Questions: Lens Combinations and Multi-Element Systems

Question 1

Two lenses, each with focal length 10 cm, are separated by 5 cm. A student calculates the effective focal length using 1/f_eff = 1/10 + 1/10 = 5 cm. What is the error in this approach?

• A. Nothing — 1/f_eff = 1/f₁ + 1/f₂ always holds for two lenses, regardless of separation
• B. This formula only applies when lenses are in contact; with separation, the intermediate image location shifts the geometry and must be tracked sequentially
• C. The student should have used 1/f_eff = 1/f₁ − 1/f₂ for separated lenses of equal focal length
• D. The formula requires the focal lengths to be different before it can be applied

Answer: (B)

The optical power additivity rule (1/f_eff = 1/f₁ + 1/f₂) is valid only when the lenses are in contact — zero separation. When lenses are separated, the image formed by lens 1 falls at a different location depending on the separation distance, shifting the object distance for lens 2. You must apply the thin lens equation sequentially to lens 1, find the intermediate image location, adjust for the separation, then apply it again to lens 2. The contact-lens formula is a special case that breaks down as soon as there is any gap.

Question 2

Lens 1 forms a virtual image 8 cm to its left (d_i1 = −8 cm). Lens 2 is placed 5 cm to the right of lens 1. What is the object distance d_o2 for lens 2?

• A. 5 cm — just the separation distance
• B. 3 cm — separation minus the image distance magnitude
• C. 13 cm — separation plus the distance to the virtual image behind lens 1
• D. −3 cm — negative because the virtual image is behind lens 2

Answer: (C)

The virtual image from lens 1 is 8 cm to the LEFT of lens 1. Lens 2 is 5 cm to the RIGHT of lens 1. So the image is 5 + 8 = 13 cm to the left of lens 2 — a real, positive object distance for lens 2. The sign of d_i1 tells you the image is virtual (behind lens 1), but its physical position is still 8 cm to lens 1's left. Option D would apply if the image fell to the right of lens 2 (inside the lens spacing), making it a virtual object for lens 2.

Question 3

The total magnification of a two-lens system is typically greater than the magnification of either individual lens.

• T. True
• F. False

Answer: False

Total magnification M = m₁ × m₂ is the product of individual magnifications, which can be greater than, equal to, or less than either factor. If m₁ = 2 and m₂ = 3, then M = 6 (greater than both). But if m₁ = 0.5 and m₂ = 0.5, then M = 0.25 (less than both). A lens with |m| < 1 reduces the image; two such lenses compound the reduction. The power of combining lenses is that magnifications multiply — both for amplification (microscopes) and reduction (camera systems).

Question 4

The formula 1/f_eff = 1/f₁ + 1/f₂ gives the correct effective focal length for any two-lens system, regardless of the distance between the lenses.

• T. True
• F. False

Answer: False

This formula applies only to lenses in contact (separation = 0). For separated lenses, the effective system behavior depends on the separation distance — the same two lenses arranged differently produce different effective focal lengths. The correct approach for separated lenses is sequential application of the thin lens equation: find the image from lens 1, use it as the object for lens 2. The contact formula is a useful shortcut for optometrists stacking trial lenses but cannot be generalized to telescope or microscope design.

Question 5

When two lenses are separated by a distance, explain why you cannot simply add their optical powers (1/f values) to find the system's effective focal length. What must you do instead?

Model answer: Adding optical powers assumes both lenses refract the same incoming ray geometry — which is only true when they share the same location. With separation, the image formed by lens 1 becomes the object for lens 2 at a distance that depends on the gap. The object distance for lens 2 changes as the separation changes, altering where the final image forms. Instead, apply the thin lens equation to lens 1 to find the intermediate image, adjust the object distance for lens 2 based on the separation, then apply the thin lens equation to lens 2 for the final result.

The sequential method is not just a computational workaround — it reflects the physics. Each lens independently refracts light based on what it 'sees' as incoming rays. The intermediate image is where lens 1's refracted rays would converge; lens 2 intercepts those rays before or after they converge, and refracts them again. Total magnification is the product of the two stages because magnification compounds multiplicatively, not additively.