Ray Optics and Optical Instruments

2.1 Core concepts

• Light is the electromagnetic radiation in the visible range (~400–750 nm); in vacuum its speed c = 2.99792458 × 10^8 m s^-1, usually taken as 3 × 10^8 m s^-1, and this is the highest attainable speed in nature (NCERT §9.1, p. 221).
• The ray picture treats a light wave as travelling along a straight line because the wavelength is much smaller than ordinary objects; a bundle of such rays is a beam (NCERT §9.1, p. 222).
• Laws of reflection: angle of reflection equals angle of incidence, and the incident ray, reflected ray, and the normal lie in the same plane; for spherical mirrors the normal is along the radius through the point of incidence (NCERT §9.2, p. 222).
• Cartesian sign convention: distances are measured from the pole/optical centre; distances along the direction of incident light are positive, those opposite are negative; heights above the principal axis are positive, below are negative (NCERT §9.2.1, p. 222–223).
• Focal length of a spherical mirror is the distance between focus F and pole P; for paraxial rays geometry gives f = R/2 (NCERT §9.2.2, p. 223).
• Mirror equation derived from similar triangles: 1/v + 1/u = 1/f; valid for both concave and convex mirrors and for real or virtual images (NCERT §9.2.3, p. 225).
• Linear magnification m = h′/h = −v/u; sign indicates whether the image is erect/virtual (positive) or inverted/real (negative) (NCERT §9.2.3, p. 226).
• Snell's law of refraction: sin i / sin r = n21, where n21 is the refractive index of medium 2 with respect to medium 1; the incident ray, refracted ray and normal are coplanar (NCERT §9.3, p. 228).
• n12 = 1/n21, and for three media n32 = n31 × n12; optical density is not the same as mass density (e.g. turpentine is less mass-dense than water but optically denser) (NCERT §9.3, p. 228–229).
• For a parallel-sided slab the emergent ray is parallel to the incident ray (no deviation, only lateral shift); apparent depth = real depth / refractive index for near-normal viewing (NCERT §9.3, p. 229).
• Total internal reflection (TIR) happens when light goes from denser to rarer medium and the angle of incidence exceeds the critical angle ic, where sin ic = n21 (n21 < 1) or n12 = 1/sin ic; no transmission occurs (NCERT §9.4, p. 229–230).
• Table 9.1 lists critical angles with respect to air: water 48.75°, crown glass 41.14°, dense flint glass 37.31°, diamond 24.41° — the small ic of diamond explains its sparkle (NCERT §9.4, Table 9.1, p. 230).
• TIR applications include 90°/180° totally reflecting prisms, optical fibres (core has higher refractive index than cladding; >95% transmission over 1 km in silica), endoscopy "light pipes" (NCERT §9.4.1, p. 231–232).
• Refraction at a single spherical surface: n2/v − n1/u = (n2 − n1)/R, derived from Snell's law in small-angle approximation (NCERT §9.5.1, p. 233).
• Lens maker's formula (lens of refractive index n in air): 1/f = (n − 1)(1/R1 − 1/R2), with R1 positive and R2 negative for a double-convex lens; f turns out negative for a concave lens (NCERT §9.5.2, p. 234).
• Thin lens formula: 1/v − 1/u = 1/f, valid for both convex and concave lenses, real and virtual images; lens magnification m = h′/h = v/u (NCERT §9.5.2, p. 235).
• Power of a lens P = 1/f (f in metres); SI unit dioptre (D), 1 D = 1 m^-1; positive for converging, negative for diverging (NCERT §9.5.3, p. 236).
• For thin lenses in contact 1/f = 1/f1 + 1/f2 + … so P = P1 + P2 + … (algebraic sum); total magnification of the combination is m = m1 m2 m3 … (NCERT §9.5.4, p. 237–238).
• Prism geometry: r1 + r2 = A and d = i + e − A; the deviation curve has a minimum where i = e, giving r1 = r2 = A/2 and i = (A + Dm)/2; the prism formula is n21 = sin[(A + Dm)/2] / sin(A/2) (NCERT §9.6, p. 239–240).
• For a thin prism (small A), Dm ≈ (n21 − 1)A — thin prisms deviate light only slightly (NCERT §9.6, p. 240).
• Simple microscope (magnifier): a short focal-length convex lens; magnification with image at near point m = 1 + D/f, with image at infinity m = D/f; D = 25 cm is the least distance of distinct vision (NCERT §9.7.1, p. 240–242).
• Compound microscope uses an objective forming a real, inverted, magnified intermediate image at/near the focal plane of the eyepiece; mo = L/fo (L = tube length between second focal point of objective and first focal point of eyepiece); total magnification m = (L/fo)(1 + D/fe) for image at near point, m = (L/fo)(D/fe) for image at infinity (NCERT §9.7.1, p. 243–244).
• Refracting telescope: objective has large focal length and aperture, eyepiece short focal length; in normal adjustment m = fo/fe, tube length fo + fe; terrestrial telescopes add inverting lenses for an erect image (NCERT §9.7.2, p. 244–245).
• Reflecting (Cassegrain) telescope uses a concave primary mirror with a convex secondary mirror that sends light back through a hole in the primary; mirrors have no chromatic aberration, are lighter, and can be supported across the back; world's largest pair is the Keck telescopes (10 m mirrors) in Hawaii; largest in India is the 2.34 m Cassegrain at Kavalur (NCERT §9.7.2, p. 245–246).

