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Re infinite square lattice of resistors



The principle of superposition can be used to solve the
problem of an infinite square array of identical resistors.
This principle states that if you apply two different
voltage sources to an array and determine what
currents will flow, then the currents that flow when
both voltage sources are applied at the same time are
just the sums of the currents found with each single
source.

We will use this to find the resistance (R) measured
across a single resistor (resistance r) of an infinite
square lattice.

Label the ends of a single resistor A and B. Take the
potential at infinity to be zero. (Imagine a conducting
ring at a great distance to be grounded.)

Source 1:
Connect the positive terminal of a battery (voltage V)
to A and the negative terminal to the conducting ring.
By symmetry, the current an all four resistors meeting
at A will be the same flowing away from A. Call this
current I, so the battery current is 4I.

Source 2:
Connect the negative terminal of another battery (also V)
to B and the positive terminal to the ring. We will now
have a current I flowing toward B through the four
resistors meeting at B.

With both sources connected at the same time, the
voltage between A and B is 2V and the current through
resistor AB will be 2I from A to B. Using Ohm's Law for
resistor AB:
r = (2V)/(2I) = V/I.

However, the battery current is 4I, so the effective
resistance of the lattice is:
R = (2V)/(4I) = (1/2)(V/I) = r/2.

The same reasoning applied to a cubic lattice gives
the result:
R = (2V)/(6I) = r/3.

Question: Can the problem of finding R for diagonal
points of a square or cubic lattice be found using
this approach?

I hope this helps.

Bob Blumenthal <blumenthal@whitman.edu>