Refraction by spherical lenses

A lens is a transparent material, usually glass, bounded by two surfaces of which at least one is spherical (curved). Two shapes matter for this chapter.

• A convex lens is thicker at the middle than at the edges. It is also called a converging lens because it bends parallel rays inward so that they meet — converge — at a point after refraction.
• A concave lens is thinner at the middle than at the edges. It is also called a diverging lens because it bends parallel rays outward, away from the axis, so that they never actually meet but appear to spread out from a point.

A set of shared landmarks describes any spherical lens, and they carry over almost unchanged from the language used for spherical mirrors earlier in the chapter:

• Optical centre ($O$): the central point of the lens; a ray aimed at $O$ passes straight through without any net bending.
• Principal axis: the imaginary line through $O$, perpendicular to both curved surfaces, passing through the centres of curvature of the two surfaces.
• Principal focus ($F$): for a convex lens, the point where rays travelling parallel to the axis actually converge after the two refractions. For a concave lens, the point from which the diverged rays appear to spread out, found by extending the outgoing rays backward. Because light can enter a lens from either face, every lens has two principal foci, one on each side of $O$, at nearly equal distances.
• Focal length ($f$): the distance from $O$ to a principal focus $F$.

The bending itself happens at both curved surfaces of the lens, not one — light refracts once entering the glass and once more leaving it, and the lens's overall converging or diverging behaviour is the combined effect of those two refractions. A useful, exam-friendly way to find a focal length without any formula uses exactly this converging property:

where: $d$ is the distance from the optical centre to the sharp, bright spot formed on a screen when parallel rays — such as sunlight, which arrives effectively parallel because the sun is so far away — are focused by a convex lens; since parallel rays converge, by definition, at the principal focus, that spot marks $F$ directly. Convex lenses appear in magnifying glasses, cameras, and correction for hypermetropia (far-sightedness); concave lenses correct myopia (near-sightedness) and appear wherever a diverging effect is needed.

There is no single formula behind "convex converges, concave diverges" — it follows directly from how a curved surface bends light, so the reasoning, not an equation, is what an exam expects. Think of a convex lens as built from tiny prism-like wedges. Near the edges of a convex lens the two surfaces are tilted like a thin prism with its base toward the rim, and any prism bends light toward its base — so a ray near the top edge of a convex lens is bent downward, and a ray near the bottom edge is bent upward, both toward the axis. Near the very centre the two surfaces are almost parallel, so a ray through the optical centre passes through with negligible net bending. The result is that every incoming parallel ray is bent toward the axis by an amount that grows with distance from the centre, and rays converge to a single point — the principal focus. A concave lens is thickest at the edges and thinnest in the middle, so its edge behaves like a prism with its base toward the axis instead. A prism always bends light toward its base, so rays near the edges are bent away from the axis. Parallel rays entering a concave lens therefore spread apart after refraction; they never physically meet, but tracing the emerging rays backward shows them appearing to originate from a common point on the incoming side — the (virtual) principal focus. This is also why every lens needs two principal foci rather than one. A mirror only ever reflects light back the way it came, so it has a single face and a single focus. A lens is a genuine window — parallel light can be sent through it from the left or from the right — and the converging or diverging behaviour is symmetric, giving a focus on each side of the optical centre at (very nearly) equal focal lengths.

A convex lens held so that sunlight falls directly on it produces a sharp, small, bright spot of light on a sheet of paper held exactly $12\ \text{cm}$ from the lens, and the paper begins to smoke at that distance. Find the approximate focal length of the lens and explain why this method works.

Step 1. Identify what the sunlight is doing optically. The sun is so far away that its rays reach the lens essentially parallel to one another and to the principal axis — the same condition assumed for the definition of a principal focus.

Step 2. Apply the definition of principal focus. A convex lens bends parallel rays inward until they meet at a single point, the principal focus $F$. The sharpest, smallest, brightest spot on the paper is exactly where the maximum concentration of refracted rays overlap, so that spot marks $F$.

Step 3. Read off the focal length. Since the bright spot is $12\ \text{cm}$ from the optical centre of the lens, $f \approx d = 12\ \text{cm}$, using the relation from the core idea section directly — no calculation beyond the definition is needed.

The focal length of the lens is approximately $f \approx 12\ \text{cm}$. The paper smokes because the concentrated sunlight at the focus delivers far more energy per unit area there than anywhere else on the paper, which is exactly why this converging property makes convex lenses useful as burning glasses as well as a quick classroom method for estimating focal length.

• Swapping convex and concave with mirror conventions. For lenses, convex means thicker in the middle (converging, like a magnifying glass); concave means thinner in the middle (diverging, like myopia-correcting spectacles) — this shape-to-name mapping is the same as for mirrors, but the shape-to-effect mapping is different, so students who reason from mirror habits often predict the wrong behaviour.
• Assuming a lens has only one focus. Because light passes through a lens from either side, every lens has two principal foci, one on each side of the optical centre, unlike a mirror which has a single principal focus.
• Treating the lens as bending light once. Refraction happens twice — entering the first curved surface and leaving the second — the lens's converging or diverging behaviour is the net result of both bends, not a single one.
• Believing every ray through the lens bends. Only a ray through the optical centre passes through essentially undeviated; all other rays are refracted, with the amount of bending increasing toward the edges.

Ask the tutor: If you are unsure why a lens needs two focal points while a mirror only needs one, ask the AI tutor to walk through parallel rays entering from both sides of a convex lens.