The Angle Where Light Gets Trapped Inside Glass

Every distance in this chapter obeys one consistent bookkeeping system, the Cartesian sign convention: all distances are measured from the pole of a mirror or the optical centre of a lens, distances measured in the same direction as the incident light are positive, distances measured against it are negative, heights measured upward from the principal axis are positive, and heights measured downward are negative. What makes this convention worth deriving carefully, rather than simply stating, is that it lets a single equation handle every possible case, concave or convex, real or virtual, without needing a separate rule for each. Tracing a ray parallel to the principal axis as it strikes a concave mirror at a point M, calling the angle it makes with the normal at M as q, and using the small-angle approximation valid for rays close to the principal axis (paraxial rays), the geometry of the triangle it forms with the centre of curvature C and the focus F shows directly that FD = CD/2; since D sits essentially at the pole P for such rays, this becomes the mirror's most basic relationship, f = R/2, the focal length is exactly half the radius of curvature. Extending the same triangle-similarity approach to a full object-image pair, comparing triangle A'B'F to MPF and triangle A'B'P to ABP, and then carefully substituting the sign convention (object distance u, image distance v and focal length f all negative for a real object and a real image formed by a concave mirror, since all three are measured backward from the pole), the two similar-triangle relationships combine algebraically into the mirror equation, 1/v + 1/u = 1/f. The very same derivation, followed through for a virtual image, produces linear magnification m = h'/h = -v/u -- and although both formulas were derived here specifically for a real, inverted image in a concave mirror, applying the sign convention correctly makes them valid for every spherical mirror, concave or convex, whether the resulting image is real or virtual.

Locating any image needs only two of four conveniently predictable rays traced from a single object point: a ray parallel to the principal axis, which reflects through the focus (or appears to diverge from it, for a convex mirror); a ray through, or aimed at, the centre of curvature, which strikes the mirror along the normal and simply retraces its own path; a ray through, or aimed at, the focus, which reflects out parallel to the principal axis; and a ray striking the pole at any angle, which reflects obeying the ordinary laws of reflection there. A genuinely surprising question follows directly from this: if the lower half of a concave mirror's reflecting surface is covered with opaque material, what happens to the image? Intuition suggests only half the object would appear, but tracing rays carefully shows this is wrong -- since every remaining point on the mirror still obeys the same laws of reflection, rays from every part of the object still find their way to the same image point via the uncovered half alone, so the COMPLETE image of the whole object still forms, merely dimmer, since only half as much light now reaches it. A second, equally revealing case: an object like a long pin or a mobile phone lying flat along the principal axis itself, rather than standing perpendicular to it, produces an image that is NOT uniformly magnified along its own length, because each point along the pin sits at a different object distance u from the mirror, and since magnification m=-v/u depends on u, points closer to the mirror and points farther from it are magnified by different amounts -- which is exactly why such an image looks visibly distorted, stretched or compressed unevenly along its length, rather than simply being a scaled-up copy of the original.

Refraction obeys two laws established experimentally by Snell: the incident ray, refracted ray and normal all lie in the same plane, and the ratio sin i / sin r is a constant for a given pair of media, called n21, the refractive index of the second medium with respect to the first -- Snell's law. When n21 is greater than one, the refracted ray bends toward the normal, and the second medium is called optically denser than the first; when n21 is less than one, the ray bends away from the normal, exactly what happens when light travels from a denser medium into a rarer one. A clean, useful relationship connects refractive indices measured in opposite directions: if n21 is medium 2's index relative to medium 1, then n12, medium 1's index relative to medium 2, is simply its reciprocal, n12 = 1/n21; extending this further, if n32 is medium 3's index relative to medium 2, then n32 = n31 x n12, letting refractive indices be chained through an intermediate medium. A genuinely important caution belongs here: optical density (refractive index) must never be confused with ordinary mass density; turpentine, for instance, has a LOWER mass density than water yet a HIGHER optical density, since optical density is purely the ratio of light's speed in the two media and has nothing to do with weight. Two everyday consequences follow directly from these laws. Passing through a rectangular slab, a ray bends twice, toward the normal entering (rarer to denser) and away from it leaving (denser to rarer); since the two faces are parallel, these exactly cancel in direction, so the emergent ray runs parallel to the original incident ray, merely displaced sideways (a lateral shift). And viewed from directly above, a tank's water-covered bottom appears raised, its apparent depth equal to its real depth divided by the water's refractive index -- exactly why a pool always looks shallower than it truly is.

