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Question

The given infinite grid consists of hexagonal cells of six resistors each of resistance R. Then R12 is equal to

A
R3
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B
2R3
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C
4R3
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D
3R4
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Solution

The correct option is B 2R3
Let point 1 is connected to a current source (source that provides current irrespective of load resistance). Since point 1 is connected to the whole circuit having resistor of equal resistance R, therefore current gets equal distributed among the branches connected to point 1 as shown.
Simillarly let point 2 is connected to a current source (source that draws current irrespective of load resistance). Since point 2 is connected to the whole circuit having resistor of equal resistance R, therefore equal current gets drawn from all the branches connected to point 2 as shown.

When both point 1 and 2 are connected to current source as shown, then current flowing across 1-2 will be the superposition of cases when individual points 1 and 2 are connected to the current source separately.
Circuit for 1-2 can be drawn as shown below.
Since the circuit is an infinite grid, therefore all the resistance are connected in parallel to resistance across 1 and 2. Let the equivalent resistance for the whole grid be R1 and current flowing through R1 will be I/3 as discussed. Therefore the equivalent circuit for the above grid can be drawn as shown.
The potential drop across R and R1 will be same as they are in parallel. Therefore equating the potential drop,
2I3×R=I3×R1
R1=2R
Hence equivalent circuit becomes,
Therefore equivalent resistance across 1 and 2, Req=2R×R2R+R=2R3

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