Why a Straight Pencil Looks Broken in a Glass of Water

Reflection's two governing laws are already familiar from earlier grades: the angle of incidence equals the angle of reflection, and the incident ray, the normal at the point of incidence, and the reflected ray all lie in the same plane, true for every reflecting surface, flat or curved. A spherical mirror, whose reflecting surface forms part of a hollow sphere, curving either inward (a concave mirror) or outward (a convex mirror), needs a small, precise vocabulary before its images can be predicted rather than just observed. The centre point of the mirror's own reflecting surface is its pole (P); the centre of the imaginary sphere the mirror's surface belongs to is its centre of curvature (C), lying in front of a concave mirror but behind a convex one; the distance between them is the radius of curvature (R); and the straight line running through both P and C is the principal axis. Aiming a concave mirror at the Sun and catching the reflected light on a sheet of paper reveals something dramatic: a small, sharp, brilliantly bright spot appears, and left in place for a few minutes, the paper begins to smoke and can genuinely catch fire, because the mirror has concentrated the Sun's parallel rays onto a single point, its principal focus (F), the same experiment must only ever be performed under adult or teacher supervision, and the Sun itself, or its reflection, should never be looked at directly. The distance from the pole to this principal focus is the focal length (f), and for a mirror of small aperture (its reflecting surface's diameter), a clean geometric relationship connects it to the radius of curvature: R = 2f, meaning the principal focus always sits exactly midway between the pole and the centre of curvature. A convex mirror has its own principal focus too, though parallel rays only APPEAR to diverge from a point behind the mirror after reflecting, rather than genuinely converging there.

Placing an object at each of six meaningful positions in front of a concave mirror, at infinity, beyond C, at C, between C and F, at F, and between P and F, and locating the image every time, reveals a strikingly systematic pattern. Far away (at infinity), the image forms right at F, real, inverted, and reduced to a mere point. Just beyond C, the image forms between F and C, still real and inverted, but only slightly diminished. Exactly at C, the image also forms exactly at C, real, inverted, and now the same size as the object. Between C and F, the image forms beyond C instead, real, inverted, and now genuinely enlarged. Right at F, something breaks down entirely: the reflected rays emerge perfectly parallel, meeting only at infinity, so no image forms at any finite distance at all. And between P and F, closer to the mirror than the focus, the image finally becomes virtual, forming behind the mirror, erect, and enlarged, exactly the magnifying, cannot-be-projected-on-a-screen image a shaving mirror or a dentist's mirror is built to produce. A convex mirror's story is far shorter: for an object anywhere between infinity and the pole, the image is always virtual, erect, and diminished, forming somewhere between the pole and F behind the mirror, shrinking toward a point as the object moves to infinity, and growing only slightly larger as the object approaches the mirror.

Locating an image precisely, rather than just describing it, only needs two carefully chosen rays traced from a single point on the object, since any two reflected rays' meeting point (or apparent meeting point) IS the image. Four rays are especially convenient because their reflected paths are simple to predict: a ray parallel to the principal axis reflects through the principal focus (concave) or appears to diverge from it (convex); a ray passing through, or aimed at, the principal focus reflects out parallel to the principal axis; a ray passing through, or aimed at, the centre of curvature strikes the mirror exactly along the normal and reflects straight back along its own path; and a ray striking the pole obliquely reflects obliquely too, obeying the ordinary laws of reflection there, both making equal angles with the principal axis. Any two of these four, traced from the same object point, cross (or their extensions cross) exactly at the image of that point. These predictable behaviours translate directly into real devices. Concave mirrors, capable of producing a magnified image, are used in torches, search-lights and vehicle headlights (to throw a powerful, near-parallel beam forward), as shaving mirrors, by dentists to see an enlarged view of teeth, and, at large scale, to concentrate sunlight in solar furnaces. Convex mirrors, always giving a smaller image with a genuinely wide field of view, are used as vehicle side-view mirrors, letting a driver see far more of the road behind than a plane mirror of the same size ever could, exactly why a full-length image of the Taj Mahal can be seen in a small convex mirror set into a wall of the Agra Fort.

Turning the image tables into exact numbers needs a consistent bookkeeping system, the New Cartesian Sign Convention: the pole is the origin, the principal axis is the x-axis, the object always sits to the left (so light travels left to right), distances measured in the direction light travels (to the right) are positive and distances measured backward (to the left) are negative, and heights measured upward from the axis are positive, downward negative. Under this convention, a single relationship, the mirror formula, 1/v + 1/u = 1/f, connects object distance (u), image distance (v), and focal length (f) for every spherical mirror, in every situation, replacing the entire lookup-table approach with one calculation. Magnification, the ratio of image height to object height, m = h'/h, can equally be written in terms of distances, m = -v/u; a negative magnification signals a real, inverted image, while a positive magnification signals a virtual, erect one, and the magnitude of m shows directly whether the image is enlarged (|m| > 1), the same size (|m| = 1), or diminished (|m| < 1). Under this convention, a concave mirror's focal length is negative (since its focus sits in front of the mirror, on the same side as incoming light), while a convex mirror's focal length is positive (its focus sits behind it); getting this sign right at the very start of a calculation is what makes the rest of the arithmetic actually mean something physical.

