Aniket's Optics Challenges (Part 1)

Classical Mechanics Level 5

Take a beaker as shown in the figure above with it's lower surface silvered so that base can act as a plane mirror . Now, pour liquids with different refractive index one above the other as shown in the above figure. Now place 2 similar lenses one on the base of the beaker and let the other float on the uppermost liquid. Now, place an object symmetrically at a distance of \(10\text{ cm}\) from the upper layer of uppermost liquid.

Let an observer see the image of the object at a distance \(x\) from the upper layer of topmost liquid .

The value of \( x \) can be expressed as \( - \frac{a}{b} \), where \(a\) and \(b\) are coprime positive integers .

Enter your answer as \( a + b\)

Details and Assumptions :

  • Consider only the paraxial rays.

  • In figure the upper curved surface of lens has radius of curvature \( {R}_{1} = 10 cm \) and lower surface of lens has radius of curvature \( {R}_{2} = 20 cm \) and and refractive index is \( \mu = 1.5 \)

  • Refer to figure for information. The base of beaker act as a plane mirror .

  • Lens are very thin (negligible small) .

  • The distance \(x\) is measured taking direction above liquid to be positive and is measured from upper layer of topmost liquid.

  • Observer is observing from air (\( \mu = 1 \) ), symmetrically (i.e. on Principle axis).

  • Lens are similar in context to optics , that is same refractive index and same geometry.

Hint : The light rays will go into the beaker then get reflected and then will come back again and the observer will see the resultant image.

As usual , It is Original. :)

This is a part of my set Aniket's Level 5 Challenges in Classical Mechanics.

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