A and B are two points on a uniform ring of resistance R the ∠AOB=θ3 where O is the centre of th

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A and B are two points on a uniform ring of resistance R the...

$\mathrm{A}$ and $\mathrm{B}$ are two points on a uniform ring of resistance $\mathrm{R}$ the $\angle \mathrm{AOB}=\theta_{3}$ where $\mathrm{O}$ is the centre of the ring. The equivalent resistance between $\mathrm{A}$ and $\mathrm{B}$ is. (A) $(\mathrm{R} \theta / 2 \pi)$ (B) $[\{\mathrm{R}(2 \pi-\theta)\} /\{4 \pi\}]$ (C) $\mathrm{R}[1-(\theta / 2 \pi)]$ (D) $\left\{\mathrm{R} /\left(4 \pi^{2}\right)\right\}(2 \pi-\theta) \theta$

$\mathrm{A}$ and $\mathrm{B}$ are two points on a uniform ring of resistance $\mathrm{R}$ the $\angle \mathrm{AOB}=\theta_{3}$ where $\mathrm{O}$ is the centre of the ring. The equivalent resistance between $\mathrm{A}$ and $\mathrm{B}$ is.
(A) $(\mathrm{R} \theta / 2 \pi)$
(B) $[\{\mathrm{R}(2 \pi-\theta)\} /\{4 \pi\}]$
(C) $\mathrm{R}[1-(\theta / 2 \pi)]$
(D) $\left\{\mathrm{R} /\left(4 \pi^{2}\right)\right\}(2 \pi-\theta) \theta$

Key Concepts

Geometric Partitioning in Circular Circuits

This concept involves dividing a circular resistor into segments based on geometrical angles or arc lengths. The relative lengths of these segments, corresponding to the angles they subtend at the center of the circle, determine their individual resistances. Understanding this partitioning is crucial for solving problems where the resistance between two points on a ring is required, as it allows one to apply the series and parallel resistance principles to these segments.

Parallel Combination of Resistances

When two or more distinct conductive paths connect the same two nodes, they are in parallel. The equivalent resistance of these paths is determined by the reciprocal sum of their individual conductances. In the context of a ring, the two different arcs connecting the same points act as parallel resistors, and the calculation of the overall equivalent resistance involves this parallel resistance formula.

Series Combination of Resistances

Resistors connected end-to-end are said to be in series, and their resistances add up directly. This principle is essential when analyzing portions of the ring that lie consecutively along a single path between two points, as the overall resistance of that path is the sum of the resistances of each constituent segment.

Uniform Resistance Distribution

This concept refers to the idea that the total resistance of a conductor, like a circular ring, is evenly spread along its entire length. When the resistance is uniform, every segment of the conductor has a resistance that is directly proportional to its length, making it possible to calculate the resistance of any arc or segment simply by knowing its fraction of the total circumference.

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