5.4 Geometrical Optics: Mirrors, Lenses & the Eye

Geometrical Optics: Mirrors, Lenses & the Eye

Geometrical optics treats light as traveling in straight-line rays, using simple geometric rules to predict how mirrors and lenses form images. This is the physics behind cameras, microscopes, corrective eyewear, and the eye itself.

Reflection from a Plane Surface

The law of reflection states that when a ray of light strikes a smooth surface, the angle of incidence equals the angle of reflection — both measured from the normal, an imaginary line perpendicular to the surface at the point of contact (not from the surface itself). A flat (plane) mirror forms a virtual image: upright, the same size as the object, and located as far behind the mirror as the object is in front of it. The image is virtual because the reflected rays only appear to diverge from a point behind the mirror — they never actually converge there.

Refraction and Snell's Law

When light passes from one transparent medium into another, its speed changes, and if it strikes the boundary at an angle other than 90°, its direction changes too — this bending is refraction. Each medium is characterized by its refractive index:

n = c / v

where c is the speed of light in vacuum and v is the speed of light in that medium. Because v ≤ c always, n ≥ 1 always (n = 1 exactly for vacuum). A higher refractive index means light travels more slowly through that medium (water: n ≈ 1.33; crown glass: n ≈ 1.5; diamond: n ≈ 2.4).

Refraction at an interface obeys Snell's Law:

n1 sinθ1 = n2 sinθ2

where θ1 and θ2 are the angles of incidence and refraction, both measured from the normal. The practical rule to memorize: light bends toward the normal when entering a medium of higher refractive index (slowing down), and bends away from the normal when entering a medium of lower refractive index (speeding up).

Dispersion

A medium's refractive index actually depends slightly on the wavelength of light passing through it — this wavelength-dependence is dispersion. Shorter-wavelength (higher-frequency, violet/blue) light typically experiences a slightly higher refractive index, and therefore bends more, than longer-wavelength (red) light in the same medium. This is why a glass prism spreads white light into a full spectrum, with violet bending most and red bending least, and why rainbows form as sunlight disperses and internally reflects inside spherical raindrops.

Total Internal Reflection

When light travels from a higher-index medium into a lower-index medium (e.g., glass into air), Snell's Law predicts that the refracted ray bends away from the normal. As the angle of incidence increases, the angle of refraction increases faster, until at some critical angle (θc) the refracted ray grazes exactly along the boundary (θ2 = 90°). Beyond this critical angle, refraction becomes impossible — all of the light reflects back into the original medium instead. This is total internal reflection (TIR), and the critical angle is found by setting θ2 = 90° in Snell's Law:

n1 sinθc = n2 sin(90°) = n2, so sinθc = n2 / n1

TIR requires two conditions simultaneously: light must be moving from a higher-index to a lower-index medium, and the angle of incidence must exceed θc. TIR is the operating principle of fiber-optic cables and medical endoscopes, where light is guided along a thin, flexible glass or plastic core by repeated total internal reflection off the core-cladding boundary, with essentially no loss of light at each bounce.

Spherical Mirrors

A spherical mirror is a small section of a sphere's surface. Its center of curvature (C) is the center of that sphere, and its focal point (f) — where parallel rays converge (or appear to diverge from) after reflecting — sits halfway between the mirror's surface and C:

f = R / 2

where R is the radius of curvature. A concave mirror (reflective surface curves inward, like the inside of a bowl) is converging: it can form either a real, inverted image (when the object is farther from the mirror than f) or a magnified, virtual, upright image (when the object is closer than f — the basis of a shaving/makeup mirror). A convex mirror (reflective surface curves outward) is diverging: it always forms a virtual, upright, reduced image, regardless of object distance — this wide field of view is exactly why convex mirrors are used for car passenger-side mirrors and store security mirrors.

The Mirror/Thin-Lens Equation and Sign Conventions

Both spherical mirrors and thin lenses obey the same governing equation, relating object distance p, image distance q, and focal length f:

1/p + 1/q = 1/f

Memorizing a consistent sign convention is essential:

Magnification is m = −q/p; its magnitude tells you the image size relative to the object (|m| > 1 magnified, |m| < 1 reduced), and its sign tells you orientation (positive = upright, negative = inverted).

