Re: Simple Question

The theory:
If you connect two resistors in parallel, the resulting resistance is
always lower than the lowest of either's values. So, if you connect an
infinite number of resistors in parallel, the value is infinitely lower,
which is, by definition, 0.

But this wasn't the case here, at least as I understood
the problem. The original post described an infinite
"grid" of resistors - i.e., imagine an infinitely large
grid of squares, with each "node" (every point within
the grid where the corners of the squares meet) connected
to the adjacent nodes by a one-ohm resistor. Now measure
the resistance between any two adjacent nodes, and see
what you get.

The answer clearly can't be zero. The situation is actually
very analogous to determining the resistance between
two points separated by a finite distance, but within an
infinite plane of some conductive (but not perfectly
conductive) material. For instance, imagine a very large
sheet of thin copper; put the probes of your ohmmeter (which
presumably has a scale for sufficiently low resistances)
an inch apart in the middle of the sheet - what should you
read?


Hmm.
Imagine an infinitely large grid of resistors. In between two points, you
have an infinite number of paths through the adjacent resistors, which are
connected in series so we can treat them as one resistor.
Now if you connect an infinite number of resistors in parallel, you will get
zero ohms, no matter how large they are.
True, if connecting the parallel resistors one by one, the drop in
resistance will be very, very small - so small that you might not even
notice with an ohmmeter. I like your analogy with the copper sheet, it's
absolutely obvious that there is a finite resistance between two points an
inch apart.
On the other hand, this is a play on our inability to conceive "infinite".
There is no such thing, so our experience is absolutely worthless.
The "knight move" is just a trick to distract our attention.
Try it like this:
Take a standard-format unetched copper PCB and measure the resistance
between two points.
Then, solder a second, identical PCB to one of its sides. Measure again
between the same points.
If the resistance becomes lower (and it will), I see no reason why it should
not go down to 0 if you take an *infinite* number of connected PCBs, or an
infinitely large copper sheet, if you like.

Regards,
Leo


.

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