A wire of length l and resistance R is bent in the form of the ring. The resistance between two points that are separated by an angle θ is ____.

Question

A wire of length l and resistance R is bent in the form of the ring. The resistance between two points that are separated by an angle θ  is ____.

Accepted Answer

A wire of length l and resistance R is bent in the form of the ring. The resistance between two points that are separated by an angle θ  is Rθ4π2(2π-θ).In this question, we are going to first take two points z and w, in the different sections to define them easily, then the resistances for both of them are calculated from the resistance per unit length, and then as they are in parallel combination, the equivalent resistance is calculated.The resistance of an element is given by, Rxwy= Resistance per unit length × Length of the elementFor two resistors   R1and R2 connected in parallel.As we know, the resistance of a wire is directly proportional to its length.Now, taking the points z and w, on the wire as given in the diagram belowLet us take the element xwy , the resistance is given by,Rxwy=  Resistance per unit length×Length of the elementPutting the values in this, we getRxwy= R2πr×rθ Rxwy= Rθ2πNow taking the length elementRxzy= R2πr×r(2π-θ)Rxzy= R(2π-θ)2πAs we look at the elements, if they are considered to be two resistors connected to each other, Then, this connection is in parallel. Now the equivalent resistance for this parallel combination is calculated as:Req=Rxwy×RxzyRxwy+RxzyNow putting the values of the resistances, as obtained above, we get  Req=R(2π-θ)2π×Rθ2πR(2π-θ)2π+Rθ2π Req=Rθ4π2(2π-θ)

A wire of length l and resistance $R$ is bent in the form of the ring. The resistance between two

points that are separated by an angle $θ$ is ____.

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