A convex lens has a focal length of 10 cm when used in air. An object is then moved from 20 cm to 50 cm in front of the lens. How does the focal length change?
AIt increases — farther objects require longer focal lengths to form sharp images
BIt decreases — the lens must bend light less for a more distant object
CIt stays at 10 cm — focal length is determined by the lens geometry and material, not by object position
DIt depends on the brightness of the object
Focal length is a fixed property of the lens determined entirely by its surface curvatures and refractive index (via the lensmaker's equation). Object distance has no effect on f. What changes when the object moves is where the image forms — governed by the thin lens equation — but the lens itself is unaltered. This is the central misconception: students often confuse image distance (which does change with object distance) with focal length (which does not).
Question 2 Multiple Choice
A contact lens prescription reads −3D. What does this tell you about the lens?
AIt is a converging lens that focuses parallel rays 3 meters away
BIt is a diverging lens; its focal length has a magnitude of approximately 33 cm
CIt is a converging lens with a focal length of 3 meters
DIt bends light less strongly than a +3D lens
Optical power P = 1/f, so f = 1/P = 1/(−3) ≈ −0.33 m. The negative sign means the lens is diverging — it causes parallel rays to spread as if emanating from a virtual focal point on the same side as the incoming light. The magnitude of the focal length is about 33 cm. Option C confuses the sign: a −3D lens is not converging. Option D is wrong because a −3D lens has the same absolute bending strength as +3D, just in the opposite sense.
Question 3 True / False
A diverging lens has negative optical power.
TTrue
FFalse
Answer: True
By convention, a diverging lens has a negative focal length (parallel rays appear to diverge from a virtual focal point on the incoming side). Since P = 1/f, a negative f directly gives negative optical power. Negative power means the lens reduces the convergence of a ray bundle — the opposite of a converging lens. This sign convention is consistent across the lensmaker's equation, the thin lens equation, and optometric prescriptions.
Question 4 True / False
Using a lens material with a higher refractive index makes the focal length longer (weaker optical power).
TTrue
FFalse
Answer: False
The lensmaker's equation is 1/f = (n−1)[1/R₁ − 1/R₂]. A higher refractive index n increases (n−1), which increases 1/f, which means a SHORTER focal length and GREATER optical power (more diopters). This is exactly why high-index lens materials allow thinner eyeglass lenses: they achieve the same correction (same diopter value) with flatter, less curved surfaces, reducing lens thickness.
Question 5 Short Answer
Why do optometrists express prescriptions in diopters rather than focal lengths, and what property of diopters makes them practically useful when combining lenses?
Think about your answer, then reveal below.
Model answer: Diopters (P = 1/f) add linearly when lenses are placed in contact, while focal lengths do not combine simply. Prescribing a +3D and −1D correction together gives +2D total, computable by simple addition. Focal lengths would require the formula 1/f_total = 1/f₁ + 1/f₂ each time, which is less intuitive. Diopters also map inversely to focal length, so stronger correction (shorter focal length) corresponds to a larger diopter number — the scale is more intuitive for clinical use.
The linear additivity of diopters is a direct consequence of the thin-lens combination formula. When two thin lenses touch, their powers add: P_total = P₁ + P₂. This makes diopters the natural unit for ophthalmic work and also for calculating the combined power of lens systems (like camera lens elements). The sign convention — positive for converging, negative for diverging — lets practitioners immediately know the correction type from the number alone.