Skip to main content

Focal lengths of achromatic lens pair

Short answer

Achromatism and minimum spherical aberration for identical materials give x = (f1+f2)/2 = f1 − f2, so f1 = 3f2. Equivalent focal length F = 1.5 f2, hence f2 = 20 cm and f1 = 60 cm (separation x = 40 cm).

Two thin lenses of focal length f1 and f2 separated by a distance 'x' have an equivalent focal length 30 cm and both lenses are of same material. The combination satisfies the condition of achromatism and minimum spherical aberration. Find the values of f1 and f2.

Conditions for two lenses of the same material separated by x:

  1. Achromatism: x = (f₁ + f₂) / 2
  2. Minimum spherical aberration: x = f₁ − f₂ (taking f₁ > f₂)

Step 1: relation between f₁ and f₂

(f₁ + f₂)/2 = f₁ − f₂   ->   f₁ = 3 f₂

Step 2: equivalent focal length

F = f₁f₂ / (f₁ + f₂ − x)
  = f₁f₂ / ((f₁ + f₂)/2)
  = 2 f₁ f₂ / (f₁ + f₂)
  = 2 (3f₂)(f₂) / (4 f₂)
  = 1.5 f₂

Step 3: values for F = 30 cm

f₂ = 30 / 1.5 = 20 cm
f₁ = 3 × 20  = 60 cm
x  = (60 + 20)/2 = 40 cm

Check: F = 2 × 60 × 20 / 80 = 30 cm; x = f₁ − f₂ = 40 cm.

Answer: f₁ = 60 cm, f₂ = 20 cm (separation x = 40 cm).

✓ Answer checked by Claude Opus 5.5 (high effort)

Related questions

All Engineering Physics chapter 5 questions →