A telescope has an objective of focal length `50cm` and an eyepiece of focal length `5cm.` The least distance of distinct vision is `25cm.` The telescope is focused for distinct vision on a scale `2m` away from the objective. Calculate (a) magnification produced and (b) separation between objective and eyepiece.

Explanation

Final image is formed at the distance of clear vision.
`f_(0) = 50cm, f_(e) = 5cm, D = 25cm`
For objective:
`u_(0) =- 200cm, f_(0) = 50cm, v_(0) = ?`
`(1)/(v_(0)) - (1)/(u_(0)) = (1)/(f_(0))`
`(1)/(v_(0)) - (1)/(-200) = (1)/(50) rArr (1)/(v_(0)) = (1)/(50) - (1)/(200) = (4-1)/(200)`
`v_(0) = (200)/(3)cm`
For eye-piece:
`(1)/(v_(e)) - (1)/(u_(e)) = (1)/(f_(e))`
`(1)/(-D) -(1)/(u_(e)) = (1)/(5)`
`-(1)/(u_(e)) = (1)/(5)+(1)/(D) = (1)/(5)+(1)/(25) = (5+1)/(25)`
`u_(e) =- (25)/(6)cm`
`m_(0) = (v_(0))/(u_(0)) = (200//3)/(-200) =- (1)/(3)`
`m_(e) = (v_(e))/(u_(e)) = (-25)/(-25//6) = 6`
`(a) m = m_(0) xxm_(e) =- (1)/(3)xx6 =- 2`
`(b) L = v_(0) +|u_(e)| = (200)/(3) +(25)/(6) = (425)/(6)cm`