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How to find the resistance between 2 corners of a square in an infinite square mesh??

How to find the resistance between 2 corners of a square in an infinite square mesh??

Grade:12th Pass

1 Answers

Aravind Bommera
36 Points
11 years ago

In an infinite square lattice consisting of identical resistances R, the
resistance between the origin and any lattice point (l , m) can be calculated
using 10
Ro (l , m) ? R[Go (0,0) ? Go (l , m)]
(1)
where Go (0,0) is the LGF of the infinite square lattice at the origin,
and Go (l , m) is the LGF at the site (l , m) .
First of all, the resistance between two adjacent points can easily be
obtained as
Ro (1,0) ? R[Go (0,0) ? Go (1,0)] .
(2)
Go (1,0) can be expressed as (see Appendix A)
1
Go (1,0) ? [tGo (0,0) ? 1] ; t ? 2 .
2
(3)
Where t is the energy.
and t=2 refers to the energy of the infinite square lattice at which the
density of states (the imaginary parts of the LGF) is singular (Van Hove
singularities) 23?25 .
Thus, Eq. (2) becomes
1
R
Ro (1,0) ? R[Go (0,0) ? Go (0,0) ? ] ? .
2
2
(4)
R
2
So, Ro (1,0) ? Ro (0,1) ? . (due to the symmetry of the lattice).
The same result was obtained by Venzian 3 , Atkinson et. al. 4 , and
Cserti 10 .
To calculate the resistance between the origin and the second nearest
neighbors (i.e. (1, 1)) then
3
Ro (1,1) ? R[G o (0,0) ? G o (1,1)] .
(5)
?
Go (1,1) can be expressed in terms of Go (0,0) and Go (0,0) as (see Appendix
A)
Go (1,1) ? (
t2
t
?
? 1)Go (0,0) ? (4 ? t 2 )Go (0,0) .
2
2
2
? E( )
2
2
t ? 1 K ( 2) .
?
Go (0,0) ? K ( ) and Go (0,0) ?
?t
t
? t (t ? 2) ? t 2
t
2
t
(6)
(7)
2
t
Where K ( ) and E ( ) are the elliptic integrals of the first kind and
second kind respectively.
Substituting the last two expressions into Eq. (4), one obtains
Ro (1,1) ?
2R
.
?
(8)
Again our result is the same as Cserti 10 and Venezain 3 .
Finally, to find the resistance between the origin and any lattice site
(l , m) one can use the above method, or we may use the recurrence
formulae presented by Cserti 10 (i.e. Eq. 32).
So, using Eq. (32) in Cserti 10 and the known values of
Ro (0,0) ? 0 ,
Ro (1,0) ?
R
2
and Ro (1,1) ?
2R
?
we calculate exactly the
resistance for arbitrary sites. The same result was obtained by Atkinson
et. al 4 , and below are some calculated values:
Ro (2,0)
R (3,0)
R (4,0)
? 0.7267, o
? 0.8606, and o
? 0.9539 .
R
R
R
For large values of l or/and m the resistance between the origin and the
site (l , m) is given as 10
Ro (l , m) ?
R
Ln8
( Ln l 2 ? m 2 ? ? ?
)
?
2
(9)
4
where ? ? 0.5772 is the Euler-Mascheroni constant 26 . Venezain obtained
the same result 3 .
Finally, as l or m goes to infinity then the resistance in a perfect infinite
square lattice divergence.

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