Match List I with List II
| List I | List II |
| A. Double Convex Lens | I. \( f = \displaystyle\frac{R}{\left(1 - n_{g}^{a}\right)} \) |
| B. Plane Convex Lens | II. \( f = \displaystyle\frac{R}{2\left(n_{g}^{a} - 1\right)} \) |
| C. Double Concave Lens | III. \( f = \displaystyle\frac{R}{\left(n_{g}^{a} - 1\right)} \) |
| D. Plano Concave Lens | IV. \( f = \displaystyle\frac{R}{2\left(1 - n_{g}^{a}\right)} \) |
Choose the correct answer from the options given below:
Answer & explanation
Correct answer: option 2
The correct answer is Option (2) → A-II, B-III, C-IV, D-I
Each focal length follows from the lens maker's formula (with \( n = n_{g}^{a} \), the refractive index of glass with respect to air, and both curved surfaces of radius \( R \)):
\( \displaystyle \frac{1}{f} = (n - 1)\left(\frac{1}{R_{1}} - \frac{1}{R_{2}}\right) \)
A plane surface has \( R = \infty \), so its term \( 1/R \) is zero. With the Cartesian sign convention, a convex lens has \( R_{1} = +R,\ R_{2} = -R \) and a concave lens has \( R_{1} = -R,\ R_{2} = +R \).
(A) Double Convex Lens — this matches (II) \( f = R/2(n_{g}^{a} - 1) \). Both surfaces bend the light towards the axis.
\( \displaystyle \frac{1}{f} = (n - 1)\left(\frac{1}{R} + \frac{1}{R}\right) = \frac{2(n - 1)}{R} \;\Rightarrow\; f = \frac{R}{2(n - 1)} \)
(B) Plane Convex Lens — this matches (III) \( f = R/(n_{g}^{a} - 1) \). Only one surface is curved.
\( \displaystyle \frac{1}{f} = (n - 1)\left(\frac{1}{R} - \frac{1}{\infty}\right) = \frac{n - 1}{R} \;\Rightarrow\; f = \frac{R}{n - 1} \)
(C) Double Concave Lens — this matches (IV) \( f = R/2(1 - n_{g}^{a}) \). The focal length is negative, which the factor \( (1 - n) \) gives.
\( \displaystyle \frac{1}{f} = (n - 1)\left(-\frac{1}{R} - \frac{1}{R}\right) = -\frac{2(n - 1)}{R} \;\Rightarrow\; f = \frac{R}{2(1 - n)} \)
(D) Plano Concave Lens — this matches (I) \( f = R/(1 - n_{g}^{a}) \). One curved surface, and a negative focal length.
\( \displaystyle \frac{1}{f} = (n - 1)\left(-\frac{1}{R} - \frac{1}{\infty}\right) = -\frac{n - 1}{R} \;\Rightarrow\; f = \frac{R}{1 - n} \)