2.2 Definitions to memorise

2.3 Diagrams / processes to remember

• Fig. 9.1 — laws of reflection at a curved surface (p. 222).
• Fig. 9.2 — Cartesian sign convention diagram (p. 222).
• Fig. 9.3 — focus of concave and convex mirrors with parallel paraxial rays (p. 223).
• Fig. 9.4 — geometry used to prove f = R/2 (p. 223–224).
• Fig. 9.5 / 9.6 — ray diagrams for image formation by concave and convex mirrors (p. 224–226).
• Fig. 9.8 — Snell's law refraction; Fig. 9.9 — lateral shift through a parallel slab; Fig. 9.10 — apparent depth (p. 228–229).
• Fig. 9.11 — refraction and TIR from a denser medium; Fig. 9.12 — laser-beam demonstration of TIR in a beaker (p. 230–231).
• Fig. 9.13 — 90°/180° prisms and inverting prisms using TIR (p. 231).
• Fig. 9.14 — multiple TIR inside an optical fibre (p. 232).
• Fig. 9.15 — refraction at a single spherical surface (p. 232–233).
• Fig. 9.16 / 9.17 — image formation by a double-convex/concave lens and standard ray diagrams (p. 234–235).
• Fig. 9.18 — power of a lens defined via deviation of a parallel ray at unit height (p. 236).
• Fig. 9.19 — image formation by two thin lenses in contact (p. 237).
• Fig. 9.21 / 9.22 — prism geometry and the d-vs-i plot (p. 239–240).
• Fig. 9.23 — simple microscope, image at near point and at infinity (p. 241).
• Fig. 9.24 — compound microscope ray diagram (p. 243).
• Fig. 9.25 — refracting telescope; Fig. 9.26 — Cassegrain reflecting telescope (p. 245–246).

2.4 Common confusions / NTA trap points

• "Optical density" is about speed of light in the medium (refractive index), not mass per unit volume — turpentine is mass-lighter than water but optically denser (p. 229).
• Mirror equation is 1/v + 1/u = 1/f, but the lens formula is 1/v − 1/u = 1/f — the signs of the u-term differ. NTA distractors swap the two.
• For a concave mirror, f is negative; for a convex mirror, f is positive — students often invert this because "concave" sounds positive (Summary §4, p. 247).
• sin ic = n21 here uses n21 = rarer w.r.t. denser (so n21 < 1); the more common written form is sin ic = 1/n where n is denser w.r.t. rarer (p. 230). Watch which n is in the question.
• In the prism formula the angle at minimum deviation is i = (A + Dm)/2, NOT (A + Dm); and at minimum deviation r1 = r2 = A/2 (p. 240).
• Simple-microscope magnification has two forms: 1 + D/f (image at near point) vs D/f (image at infinity). They differ by 1; pick the one matching the question's stated condition (p. 241–242).
• For combination of lenses, P = P1 + P2 … is algebraic — convex contributes positive, concave negative; sign errors are the most common trap (p. 238).
• Cassegrain uses a CONVEX secondary mirror (not concave) to send light back through a hole in the primary (p. 246).