Light: Reflection & Refraction

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1. Reflection of Light Laws of Reflection The angle of incidence is equal to the angle of reflection ($\angle i = \angle r$). The incident ray, the normal to the mirror at the point of incidence, and the reflected ray all lie in the same plane. Plane Mirrors Image is always virtual and erect. Size of image equals size of object. Image is as far behind the mirror as the object is in front. Image is laterally inverted. Spherical Mirrors Concave Mirror: Reflecting surface curved inwards (converging mirror). Convex Mirror: Reflecting surface curved outwards (diverging mirror). Key Terms for Spherical Mirrors Pole (P): Center of the reflecting surface. Centre of Curvature (C): Center of the sphere from which the mirror is part. Radius of Curvature (R): Distance PC. Principal Axis: Straight line passing through P and C. Principal Focus (F): Concave Mirror: Point on the principal axis where parallel rays converge after reflection. Convex Mirror: Point on the principal axis from which parallel rays appear to diverge after reflection. Focal Length (f): Distance PF. Aperture: Diameter of the reflecting surface (MN in diagrams). Relationship: $R = 2f$ (for mirrors with small apertures). Ray Diagrams for Spherical Mirrors Ray parallel to principal axis: Concave: Passes through F after reflection. Convex: Appears to diverge from F after reflection. Ray passing through F (concave) or directed towards F (convex): Emerges parallel to the principal axis after reflection. Ray passing through C (concave) or directed towards C (convex): Reflected back along the same path. Ray incident obliquely towards P: Reflected obliquely, making equal angles with the principal axis. Image Formation by Concave Mirror Object Position Image Position Image Size Image Nature At infinity At F Highly diminished, point-sized Real and inverted Beyond C Between F and C Diminished Real and inverted At C At C Same size Real and inverted Between C and F Beyond C Enlarged Real and inverted At F At infinity Highly enlarged Real and inverted Between P and F Behind the mirror Enlarged Virtual and erect Uses of Concave Mirrors Torches, search-lights, vehicle headlights (to get parallel beams). Shaving mirrors, dentists' mirrors (to see larger images). Solar furnaces (to concentrate sunlight). Image Formation by Convex Mirror Object Position Image Position Image Size Image Nature At infinity At F, behind the mirror Highly diminished, point-sized Virtual and erect Between infinity and P Between P and F, behind the mirror Diminished Virtual and erect Uses of Convex Mirrors Rear-view/wing mirrors in vehicles (always give erect, diminished image, wider field of view). New Cartesian Sign Convention for Reflection Pole (P) is the origin (0,0). Principal axis is the x-axis. Object always placed to the left of the mirror (light falls from left). All distances parallel to principal axis measured from P. Distances to the right of origin ($+x$-axis) are positive. Distances to the left of origin ($-x$-axis) are negative. Distances perpendicular to and above principal axis ($+y$-axis) are positive. Distances perpendicular to and below principal axis ($-y$-axis) are negative. Mirror Formula and Magnification Mirror Formula: $\frac{1}{v} + \frac{1}{u} = \frac{1}{f}$ $u$: object distance, $v$: image distance, $f$: focal length. Magnification (m): Ratio of image height ($h'$) to object height ($h$). $m = \frac{h'}{h} = -\frac{v}{u}$ Negative $m$ indicates real and inverted image. Positive $m$ indicates virtual and erect image. 2. Refraction of Light Bending of light when it passes obliquely from one transparent medium to another. Laws of Refraction The incident ray, the refracted ray, and the normal to the interface at the point of incidence, all lie in the same plane. Snell's Law: The ratio of sine of angle of incidence ($\angle i$) to sine of angle of refraction ($\angle r$) is a constant for a given pair of media and color. $ \frac{\sin i}{\sin r} = \text{constant} $ Refractive Index The constant in Snell's Law is called the refractive index ($n$). $n_{21} = \frac{\text{Speed of light in medium 1}}{\text{Speed of light in medium 2}} = \frac{v_1}{v_2}$ (Refractive index of medium 2 with respect to medium 1). Absolute Refractive Index (n): Refractive index with respect to vacuum or air. $ n_m = \frac{\text{Speed of light in air/vacuum}}{\text{Speed of light in the medium}} = \frac{c}{v} $ Optical Density: A measure of a medium's ability to refract light. Higher refractive index means higher optical density. Light bends towards the normal when entering a denser medium ($v$ decreases). Light bends away from the normal when entering a rarer medium ($v$ increases). For a rectangular glass slab, the emergent ray is parallel to the incident ray. 3. Refraction by Spherical Lenses A transparent material bound by two spherical surfaces or one spherical and one plane surface. Convex Lens (Converging): Thicker in the middle, converges parallel light rays. Concave Lens (Diverging): Thicker at the edges, diverges parallel light rays. Key Terms for Spherical Lenses Optical Centre (O): Central point of the lens. A ray of light through O passes undeviated. Principal Axis: Imaginary straight line passing through the two centers of curvature. Principal Foci ($F_1, F_2$): Two focal points for lenses. Focal Length (f): Distance from O to F. Aperture: Effective diameter of the circular outline of a spherical lens. Ray Diagrams for Spherical Lenses Ray parallel to principal axis: Convex: Passes through $F_2$ after refraction. Concave: Appears to diverge from $F_1$ after refraction. Ray passing through $F_1$ (convex) or directed towards $F_2$ (concave): Emerges parallel to the principal axis after refraction. Ray passing through optical center (O): Emerges without any deviation. Image Formation by Convex Lens Object Position Image Position Image Size Image Nature At infinity At $F_2$ Highly diminished, point-sized Real and inverted Beyond $2F_1$ Between $F_2$ and $2F_2$ Diminished Real and inverted At $2F_1$ At $2F_2$ Same size Real and inverted Between $F_1$ and $2F_1$ Beyond $2F_2$ Enlarged Real and inverted At $F_1$ At infinity Infinitely large or highly enlarged Real and inverted Between $F_1$ and O On the same side of the lens as object Enlarged Virtual and erect Image Formation by Concave Lens Object Position Image Position Image Size Image Nature At infinity At $F_1$ Highly diminished, point-sized Virtual and erect Between infinity and O Between $F_1$ and O Diminished Virtual and erect Sign Convention for Spherical Lenses Similar to mirrors, but all measurements are taken from the optical center (O). Focal length of convex lens is positive. Focal length of concave lens is negative. Lens Formula and Magnification Lens Formula: $\frac{1}{v} - \frac{1}{u} = \frac{1}{f}$ $u$: object distance, $v$: image distance, $f$: focal length. Magnification (m): $m = \frac{h'}{h} = \frac{v}{u}$ Negative $m$ indicates real and inverted image. Positive $m$ indicates virtual and erect image. Power of a Lens (P) Ability of a lens to converge or diverge light rays. $P = \frac{1}{f}$ (where $f$ is in meters). SI unit: Dioptre (D). $1 \text{ D} = 1 \text{ m}^{-1}$. Power of a convex lens is positive. Power of a concave lens is negative. For multiple lenses in contact, total power $P = P_1 + P_2 + P_3 + ...$