A transparent thin film of uniform thickness and refractive index n...

A transparent thin film of uniform thickness and refractive index ${n}_{1} = 1.4$ is coated on the convex spherical surface of radius R at one end of a long solid glass cylinder of refractive index ${n}_{2} = 1.5$ , as shown in the figure. Rays of light parallel to the axis of the cylinder traversing through the film from air to glass get focused at distance f₁ from the film, while rays of light traversing from glass to air get focused at distance f₂ from the film. Then

$A transparent thin film of uniform thickness and refractive index n _ 1 = 1.4 is coated on the convex spherical surface$

Options

A $\left|{f}_{1}\right| = 3R$
B $\left|{f}_{1}\right| = 2.8R$
C $\left|{f}_{2}\right| = 2R$
D $\left|{f}_{2}\right| = 1.4R$

Correct answer

A. $\left|{f}_{1}\right| = 3R$

Step-by-step solution

For air to glass

\dfrac{1.5}{{f}_{1}} = \dfrac{1.4 - 1}{R} + \dfrac{1.5 - 1.4}{R}

∴ {f}_{1} = 3R

For glass to air.

\dfrac{1}{{f}_{2}} = \dfrac{1.4 - 1.5}{- R} + \dfrac{1 - 1.4}{- R}

∴ {f}_{2} = 2R

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