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Principal points of 3 cm and 4 cm lenses

Short answer

Equivalent focal length F = 2.4 cm. The first principal point H1 is 1.2 cm to the right of the first lens; the second principal point H2 is 1.6 cm to the left of the second lens. The final image is located 2.86 cm to the right of the second lens (real, inverted).

Two thin converging lenses of focal lengths 3 cm and 4 cm respectively are placed coaxially in air separated by a distance of 2 cm. An object is placed at 4 cm in front of the first lens. Locate the positions of the principal points and final image.

Use the Cartesian sign convention with all distances measured from the lens centre.

Given: f₁ = 3 cm, f₂ = 4 cm, d = 2 cm, object at u = −4 cm from L1.

Equivalent system and principal points

1/F = 1/f₁ + 1/f₂ − d/(f₁f₂)
    = 1/3 + 1/4 − 2/12 = 5/12
F   = 2.4 cm

α = F d / f₂ = 2.4 × 2 / 4 = 1.2 cm   (H₁ is 1.2 cm right of L1)
β = F d / f₁ = 2.4 × 2 / 3 = 1.6 cm   (H₂ is 1.6 cm left of L2)

H₂ lies 2 − 1.6 = 0.4 cm to the right of L1.

 L1   H2        H1           L2
 |----.---------.------------|
 0   0.4       1.2          2.0   (cm from L1)

Final image (lens by lens)

Lens 1:  1/v₁ = 1/3 − 1/4 = 1/12,  v₁ = +12 cm
Lens 2:  object I₁ is 12 − 2 = 10 cm beyond L2, u₂ = +10 cm
         1/v₂ = 1/4 + 1/10 = 7/20,  v₂ = +2.857 cm

Check using the principal points: the object is 4 + 1.2 = 5.2 cm from H₁ (U = −5.2 cm). 1/V = 1/2.4 − 1/5.2 gives V = 4.457 cm from H₂, which is 4.457 − 1.6 = 2.857 cm from L2.

Answer: F = 2.4 cm; H₁ is 1.2 cm to the right of L1; H₂ is 1.6 cm to the left of L2 (0.4 cm to the right of L1); the final image is 2.86 cm to the right of L2 (real, inverted).

✓ Answer checked by Claude Opus 5.5 (high effort)

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