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Chapter 12: Problem 1800

A wire of resistor \(R\) is bent into a circular ring a circular ring of radius \(\mathrm{r}\) Equivalent resistance between two points \(\mathrm{X}\) and \(\mathrm{Y}\) on its circumference, when angle xoy is \(\alpha\), can be given by (A) $\left\\{(\mathrm{R} \alpha) /\left(4 \pi^{2}\right)\right\\}(2 \pi-\alpha)$ (B) \((\mathrm{R} / 2 \pi)(2 \pi-\alpha)\) (C) \(\mathrm{R}(2 \pi-\alpha)\) (D) \((4 \pi / \mathrm{R} \alpha)(2 \pi-\alpha)\)

Short Answer

Expert verified
The short answer based on the given step-by-step solution is: The correct answer is (B) \(\frac{R}{2 \pi}(2 \pi - \alpha)\).

Step by step solution

01

Calculate the length of each segment

To find the length of each segment between points X and Y, first, we need to determine the circumference of the entire circle, which can be given by: C (Circumference) = 2πr Now, since angle XOY = α, we can use the fact that the ratio of the arc length to the total circumference equals the ratio of the angle (in radians) to 2π. Therefore, the length of segments XY and YX can be calculated as: Length of XY = (α/2π) * C = (α/2π) * 2πr = αr Length of YX = (2π - α)/2π * C = ((2π - α)/2π) * 2πr = (2π - α)r
02

Calculate the resistance of each segment

Now, using the formula R = ρ * (Length/Area), where ρ is the resistivity of the wire, we can find the resistance of each segment. Since we are given only the total resistance R of the wire and we know that resistivity and wire area will remain constant throughout the wire, we can say: Resistance of XY = (α/2π) * R Resistance of YX = ((2π - α)/2π) * R
03

Determine the equivalent resistance between points X and Y

Now, we can use the concept of resistances in parallel to find the equivalent resistance REQ between the two points X and Y. The formula is: 1/REQ = 1/RXY + 1/RYX Now replacing RXY and RYX with calculated resistance values from step 2, we get: 1/REQ = 1/((α/2π) * R) + 1/(((2π - α)/2π) * R) Simplifying: REQ = \(\frac{R \alpha(2 \pi - \alpha)}{2\pi [(\alpha)(2 \pi - \alpha)]}\) After cancellation: REQ = \(\frac{R(2 \pi - \alpha)}{2\pi}\) Comparing to the given options, we see that this expression for equivalent resistance matches option (B). So, the correct answer is (B) \(\frac{R}{2 \pi}(2 \pi - \alpha)\).

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Most popular questions from this chapter

The reading of ammeter shown in figure is.... (A) \(2.18 \mathrm{~A}\) (B) \(3.28 \mathrm{~A}\) (C) \(6.56 \mathrm{~A}\) (D) \(1.09 \mathrm{~A}\)

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