Solution: Resistance of Infinite Grid (Midpoint Connection)
Step 1: Isolate the Edge AB
We know that for an infinite grid of resistors $R$, the equivalent resistance between adjacent nodes (like A and B) is $R_{eq(AB)} = R/2$.
This total resistance consists of the direct edge $AB$ (resistance $R$) in parallel with the “rest of the grid” ($R_{rest}$).
$$ \frac{1}{R/2} = \frac{1}{R} + \frac{1}{R_{rest}} $$
$$ \frac{2}{R} – \frac{1}{R} = \frac{1}{R_{rest}} \implies \frac{1}{R_{rest}} = \frac{1}{R} $$
So, the resistance of the entire infinite grid excluding the direct connection $AB$ is exactly $R$.
Step 2: Analyze the Path A to C
The connection is made between node A and the midpoint C of the edge AB. The circuit divides into two parallel branches between A and C:
- Direct Path (A to C): This is half the edge length. Resistance = $R/2$.
- Indirect Path (A to B to C): This involves going from A to B via the “rest of the grid” (Resistance $R$) and then from B back to C (Resistance $R/2$).
Total Indirect Resistance = $R + R/2 = 3R/2$.
Step 3: Calculate Equivalent Resistance
We now have two resistors in parallel: $R/2$ and $3R/2$.
$$ \frac{1}{R_{total}} = \frac{1}{R/2} + \frac{1}{3R/2} $$
$$ \frac{1}{R_{total}} = \frac{2}{R} + \frac{2}{3R} = \frac{6+2}{3R} = \frac{8}{3R} $$
$$ R_{total} = \frac{3R}{8} $$