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Equivalent focal length and principal points

Short answer

Using 1/F = 1/10 + 1/4 - 5/(10×4) gives F = 4.44 cm. The first principal point is 5.56 cm right of the first lens (0.56 cm beyond the second lens); the second principal point is 2.22 cm left of the second lens (2.78 cm from the first lens).

Two thin convex lenses having focal lengths 10cm and 4cm are coaxially separated by a distance of 5cm. Find the equivalent focal length of the combination. Determine also the positions of the principal points.

Given: f₁ = 10 cm, f₂ = 4 cm, d = 5 cm (both convex, in air).

Equivalent focal length

1/F = 1/f₁ + 1/f₂ − d/(f₁f₂)
    = 1/10 + 1/4 − 5/40
    = 0.100 + 0.250 − 0.125 = 0.225
F   = 4.44 cm

Principal points

α = F d / f₂ = 4.444 × 5 / 4  = 5.56 cm  (H₁ from L1, towards L2)
β = F d / f₁ = 4.444 × 5 / 10 = 2.22 cm  (H₂ from L2, towards L1)

 L1                H2     L2 H1
 |-----------------.------|--.
 0                2.78    5.0 5.56   (cm from L1)

H₂ is 5 − 2.22 = 2.78 cm from L1; H₁ lies 0.56 cm beyond L2.

Answer: F = 4.44 cm; the first principal point is 5.56 cm to the right of L1 (0.56 cm beyond L2); the second principal point is 2.22 cm to the left of L2 (2.78 cm from L1).

✓ Answer checked by Claude Opus 5.5 (high effort)

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