Question

Consider an infinite mesh as shown in figure. Each side of the mesh has resistance $R$. Consider the hexagonal part ABCDEF of the infinite mesh. Equivalent resistance between any two points of the hexagon is measured. Equivalent resistance between A and B is $\frac{\alpha R}{\beta }$. Find the least value of $\alpha +\beta$.

Answer 1

If we connect the positive terminal of a battery to A and negative terminal to infinity, then the current through the branches will be as shown.


If we connect the negative of the battery to B and positive to infinity, current will be in the opposite direction through branch AB.

By superposition of the two situations, net current in the AB branch will be$\frac{2I}{3}$. Potential difference between A and B will be $\frac{2I}{3}R$. Effective resistance between A and B will be $\frac{2R}{3}$.