Thin Lenses

A converging (convex) lens is thicker in the middle than at the edges and bends parallel rays inward to a real focal point on the far side (positive f); it can form either real, inverted images (object beyond f) or magnified, virtual, upright images (object within f, as in a magnifying glass). A diverging (concave) lens is thinner in the middle than at the edges, spreads parallel rays apart as if they originated from a virtual focal point on the same side as the incoming light (negative f), and always produces a virtual, upright, reduced image, regardless of object distance — mirroring the behavior of a convex mirror.

Lens strength (power) is measured in diopters (D), the reciprocal of the focal length in meters:

D = 1 / f(m)

A short focal length means a strong, high-diopter lens; a long focal length means a weak lens close to zero diopters. When two thin lenses are placed in contact (combination of lenses), their powers simply add:

D(total) = D1 + D2

This additive rule is exactly how prescription eyewear and compound optical systems are designed and specified — a single equivalent lens can replace a stack of individual lenses in contact.

Lens Aberration and Optical Instruments

Real lenses don't focus light perfectly. Chromatic aberration arises because a lens's refractive index (like any medium) varies with wavelength (dispersion) — different colors of light focus at slightly different points, producing color fringing at the edges of an image. Spherical aberration arises because a lens or mirror with a perfectly spherical shape does not focus rays striking near its edge to the same point as rays striking near its center (paraxial rays) — only a parabolic shape focuses all parallel rays to a single point exactly. Both aberrations are minimized in precision optical instruments using compound (multi-element) lens systems or aspheric lens shapes.

The human eye is the MCAT's favorite optical instrument. Light first refracts at the cornea, the curved, fixed-power outer surface that contributes roughly two-thirds of the eye's total refractive power, then passes through the pupil and is fine-tuned by the crystalline lens, whose shape the ciliary muscles adjust (accommodation) to focus objects at different distances onto the retina, where photoreceptors convert the focused image into neural signals.

Two common refractive errors:

• Myopia (nearsightedness): the eyeball is too long (or the lens/cornea too strong), so parallel rays from distant objects focus in front of the retina. Corrected with a diverging (concave, negative-diopter) lens that spreads rays slightly before they enter the eye, pushing the focal point back onto the retina.
• Hyperopia (farsightedness): the eyeball is too short (or the lens/cornea too weak), so rays focus behind the retina. Corrected with a converging (convex, positive-diopter) lens that adds convergence before light enters the eye.

Presbyopia, the age-related stiffening of the lens and weakening of accommodation, reduces the eye's ability to focus on near objects and is why most people eventually need reading glasses regardless of their distance-vision prescription.

Worked Example: The Thin-Lens Equation

An object is placed 30 cm in front of a converging lens with a focal length of 10 cm. Find the image distance, magnification, and describe the image.

Step 1 — Solve for image distance using 1/p + 1/q = 1/f:

1/30 + 1/q = 1/10

1/q = 1/10 − 1/30 = 3/30 − 1/30 = 2/30 = 1/15

q = 15 cm

Because q is positive, the image is real, forming on the opposite side of the lens from the object.

Step 2 — Solve for magnification: m = −q/p = −15/30 = −0.5

The negative sign means the image is inverted; the magnitude 0.5 means the image is reduced to half the object's height.

Step 3 — Convert the focal length to lens power in diopters: f = 10 cm = 0.10 m, so D = 1/f = 1/0.10 = 10 diopters — a fairly strong lens.

Summary: this converging lens forms a real, inverted, reduced image 15 cm behind the lens, at half the object's size — exactly the kind of image a camera lens forms on its sensor when photographing a distant object. As a sanity check, note the object (30 cm) sits beyond the focal length (10 cm) but not extremely far beyond it, consistent with a real image that is reduced but not by an enormous factor.

Common MCAT Traps

• Measuring angles from the surface instead of the normal. Both the law of reflection and Snell's Law define angles relative to the normal (perpendicular to the surface), not the surface itself.
• Forgetting TIR requires higher-to-lower index travel. Total internal reflection can never occur when light moves from a lower-index into a higher-index medium — Snell's Law never produces an undefined sine in that direction.
• Mixing up convex mirror and convex lens behavior. A convex mirror always diverges light (like a diverging lens), while a convex lens always converges light — "convex" alone doesn't tell you converging vs. diverging without specifying mirror or lens.
• Dropping the sign convention in the lens/mirror equation. A negative f (diverging lens/convex mirror) or a negative q (virtual image) carries essential information about image type; treating all values as positive gives a numerically wrong answer, not just a labeling error.
• Assuming the cornea does the fine-focusing. The cornea's power is fixed; it's the crystalline lens, adjusted through accommodation, that performs the moment-to-moment focusing adjustments for near versus far objects.