Question

A uniform wire of resistance R is shaped into a regular in sided polygon where n is even. Find the equivalent resistance between (i) opposite corners of polygon (ii) adjacent corners of polygon .

Answer 1

Let the polygon be as shown in figure
The resistance of each side of polygon `= R//n`.
(i) For resistance between opposite corners C and G, we have two resistance in parallel each of value `R//2`.
Therefore, the equivalent resistance between opposite corners is `((R//2)(R//2))/((R//2)+(R//2)) =R/4`
(ii) For resistance between adjacent corners A and H, we have two resistance of `R//n` and `(n-1)(R//n)` in parallel. The equivalent resistance is
`((R//n)(n-1) R//n)/((R//n)+(n-1)R//n)= R ((n-1)/n^(2))`