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A thin film of uniform thickness with refractive index n1 = 1.4 is coated on the convex spherical surface (radius R) at one end of a long solid glass cylinder of refractive index n2 = 1.5. Parallel rays enter from air to glass through the film and focus at distance |f1| from the film. Parallel rays entering from glass to air (through the film) focus at |f2|. Which of the following is/are correct?

1. |f1| = 3R
2. |f1| = 2.8R
3. |f2| = 2R
4. |f2| = 1.4R

Correct answer: |f1| = 3R

Solution

The thin film (uniform thickness) does not affect focusing since both surfaces contribute zero net power for a parallel beam. For air to glass (n1=1, n2=1.5) at convex surface of radius R: n2/v - n1/u = (n2-n1)/R; u = infinity; 1.5/v = 0.5/R; v = 3R. So |f1| = 3R. For glass to air: n1=1.5, n2=1, surface appears concave from glass side (R negative for glass-to-air going left through the same surface). n2/v - n1/u = (n2-n1)/R; here light travels from glass to air so n1=1.5, n2=1, and the surface has radius R but now it's diverging. 1/v - 1.5/infinity = (1-1.5)/R; 1/v = -0.5/R; v = -2R (virtual, behind surface). |f2| = 2R.

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