Question

Two thin convex lenses of focal lengths 2 cm and 6 cm are separated by a distance of 4 cm in air. Arrange the following cardinal points in ascending order on basis of their distance from second lens:
A. First Principal Point
B. First Focal Point
C. Second Focal Point
D. Second Nodal Point

• A. A, B, C, D
• B. A, C, B, D
• C. B, A, D, C
• D. C, B, D, A

Hint:

For a two-lens system, always establish a coordinate system first (e.g., first lens at the origin). Calculate the positions of the principal points relative to the physical lenses, then use these principal points as the reference for locating the focal points.

Answer 1

The Correct Option is B

Step 1: Define the system. Let the first lens (\(L_1\)) be at the origin (\(x=0\)) and the second lens (\(L_2\)) be at \(x=4\) cm. We have \(f_1 = 2\) cm, \(f_2 = 6\) cm, and \(d = 4\) cm.
Step 2: Calculate the equivalent focal length (\(F\)). \[ \frac{1}{F} = \frac{1}{f_1} + \frac{1}{f_2} - \frac{d}{f_1 f_2} = \frac{1}{2} + \frac{1}{6} - \frac{4}{12} = \frac{6+2-4}{12} = \frac{4}{12} \implies F = 3 \, \text{cm} \]
Step 3: Calculate the positions of the Principal Points (\(P_1, P_2\)). - The position of \(P_1\) relative to \(L_1\) is \(\alpha_1 = \frac{dF}{f_2} = \frac{4 \times 3}{6} = 2\) cm. So, the absolute position of \(P_1\) is at \(x=2\). - The position of \(P_2\) relative to \(L_2\) is \(\alpha_2 = -\frac{dF}{f_1} = -\frac{4 \times 3}{2} = -6\) cm. So, the absolute position of \(P_2\) is at \(x = 4 - 6 = -2\).
Step 4: Calculate the positions of the Focal Points (\(F_1, F_2\)). - The position of \(F_1\) is at a distance \(-F\) from \(P_1\). Absolute position of \(F_1\) is \(x = 2 - 3 = -1\). - The position of \(F_2\) is at a distance \(+F\) from \(P_2\). Absolute position of \(F_2\) is \(x = -2 + 3 = 1\).
Step 5: Determine the positions of the Nodal Points (\(N_1, N_2\)). Since the medium is air on both sides, the nodal points coincide with the principal points. So, \(N_1\) is at \(x=2\) and \(N_2\) is at \(x=-2\).
Step 6: Calculate the distance of each point from the second lens (\(L_2\) at \(x=4\)) and order them. - A (First Principal Point at \(x=2\)): Distance = \(|2-4| = 2\) cm. - B (First Focal Point at \(x=-1\)): Distance = \(|-1-4| = 5\) cm. - C (Second Focal Point at \(x=1\)): Distance = \(|1-4| = 3\) cm. - D (Second Nodal Point at \(x=-2\)): Distance = \(|-2-4| = 6\) cm. Arranging these distances in ascending order: A (2 cm)<C (3 cm)<B (5 cm)<D (6 cm). The correct order is A, C, B, D.