obtain the effective resistance between the origin and the point 1/2, 1/2 in an infinite square lattice
the point 1/2, 1/2 is not a lattice point
my question is pure math
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