Power & Lens Combinations
Add powers for lenses in contact; account for the gap when they're apart.
Lesson at a glanceShow the key points
- Power P = 1/f, measured in dioptres (m⁻¹); positive for converging, negative for diverging lenses.
- Lenses in contact: 1/F = 1/f₁ + 1/f₂ (18.19), so P = P₁ + P₂ (18.20).
- A thin lens deviates a ray striking at height h by δ = h/f (18.21).
- For a parallel beam, two lenses a distance d apart act like one lens with 1/F = 1/f₁ + 1/f₂ − d/(f₁f₂) (18.22), placed dF/f₁ behind the second lens (18.23).
- For an object at a finite distance, find the image lens by lens instead.
Opticians don’t talk about focal lengths: a prescription says “+2.5” or “−1.75”. That number is the lens’s power, and powers of lenses in contact simply add.
Power of a lens
A lens of focal length +25 cm has power 1/0.25 m = +4 D; one of −50 cm has power −2 D. A short focal length means a strong lens.
Two thin lenses in contact
Lens L₁ forms an image of the object at v₁; this image is the object for L₂, which forms the final image at v. Since the lenses touch and are thin, the same distance v₁ applies to both:
So the pair behaves exactly like one equivalent lens:
A convex lens of focal length 25 cm is placed in contact with a concave lens of focal length 40 cm. Find the power and focal length of the combination.
How much a lens bends a ray
A ray from an axial object O strikes a thin lens at height h and crosses the axis again at the image I. Its direction changes by δ = α + β, where α ≈ h/PO and β ≈ h/PI for small angles. With PO = −u, PI = v:
Two lenses a distance apart
Separated lenses are not equivalent to one thin lens in general. But for a parallel incident beam they are. A ray at height h₁ is turned by δ₁ = h₁/f₁ at the first lens, travels distance d, reaches the second lens at height h₂ = h₁ − dδ₁, and is turned by δ₂ = h₂/f₂. An equivalent lens of focal length F would turn it by δ = h₁/F = δ₁ + δ₂. Substituting:
The equivalent lens sits where the incident ray and the final emergent ray, extended, cross: a distance dF/f₁ from the second lens, on the side the light comes from (18.23).
Two convex lenses of focal length 20 cm are 10 cm apart. A parallel beam enters. Find the equivalent focal length and where the equivalent lens is.
- 1Power P = 1/f in dioptres (f in m); converging +, diverging −.
- 2Thin lenses in contact: 1/F = 1/f₁ + 1/f₂, P = P₁ + P₂.
- 3A thin lens turns a ray at height h by δ = h/f.
- 4Separated by d (parallel beam): 1/F = 1/f₁ + 1/f₂ − d/f₁f₂; equivalent lens dF/f₁ before the second lens.
Check yourself
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