-->

Wave Optics

Question
CBSEENPH12039180

A biconvex lens with its two faces of equal radius of curvature R is made of a transparent medium of refractive index straight mu1. It is kept in contact with a medium of refractive index as shown in the figure.

                                                  

a) Find the equivalent focal length of the combination.

b) Obtain the condition when this combination acts as a diverging lens.

c) Draw the ray diagram for the case mu subscript 1 space greater than thin space left parenthesis mu subscript 2 space plus space 1 right parenthesis divided by 2 space comma spacewhen the object is kept far away from the lens. Point out the nature of the image formed by the system.

Solution

a) Using the Len’s maker’s formula, we have

 1 over straight f space equals space open parentheses mu space minus space 1 close parentheses space open square brackets fraction numerator 1 over denominator space R subscript 1 end fraction space minus space 1 over R subscript 2 close square brackets

L e t space f subscript 1 space a n d space f subscript 2 space b e space t h e space f o c a l space l e n g t h s space o f space t h e space
t w o space m e d i u m. space

1 over straight f subscript 1 space equals space open parentheses mu space minus space 1 close parentheses space open square brackets fraction numerator 1 over denominator space R end fraction space minus space open parentheses negative 1 over R subscript 2 close parentheses close square brackets

rightwards double arrow space 1 over straight f space equals space open parentheses mu space minus space 1 close parentheses open parentheses 2 over straight R close parentheses

space 1 over straight f subscript 2 space equals space open parentheses mu subscript italic 2 space minus space 1 close parentheses space open square brackets open parentheses negative fraction numerator 1 over denominator space R end fraction close parentheses space minus space 1 over infinity close square brackets

rightwards double arrow space 1 over straight f subscript 2 space equals space open parentheses mu subscript italic 2 space minus space 1 close parentheses space open parentheses negative 1 over straight R close parentheses
Let feq is the equivalent focal length of the combination, then
1 over f subscript e q end subscript space equals space 1 over f subscript 1 plus 1 over f subscript 2

rightwards double arrow space 1 over straight f subscript eq space equals space fraction numerator 1 space left parenthesis straight mu subscript 1 space minus space 1 right parenthesis over denominator straight R end fraction space minus space fraction numerator left parenthesis straight mu subscript 2 space minus space 1 right parenthesis over denominator straight R end fraction

rightwards double arrow space 1 over straight f subscript eq space equals space fraction numerator 2 straight mu subscript 1 space minus space straight mu subscript 2 space minus space 1 over denominator straight R end fraction

rightwards double arrow space straight f subscript eq space equals space fraction numerator straight R over denominator 2 straight mu subscript 1 space minus space straight mu subscript 2 space minus space 1 end fraction 
b) 

For, the combination of lenses to behave as diverging lens, equivalent focal length < 0.

 rightwards double arrow space fraction numerator R over denominator 2 mu subscript 1 space minus space mu subscript 2 minus 1 end fraction less than space 0

rightwards double arrow space space 2 straight mu subscript 1 space minus space straight mu subscript 2 space minus space 1 space less than 0

rightwards double arrow space straight mu subscript 1 space less than space fraction numerator left parenthesis straight mu subscript 2 space plus space 1 right parenthesis over denominator 2 end fraction 
c) 
For mu subscript 1 space greater than thin space left parenthesis mu subscript 2 space plus space 1 right parenthesis divided by 2 comma spacethe combination of lenses will behave as converging lens. An object placed far from the lens will form image at the focus of the lens.


Image formed is real and diminished in nature.

Some More Questions From Wave Optics Chapter