Question

There is an infinite wire grid with cells in the form of equilateral triangles. The resistance of each wire between neighbouring joint connections is R${}_0$. The net resistance of the whole grid between the points A and B as shown is

• A. $R_0$
• B. $\frac{R_0}{2}$
• C. $\frac{R_0}{3}$
• D. $\frac{R_0}{4}$

Answer 1

Answer: (C)

$\frac{R_0}{3}$

By the superposition principle. Let current I enter at A, then current $\frac{I}{6}$ will flow through AB.

Now, in the second case, let the current be extracted from B, again current $\frac{I}{6}$ will flow through AB.

Now, if simultaneously current enters at A and leaves at B, then the current $\frac{I}{6}+\frac{1}{6}=\frac{I}{3}$ should flow through AB

and current $I-\frac{I}{3}=\frac{2I}{3}$ will flow through the remaining part. Let the resistance fo the remaining part be $R_1$

Now $\frac{R_{0}I}{3}=\frac{R_{1}2I}{3}{R}_0=2R_1$

$R_{eq}=\frac{R_1{R}_0}{R_1+R_0}=\frac{R_{1}2R_1}{R_1+2R_1}=\frac{2R_1}{3}=\frac{R_0}{3}$

Hence option 'C' is correct.