Questions: Compound Optical Systems: Lenses and Mirrors in Combination

Question 1

A two-lens system has magnifications M₁ = −3 and M₂ = −2. What is the total magnification of the system?

• A. −5 (magnifications add)
• B. +6 (product, two sign flips)
• C. −6 (product, sign retained)
• D. +1 (they cancel)

Answer: (B)

Total magnification in a compound system is the *product* of individual magnifications: M_total = M₁ × M₂ = (−3)(−2) = +6. The positive sign means the final image is upright relative to the original object — two inversions cancel. A common error is to add the magnifications (−3 + −2 = −5), but magnifications multiply because each lens applies its transformation to the output of the previous one.

Question 2

Two thin lenses with focal lengths f₁ and f₂ are placed in contact (separation d = 0). What is the effective focal length of the combined system?

• A. f_eff = f₁ + f₂
• B. f_eff = (f₁ + f₂) / 2
• C. 1/f_eff = 1/f₁ + 1/f₂
• D. 1/f_eff = 1/f₁ − 1/f₂

Answer: (C)

The general formula is 1/f_eff = 1/f₁ + 1/f₂ − d/(f₁f₂). When d = 0, the last term vanishes, giving 1/f_eff = 1/f₁ + 1/f₂. This is the optical power additive rule: powers (P = 1/f, measured in diopters) add directly when lenses are in contact. Note that focal lengths do NOT add directly — it is the reciprocals (powers) that add.

Question 3

In a compound microscope, the intermediate image formed by the objective lens is a real, magnified image that serves as the object for the eyepiece.

• T. True
• F. False

Answer: True

This is exactly the chain rule of compound optics: the objective forms a real, inverted, magnified image of the specimen somewhere inside the instrument body. The eyepiece then acts as a magnifying glass viewing that intermediate image. If the intermediate image were virtual, the eyepiece could not form a final real image. The two lenses multiply their magnifications precisely because one's output is the other's input.

Question 4

In a compound microscope, the total magnification equals the sum of the objective magnification and the eyepiece magnification.

• T. True
• F. False

Answer: False

Magnifications in a compound optical system *multiply*, not add. A 10× objective combined with a 10× eyepiece gives 100× total, not 20×. This multiplicative behavior is the whole point of compound systems — it is why a microscope can achieve magnifications that a single lens of any practical focal length could not.

Question 5

Why does a compound microscope achieve much greater magnification than a single lens with the same objective focal length?

Model answer: The objective forms a magnified real intermediate image, which the eyepiece then re-magnifies. Because total magnification is the product of the two individual magnifications, the system multiplies the gains rather than adding them. A single lens producing the same total magnification would require either an impractically short focal length or an impractically large image distance.

The key is that each lens's output becomes the next lens's input, so magnifications compound multiplicatively. The intermediate image also allows each lens to operate in a regime where it performs well — the objective at high magnification over short distances, the eyepiece as a comfortable magnifying glass — rather than asking one lens to do everything at once.