Question

A wire of resistor R is bent into a circular ring a circular ring of radius r Equivalent resistance between two points X and Y on its circumference, when angle xoy is α, can be given by

(A) {(Rα)/(4π²)} (2π – α)

(B) (R/2π) (2π – α)

(C) R(2π – α)

 (D) (4π/Rα) (2π – α)

Answer 1

The correct option (A) {(Rα)/(4π²)} (2π – α)

Explanation:

For total circumference 2πr, resistance is R.

For arc xωγ, length = αr

for arc xzy, length = (2π – α)r

∴ R_xωy = {R/(2πr)} × rα = (Rα / 2π)

R_xzy  = {R/(2πr)} × r(2π – α) = (R/2π) (2π – α)

R_eq = {(R_xωy ∙ R_xzy)/(R_xωy + R_xzy)}

= [{(Rα/2π) × (R/2π) (2π – α)}/{(Rα/2π) + (R/2π) (2π – α)}]

= [({R²α/(2π)²} (2π – α))/R]

 = {Rα/4π²} (2π – α)