Aniket's Optics Challenges (Part 1)

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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 $a + b$

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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.

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As usual , It is Original . :)

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