A pond's bottom looks raised, a coin at the bottom of an opaque bowl reappears the instant water is poured in without moving it at all, and a pencil dipped halfway into a glass of water appears to snap sharply at the water's surface. Every one of these familiar sights traces back to the same cause: light bends whenever it crosses obliquely from one transparent medium into another, a phenomenon called refraction. Tracing a ray's exact path through a rectangular glass slab, entering at one face and exiting at the opposite, parallel face, shows two separate bends, toward the normal on entering the glass (moving from air, a rarer medium, into glass, a denser one) and away from the normal on exiting (moving back from glass into air); because these two faces are parallel, the two bends exactly cancel in direction, so the ray emerges travelling parallel to how it entered, merely shifted sideways. Careful measurement of the angle of incidence and the angle of refraction, for many different entry angles, confirms two precise laws: the incident ray, refracted ray, and normal all lie in the same plane, and the ratio of the sine of the angle of incidence to the sine of the angle of refraction is a genuine constant, for any one pair of media and any one colour of light. This second rule is Snell's law of refraction, sin i / sin r = constant, and that constant is called the refractive index of the second medium with respect to the first.

Light does not travel at the same speed in every medium: it moves fastest in vacuum, at almost exactly 3x10⁸ metres per second, only marginally slower in air, and considerably slower in water or glass, and it is precisely this change in speed that causes the bending. If v1 is light's speed in one medium and v2 its speed in another, the refractive index of the second medium relative to the first is n21 = v1/v2; when the first medium is specifically air (or vacuum), this ratio is called the medium's absolute refractive index, nm = c/v, where c is light's speed in air. Water's refractive index is about 1.33, crown glass's about 1.52, and diamond's a striking 2.42, meaning light travels through diamond at barely 41 percent of its speed in air. A genuinely important distinction is worth holding onto here: a medium's refractive index (or optical density) is NOT the same thing as its ordinary mass density; kerosene, for instance, has a higher refractive index than water despite being less mass-dense, so kerosene counts as optically denser even though it would float on water. The general rule connecting speed and bending direction follows directly: light entering an optically denser medium (where it travels more slowly) bends toward the normal, while light entering an optically rarer medium (where it speeds back up) bends away from the normal, exactly the two opposite bends already seen crossing a glass slab.

A lens is a transparent material bound by at least one curved (spherical) surface; a lens whose two surfaces both bulge outward, thicker at the middle than at its edges, is a double convex (or simply convex) lens, while one whose two surfaces both curve inward, thicker at its edges than the middle, is a double concave (or simply concave) lens. Every lens has two centres of curvature, one for each surface, an optical centre (the lens's own midpoint, through which light passes without any deviation at all), and a principal axis running through both centres of curvature. Aiming a convex lens at the Sun and catching the light on paper reproduces the concave mirror's own trick exactly: the lens converges the Sun's parallel rays to a small, brilliant point, concentrating enough heat there to smoke, and eventually burn, the paper, exactly why convex lenses are also called converging lenses; a concave lens, tested the same way, instead spreads parallel rays apart, so they only appear to diverge from a point on the near side, exactly why concave lenses are called diverging lenses. That convergence or divergence point is the lens's principal focus, and, since light can enter from either face, a lens genuinely has two principal foci, one on each side, equally distant from the optical centre; that shared distance is the lens's focal length.

Testing a convex lens at six object positions, mirrored around its two foci, at infinity, beyond 2F1, at 2F1, between F1 and 2F1, at F1, and between F1 and the optical centre, produces a pattern remarkably similar to a concave mirror's: real, inverted images that shrink from point-sized to same-size to enlarged as the object moves inward from infinity toward 2F1, no image at all at F1, and a virtual, erect, enlarged image once the object sits between F1 and the lens itself, exactly the magnifying-glass configuration. A concave lens is once again simpler, giving a virtual, erect, diminished image for an object at any distance whatsoever. Predicting these images with ray diagrams uses the same two-rays-from-a-point idea already used for mirrors, choosing from three convenient rays: one parallel to the axis, refracting through the far focus (convex) or appearing to diverge from the near focus (concave); one aimed at a focus, emerging parallel to the axis after refraction; and one through the optical centre, passing straight through with no bend at all. A sign convention closely matching the mirror one applies here too, except all distances are measured from the optical centre rather than the pole, and, notably, a convex lens's focal length is positive while a concave lens's is negative, exactly the OPPOSITE sign pattern from mirrors. Under this convention, the lens formula, 1/v - 1/u = 1/f, connects object distance, image distance and focal length for any spherical lens in any situation, and magnification is m = h'/h = v/u, with the very same sign rules as before: negative for real and inverted, positive for virtual and erect.

A lens with a short focal length bends light through sharper angles than one with a long focal length, converging (or diverging) it much closer to the optical centre; this genuinely useful strength is captured in a single number, the power of a lens, defined simply as the reciprocal of its focal length, P = 1/f. Measured in dioptres (D), with focal length expressed in metres, one dioptre is exactly the power of a lens with a one-metre focal length; matching the sign convention already established, a convex (converging) lens always has positive power, and a concave (diverging) lens always has negative power. This is precisely the language an eyeglass prescription actually speaks: a prescription reading +2.0D calls for a convex lens of focal length 0.50 m, while -2.5D calls for a concave lens of focal length 0.40 m, the negative sign telling an optician exactly which kind of lens to reach for before even considering the number. When two or more thin lenses are placed directly in contact, their powers simply add algebraically, P = P1 + P2 + ..., a genuinely convenient property, since it lets an optician combine several known, standard lenses during an eye test and calculate the required total power with nothing more than simple addition, rather than needing to work with focal lengths (which do not add so simply) at all; this same additive trick underlies the multi-lens systems built into cameras, microscopes, and telescopes.