Light travelling from an optically denser medium into a rarer one always bends away from the normal, and increasing the angle of incidence increases the angle of refraction faster still, until, at one particular angle of incidence, the refracted ray bends so far away from the normal that it grazes the interface itself, running exactly along the boundary between the two media -- the angle of refraction has reached 90 degrees. This special angle of incidence is the critical angle, ic, defined by sin ic = n21 (the refractive index of the rarer medium with respect to the denser one). Push the angle of incidence past ic, and something dramatic happens: refraction becomes geometrically impossible altogether, since Snell's law would demand a sine greater than one, and the incident ray is instead reflected ENTIRELY back into the denser medium -- total internal reflection, genuinely total, unlike ordinary reflection off any surface, which always leaks some fraction of light through. Diamond, with a refractive index of 2.42, has an unusually small critical angle of only about 24.4 degrees, meaning light entering a cut diamond is very likely to strike an internal face beyond this angle and be totally reflected, bouncing repeatedly inside the stone before finally escaping -- exactly the multiple internal reflections responsible for a diamond's brilliant sparkle. The same effect, deliberately engineered, gives prisms designed with precise angles the ability to bend light by exactly 90 degrees or 180 degrees, or to invert an image without changing its size, using total internal reflection rather than a silvered surface at all. And optical fibres, a core of glass with a higher refractive index surrounded by a cladding of lower refractive index, guide a light signal along their entire length, however bent, through thousands of successive total internal reflections, without any appreciable loss of intensity -- the working principle behind long-distance data transmission and the medical endoscopes used to examine the inside of the stomach or esophagus.

Before reaching the familiar thin lens formula, it helps to first work out how refraction bends light at a single spherical surface, separating a medium of refractive index n1 from one of index n2. Applying Snell's law with the small-angle approximation to a ray travelling from an object point O, striking such a surface at a point M, and using the same paraxial-ray triangle geometry already used for mirrors, produces a general relationship, n2/v - n1/u = (n2-n1)/R, connecting object distance, image distance, the two refractive indices, and the surface's radius of curvature R -- valid for any single curved refracting surface. A thin lens is simply two such curved surfaces close together, bounding a transparent medium; applying this same single-surface relationship first at the entry surface, then again at the exit surface (where the image formed by the first surface now acts as the object for the second), and adding the two resulting equations together, the intermediate image distance cancels out entirely, leaving a clean relationship between only the original object distance, the final image distance, and the two surfaces' radii of curvature. Setting the object at infinity, so the resulting image forms exactly at the lens's own focus, defines the focal length f -- and, after applying the sign convention (u negative, v positive for a real image), this same chain of substitutions collapses to the thin lens formula, 1/v - 1/u = 1/f, true for BOTH convex and concave lenses and for BOTH real and virtual images, exactly mirroring how the single mirror equation handled every mirror case. Locating an image needs, once again, just two of three convenient rays: parallel to the axis (refracting through the second focus, or appearing to diverge from the first, for a concave lens); through the optical centre (passing straight through undeviated); and through, or aimed at, the first focus (emerging parallel to the axis). Magnification follows the same logic as before, m = h'/h = v/u, positive for an erect virtual image and negative for an inverted real one.

Continuing the same two-surface derivation one step further, rather than stopping at the general thin lens formula, gives a relationship connecting a lens's focal length directly to the physical shape it is ground into: 1/f = (n21-1)(1/R1 - 1/R2), where n21 is the lens material's refractive index relative to its surroundings, and R1, R2 are the radii of curvature of its two surfaces -- the lens maker's formula, genuinely useful for designing a lens of a desired focal length from raw glass of known refractive index. The formula holds equally for a concave lens, where the sign convention simply makes R1 negative and R2 positive, correctly producing a negative focal length. A striking demonstration of exactly how much a lens's power depends on the surrounding medium, not just its own shape: a glass lens (refractive index 1.47) can be made to completely DISAPPEAR when dropped into a liquid of matching refractive index, since if n1 (the liquid) equals n2 (the glass), the lens maker's formula gives 1/f = 0, meaning f becomes infinite -- an infinitely-weak lens behaves exactly like a flat sheet of glass, bending light not at all. Since water's refractive index (1.33) does not match ordinary glass (about 1.5), water alone cannot produce this vanishing trick; a liquid like glycerine, whose refractive index sits much closer to glass, is needed instead. This same dependence on surrounding medium has a real, measurable consequence: an identical glass lens of focal length 20 cm in air stretches to a focal length of roughly 78 cm when fully submerged in water, since the lens maker's formula's (n21-1) term shrinks dramatically once the surrounding medium's own refractive index rises closer to the glass's -- exactly why swim goggles are necessary to focus properly underwater, and why the eye's own lens needs the cornea's much larger curvature to compensate for sitting permanently in a watery medium.

