Questions: Thin Lenses and Focal Length

Question 1

A lens has focal length f = −20 cm. What type of lens is it, and what happens to parallel rays that pass through it?

• A. Converging lens; parallel rays focus to a real point 20 cm behind the lens
• B. Diverging lens; parallel rays spread outward and appear to diverge from a virtual point 20 cm in front of the lens
• C. Converging lens; parallel rays bend toward the axis but never actually converge
• D. Diverging lens; parallel rays are absorbed and no image is formed

Answer: (B)

Negative focal length is the defining property of a diverging (concave) lens. Parallel rays entering such a lens are bent away from the optical axis, never converging on the far side. Extending those diverging rays backward reveals they appear to originate from a virtual focal point on the same side as the incoming light, at a distance |f| = 20 cm from the lens. Option A is wrong because positive f is converging; negative f is always diverging.

Question 2

An optometrist prescribes lenses of +2.5 diopters. What do you know about these lenses?

• A. They are converging lenses with focal length 40 cm, used to correct farsightedness
• B. They are converging lenses with focal length 2.5 cm, used to correct nearsightedness
• C. They are diverging lenses with focal length 40 cm, used to correct farsightedness
• D. They are diverging lenses with focal length 0.4 mm, used to correct astigmatism

Answer: (A)

Power P = 1/f, so f = 1/P = 1/2.5 = 0.4 m = 40 cm. Positive power means a converging lens, which adds focusing power to an eye that cannot converge parallel rays onto the retina — the definition of farsightedness (hyperopia). Nearsightedness requires a diverging (negative power) lens. The common error in option B is forgetting to convert diopters to meters before computing focal length.

Question 3

A converging lens usually forms a real, inverted image of any object placed in front of it.

• T. True
• F. False

Answer: False

When an object is placed inside the focal length of a converging lens (object distance < f), the lens acts as a magnifying glass and forms a virtual, upright, magnified image on the same side as the object. A real, inverted image is only formed when the object is beyond the focal point. This is one of the most important subtleties of converging lenses: the same lens can produce fundamentally different image types depending on object position.

Question 4

Lens power is defined as P = 1/f (in diopters) rather than just using focal length because powers of lenses in contact add directly, making compound lens calculations simple.

• T. True
• F. False

Answer: True

When two thin lenses are placed in contact, the combined focal length satisfies 1/f_total = 1/f₁ + 1/f₂, which means P_total = P₁ + P₂. This additive property makes diopters the natural unit for optometrists and optical engineers who combine multiple elements. If focal lengths were used directly, the combined focal length would require a more complex formula. The diopter system also makes it immediately obvious whether a combination is converging (net positive power) or diverging (net negative).

Question 5

Why is lens power defined as P = 1/f rather than simply using focal length f to describe a lens's strength? What physical relationship motivates this definition?

Model answer: Power measures the bending strength of a lens per unit distance. A shorter focal length means light is bent more strongly — the lens converges (or diverges) parallel rays more sharply. Because bending strength is inversely proportional to focal length, P = 1/f captures this directly: doubling the bending strength halves the focal length and doubles the power. The diopter definition also enables simple addition of powers when lenses are combined in contact, reflecting the fact that refractive deflections from successive surfaces accumulate.

The underlying physics is that each lens surface deflects a ray by an angle proportional to the surface's curvature and the refractive index difference. Total deflection determines focal length. A 'stronger' lens deflects more, producing a shorter f and higher P = 1/f. This inverse relationship makes P the natural measure of lens strength, just as spring constant k (not 1/k) is the natural measure of spring stiffness because force is directly proportional to k.