1. New Cartesian Sign Convention (The Non-Negotiable Rules)

Sign errors ruin numerical answers in optics. Memorize these four fundamental rules: (1) All distances are measured from the Pole (P) of the mirror or the Optical Center (O) of the lens; (2) Distances measured in the direction of the incident light are taken as POSITIVE (+); (3) Distances measured opposite to the direction of the incident light are taken as NEGATIVE (-); (4) Heights measured perpendicular to and upward from the principal axis are POSITIVE (+), while downwards are NEGATIVE (-). Consequence: Focal length f is ALWAYS NEGATIVE for concave mirrors and concave lenses; focal length f is ALWAYS POSITIVE for convex mirrors and convex lenses.

2. Spherical Mirror Formula & Magnification

For spherical mirrors of radius of curvature R, focal length f = R/2. The Mirror Equation relates object distance u, image distance v, and focal length f: 1/f = 1/v + 1/u. Linear lateral magnification: m = height of image (h_i) / height of object (h_o) = -v/u = (f - v)/f = f / (f - u). Real and inverted images have m < 0 (negative). Virtual and erect images have m > 0 (positive). If |m| > 1 the image is magnified; if |m| < 1 the image is diminished; if |m| = 1 the image is identical in size to the object.

3. Refraction, Snell's Law & Total Internal Reflection (TIR)

When light travels from medium 1 to medium 2: Absolute refractive index μ = c/v (speed of light in vacuum / speed in medium). Snell's Law: μ1 sin i = μ2 sin r, or 1_μ_2 = sin i / sin r = v1/v2 = λ1/λ2. Frequency (ν) remains constant during refraction; velocity and wavelength change. Total Internal Reflection occurs when: (1) Light travels from an optically denser medium to a rarer medium; (2) Angle of incidence i exceeds the Critical Angle i_c. Relation: sin i_c = μ_rarer / μ_denser. For glass-air (μ = 1.5), i_c ≈ 41.8°; for water-air (μ = 1.33), i_c ≈ 48.75°; for diamond-air (μ = 2.42), i_c ≈ 24.4°.

4. Thin Lens Formula & Lens Maker's Equation

Thin Lens Formula: 1/f = 1/v - 1/u (notice the MINUS sign, contrasting with the mirror formula). Linear magnification for lenses: m = h_i / h_o = +v/u = f / (f + u) = (f - v)/f. Lens Maker's Equation for a lens of refractive index μ2 placed in medium μ1 with surfaces of radii R1 and R2: 1/f = (μ2/μ1 - 1) [1/R1 - 1/R2]. For an equiconvex glass lens in air (R1 = +R, R2 = -R, μ = 1.5): 1/f = (1.5 - 1) [1/R - (-1/R)] = 0.5 [2/R] = 1/R -> f = R. If immersed in water (μ_w = 4/3), focal length increases to 4 times its value in air (f_w = 4 f_air).

5. Power of Lens & Combination of Thin Lenses

Power of a lens measures its converging or diverging ability: P = 1 / f(in meters) = 100 / f(in cm). SI unit: Diopter (D = m^-1). Converging (convex) lens: P > 0 (positive); diverging (concave) lens: P < 0 (negative). When two thin lenses of focal lengths f1 and f2 are placed in coaxial contact: Equivalent focal length: 1/F = 1/f1 + 1/f2. Net power: P = P1 + P2. Net magnification: m = m1 × m2. If separated by distance d in air: 1/F = 1/f1 + 1/f2 - d/(f1 f2), and P = P1 + P2 - d P1 P2.

6. Optical Instruments: Simple Microscope, Compound Microscope & Telescope

Magnifying power (angular magnification M): (1) Simple Microscope (Magnifying Glass): At near point (image at D = 25 cm): M = 1 + D/f; at normal adjustment (image at infinity): M = D/f. (2) Compound Microscope: Objective focal length fo, eyepiece fe (fo < fe): At near point: M = -(L/fo)(1 + D/fe); at infinity: M = -(L/fo)(D/fe), where L is the tube length (distance between focal points). (3) Astronomical Telescope: fo >> fe: At normal adjustment (image at infinity): M = -fo/fe, tube length L = fo + fe; at near point: M = -(fo/fe)(1 + fe/D), tube length L = fo + ue.