The eyepiece of a telescope is a system of two biconvex lenses L1 and L2, of focal length 30 cm and 20 cm respectively, with a separation of 2 cm between them. Lens L2 faces the eye. This eyepiece is to be replaced by a single bi-convex lens L. Which one among the following is the correct combination of the focal length (F in cm) of the lens L and its distance (\u03b4 in cm) from the position of lens L1 ?<\/p>\n
[amp_mcq option1=”F = 12\u00b75 and \u03b4 = 1\u00b717″ option2=”F = 12\u00b75 and \u03b4 = 1\u00b715″ option3=”F = 11\u00b75 and \u03b4 = 1\u00b717″ option4=”F = 11\u00b75 and \u03b4 = 1\u00b715″ correct=”option1″]<\/p>\n
Equivalent focal length F:
\n1\/F = 1\/f\u2081 + 1\/f\u2082 – d\/(f\u2081f\u2082)
\n1\/F = 1\/30 + 1\/20 – 2\/(30 * 20)
\n1\/F = (2 + 3)\/60 – 2\/600
\n1\/F = 5\/60 – 1\/300
\n1\/F = 1\/12 – 1\/300
\n1\/F = (25 – 1)\/300 = 24\/300 = 2\/25
\nF = 25\/2 = 12.5 cm.<\/p>\n
This confirms F = 12.5 cm, eliminating options C and D.<\/p>\n
The question asks for the distance (\u03b4) of the equivalent lens L from the position of lens L\u2081. For a system of lenses, the equivalent single lens producing the same effect as the combination is placed at one of the principal planes. Since L\u2082 faces the eye, the light exits towards the eye after passing through L\u2082. The equivalent lens is typically placed at the second principal plane (H\u2082) of the system, as measured from the lens where light exits (L\u2082). The position of H\u2082 is measured relative to L\u2082.
\nThe distance of the second principal plane H\u2082 from L\u2082 is given by \u03b2 = -d * (F \/ f\u2081).
\n\u03b2 = -2 cm * (12.5 cm \/ 30 cm) = -2 * (12.5\/30) = -25\/30 = -5\/6 cm.
\nThe negative sign indicates that H\u2082 is to the left of L\u2082 (towards L\u2081).
\nThe distance of H\u2082 from L\u2081 is the distance L\u2081-L\u2082 plus the distance L\u2082-H\u2082.
\nDistance L\u2081-H\u2082 = d + \u03b2 = 2 cm + (-5\/6 cm) = 2 – 5\/6 = (12 – 5)\/6 = 7\/6 cm.
\n7\/6 cm \u2248 1.1667 cm.
\nThe distance \u03b4 from L\u2081 is approximately 1.17 cm.<\/p>\n
This matches option A (F = 12.5 and \u03b4 = 1.17).
\n<\/section>\n","protected":false},"excerpt":{"rendered":"
The eyepiece of a telescope is a system of two biconvex lenses L1 and L2, of focal length 30 cm and 20 cm respectively, with a separation of 2 cm between them. Lens L2 faces the eye. This eyepiece is to be replaced by a single bi-convex lens L. Which one among the following is … <\/p>\n