A and B are two points on a uniform ring of radius r. The resistance of the ring is R.`angle AOB = theta` as shown in the figure. The equivalent resistance between points A and B is ...... .

Options

A. `(R theta)/(2 pi ) `
B. `R (1 - (theta)/(2pi))`
C. `(R (2pi - theta))/(4 pi)`
D. `(R)/(4 pi^(2))(2 pi - theta)theta`

Answer

D

Explanation

`(R)/(4 pi^(2))(2 pi - theta)theta`
Resistance of smaller chord xwy

`R_(1) = gamma xx ` Length of
smaller chord
= `(R)/(2 pi r) xx r theta = (R theta)/(2 pi)` .... (1)
Resistance of larger chord
`R_(1) = gamma xx ` length of larger chord
` = (R)/(2 pi r ) xx r (2 pi - theta) `
` R_(2) = (R(2pi- theta))/(2 pi ) `.... (2)
equivalent resistance between x and y
`(1)/(R) = (1)/(R_(1)) + (1)/(R_(2))`
`(1)/(R) = (2 pi)/(R theta) + (2 pi)/(R (2 pi - theta)) `
`(1)/(R) = (2 pi(2pi- theta) + 2pi theta)/(R theta (2pi - theta))`
`= (4pi^(2) - 2pi theta + 2 pi theta)/(R theta (2 pi - theta))`
= ` (4 pi^(2))/(R theta ( 2pi - theta))`
` therefore R = (R theta(2pi - theta))/(4 pi^(2)) `