Question

The given infinite grid consists of hexagonal cells of six resistors each of resistance R. Then $R_{12}$ is equal to

• A. $\frac{R}{3}$
• B. $\frac{2R}{3}$
• C. $\frac{4R}{3}$
• D. $\frac{3R}{4}$

Answer 1

The correct option is B $\frac{2R}{3}$

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 $R_1$ and current flowing through $R_1$ 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 $R_1$ will be same as they are in parallel. Therefore equating the potential drop,
$\frac{2I}{3}\times R=\frac{I}{3}\times R_1$
$R_1=2R$
Hence equivalent circuit becomes,
Therefore equivalent resistance across 1 and 2, $R_{eq}=\frac{2R\times R}{2R+R}=\frac{2R}{3}$