Equivalent Resistence

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0 0 0 0 0 0 0 0
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r r r r r r r r
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0---r---....---r---r---r---r---r---r---....---r---r---0
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r r r r r r r r
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. . . . . . . .
. . . . . . . .
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r r r r r r r r
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0---r---....---r---r---r---r---r---r---....---r---r---0
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r r r r r r r r
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0---r---....---r---r---r---r---r---r---....---r---r---0
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r r r r r r r r
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0---r---....---r---r-A-r-B-r---r---r---....---r---r---0
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r r r r r r r r
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0---r---....---r---r---r---r---r---r---....---r---r---0
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r r r r r r r r
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0---r---....---r---r---r---r---r---r---....---r---r---0
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r r r r r r r r
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. . . . . . . .
. . . . . . . .
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r r r r r r r r
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0---r---....---r---r---r---r---r---r---....---r---r---0
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r r r r r r r r
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0 0 0 0 0 0 0 0

This is a 2D resistive grid which expands in all directions. The
potential at infinite is 0 as indicated in above schematic. Find the
equivalent resistence between point A and B as shown in figure above.
I found out experimentally by constructing a huge grid and applying a
voltage source between A and B. Then I measured the current and took
ratio of voltage and current to give me equvivalent resistence. It
comes out to be approximately r/2.
But how can we do it analytical. does anyone have any clue how to
start?
Thanks.

 

[PN2222A]

0 0 0 0 0 0 0 0
| | | | | | | |
| | | | | | | |
r r r r r r r r
| | | | | | | |
| | | | | | | |
0---r---....---r---r---r---r---r---r---....---r---r---0
| | | | | | | |
r r r r r r r r
| | | | | | | |
. . . . . . . .
. . . . . . . .
| | | | | | | |
r r r r r r r r
| | | | | | | |
0---r---....---r---r---r---r---r---r---....---r---r---0
| | | | | | | |
r r r r r r r r
| | | | | | | |
0---r---....---r---r---r---r---r---r---....---r---r---0
| | | | | | | |
r r r r r r r r
| | | | | | | |
0---r---....---r---r-A-r-B-r---r---r---....---r---r---0
| | | | | | | |
r r r r r r r r
| | | | | | | |
0---r---....---r---r---r---r---r---r---....---r---r---0
| | | | | | | |
r r r r r r r r
| | | | | | | |
0---r---....---r---r---r---r---r---r---....---r---r---0
| | | | | | | |
r r r r r r r r
| | | | | | | |
. . . . . . . .
. . . . . . . .
| | | | | | | |
r r r r r r r r
| | | | | | | |
0---r---....---r---r---r---r---r---r---....---r---r---0
| | | | | | | |
| | | | | | | |
r r r r r r r r
| | | | | | | |
| | | | | | | |
0 0 0 0 0 0 0 0

This is a 2D resistive grid which expands in all directions. The
potential at infinite is 0 as indicated in above schematic. Find the
equivalent resistence between point A and B as shown in figure above.
I found out experimentally by constructing a huge grid and applying a
voltage source between A and B. Then I measured the current and took
ratio of voltage and current to give me equvivalent resistence. It
comes out to be approximately r/2.
But how can we do it analytical. does anyone have any clue how to
start?
Thanks.

By inspection, it's 35 ohms.

Unless the r's are negative.

PN2222A

 

[Jonathan Kirwan]

By inspection, it's 35 ohms.
Unless the r's are negative.

Regardless of r, such that r>0? I don't think so.

(1/2)*r is right.

Jon

 

[Jonathan Kirwan]

I found out experimentally by constructing a huge grid and applying a
voltage source between A and B. Then I measured the current and took
ratio of voltage and current to give me equvivalent resistence. It
comes out to be approximately r/2.

Since you have this beast, would you mind trying a few different
location pairs?

if (0,0) and (1,0) gives .500 r, as you've already said, then:
(0,0) and (1,1) gives .636 r 2/PI
(0,0) and (2,0) gives .726 r 2-4/PI
(0,0) and (2,1) gives .773 r 4/PI-1/2
(0,0) and (2,2) gives .849 r 8/(3*PI)
(0,0) and (3,0) gives .861 r 17/2-24/PI
(0,0) and (3,1) gives .881 r 46/(3*PI)-4
(0,0) and (3,2) gives .924 r 4/(3*PI)+1/2
(0,0) and (3,3) gives .976 r 46/(15*PI)
(0,0) and (4,0) gives .954 r 40-368/(3*PI)
(0,0) and (4,1) gives .965 r 80/PI-49/2
(0,0) and (4,2) gives .992 r 6-236/(15*PI)
(0,0) and (4,3) gives 1.028 r 24/(5*PI)-1/2
(0,0) and (4,4) gives 1.067 r 352/(105*PI)

It should match these values, I hope!

Jon