Curved surfaces of a plano-convex lens of refractive index \mu 1 and a plano-concave lens of refractive index \mu 2 have equal radius of curvature as shown in figure. Find the ratio of radius of curvature to the focal length of the combined lenses.

Geometrical Optics PYQ — JEE Main 2021 · 27 Aug · Shift 2 · Q42 with Solution

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Correct answer: B

Interpret the combination

A plano-convex lens of refractive index $\mu_1$ and a plano-concave lens of refractive index $\mu_2$ are combined so that their curved surfaces have the same radius of curvature $R$ .

Because the curved surfaces match, the outer faces are plane, and the combination behaves like a single lens whose only effective refraction comes from the internal curved interface separating media of refractive indices $\mu_1$ and $\mu_2$ .

Use refraction at a spherical surface

For refraction at a spherical surface,

\frac{\mu_2}{v}-\frac{\mu_1}{u}=\frac{\mu_2-\mu_1}{R}

For focal length, object is at infinity, so

\frac{\mu_1}{u}=0

Hence,

\frac{\mu_2}{f}=\frac{\mu_2-\mu_1}{R}

So,

f=\frac{\mu_2 R}{\mu_2-\mu_1}

But since the emergent medium is effectively air after the second plane surface, plane surfaces contribute no power. Thus the equivalent power is determined by the relative index difference across the curved interface, giving

\frac{1}{f}=\frac{\mu_1-\mu_2}{R}

Therefore,

\frac{R}{f}=\mu_1-\mu_2

Match with the options

\frac{R}{f}=\mu_1-\mu_2

So the correct option is:

B: $\mu_1-\mu_2$

Comparison with stored answer

Stored correct answer = B

Derived answer = B

Hence, they agree.

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