A lens's power, P = 1/f, measures exactly how strongly it converges or diverges a parallel beam, in dioptres (D), where one dioptre is the power of a lens with a one-metre focal length; power is positive for a converging (convex) lens and negative for a diverging (concave) lens, so an optician's prescription of +2.5D calls for a convex lens of focal length 40 cm, while -4.0D calls for a concave lens of focal length 25 cm. Placing two thin lenses of focal lengths f1 and f2 directly in contact, and working through the same two-step image-formation logic (the first lens's image serving as the second lens's object), the two lens equations add together directly, and, treated as a single equivalent lens, their combination obeys 1/f = 1/f1 + 1/f2, or, in terms of power, the wonderfully simple P = P1 + P2 -- an algebraic sum, so a converging lens's positive power and a diverging lens's negative power partially cancel when combined. This addition rule extends to any number of thin lenses in contact, and the combination's overall magnification is the PRODUCT of each individual lens's own magnification, m = m1 x m2 x m3 x ..., since each lens's image becomes the next lens's object in turn. Combining lenses this way is not merely a mathematical curiosity: it is the everyday engineering behind cameras, microscopes and telescopes, letting designers combine several standard, simpler lenses to sharpen an image or reach a magnification that no single practical lens could achieve alone, since grinding one single lens to an extreme, very short focal length becomes progressively harder to manufacture without serious distortion.

A triangular prism bends light at two faces in succession: entering at angle i and refracting to r1 at the first face, then travelling to the second face and refracting again, from angle r2 inside the glass out to angle e in air. Simple geometry involving the prism's own apex angle A shows that r1 + r2 = A exactly, and the TOTAL angle of deviation, the angle between the original incident ray and the final emergent ray, works out to d = i + e - A. Since this deviation depends on the angle of incidence, plotting d against i for a range of incidence angles produces a curve with a genuine minimum, the angle of minimum deviation, Dm; at exactly this minimum, remarkably, the incidence and emergence angles become equal (i=e), which in turn forces r1=r2=A/2, since the light ray inside the prism runs exactly parallel to its base at this specific angle. Substituting these equalities back into Snell's law at the minimum-deviation condition yields a clean, experimentally powerful formula for the prism material's refractive index: n21 = sin[(A+Dm)/2] / sin(A/2) -- since both A (the prism's own known angle) and Dm (found simply by rotating the prism and locating the incidence angle where the deviation stops decreasing and starts increasing again) can be measured directly in a laboratory, this equation gives a genuinely practical, precise way of determining an unknown material's refractive index without needing to measure i and r separately at all. For a thin prism, where A itself is small, this same relationship simplifies further, to Dm = (n21-1)A, directly showing that thin prisms deviate light only slightly, exactly the property that makes eyeglass prisms (used to correct certain vision defects) practical to wear without visibly distorting the world.

A simple microscope is nothing more than a single converging lens of short focal length, held close to a small object so that its enlarged, erect, virtual image forms comfortably at the near point (the closest distance, about 25 cm, at which the eye can focus without strain); its magnification works out to m = 1 + D/f, or, if the image instead forms at infinity for more relaxed viewing, the slightly smaller m = D/f -- either way, a lens of focal length 5 cm gives roughly six times magnification, and a single lens's practical magnification rarely usefully exceeds about nine times, since shorter and shorter focal lengths become progressively difficult to grind without serious distortion. A compound microscope overcomes this limit by using two lenses together: an objective of very short focal length forms a real, inverted, already-magnified image of the object, which then serves as the object for a second lens, the eyepiece, functioning exactly like a simple microscope, magnifying that first image further into a final, enlarged, virtual image. The objective's own magnification is mO = L/fo (L being the tube length, the distance between the objective's focal point and the eyepiece's own focal point), the eyepiece contributes its familiar me = 1 + D/fe (or D/fe for a relaxed eye), and since each stage's image becomes the next stage's object, the TOTAL magnification is the product, m = mO x me -- a real compound microscope built with a 1 cm objective, a 2 cm eyepiece, and a 20 cm tube length reaches a combined magnification of roughly 250 times, far beyond any single lens's reach. A telescope tackles the opposite problem, a distant rather than a nearby object, using an objective with a LARGE focal length and aperture (rather than a short one) to form a real image of a far-off object, which the eyepiece then magnifies; its magnifying power is simply m = fo/fe, and, since a telescope's light-gathering power and resolving power both depend on its objective's diameter, the very largest telescopes replace the objective LENS with an objective MIRROR instead (a reflecting telescope), avoiding the colour-distorting chromatic aberration a large lens would introduce and the sheer weight problem of supporting a huge lens only at its rim -- the Cassegrain design, using a secondary mirror to fold the light path back through a hole in the primary mirror, is used in India's own largest telescope, the 2.34 m reflector at Kavalur, Tamil Nadu, and in the 10-metre Keck telescopes in Hawaii, among the largest reflecting telescopes on Earth.