Reflection by Plane & Spherical Mirrors Reflection by Plane & Spherical Mirrors Light reflecting from a smooth surface follows simple rules that let you predict where an image will appear and what it looks like. In a plane mirror, the image appears the same size and as far behind the mirror as the object is in front. In spherical mirrors (concave or convex), the curvature makes images magnified or diminished, real or virtual, depending on where the object is placed. The mirror formula ties together object distance u , image distance v , and focal length f , while magnification m tells you how big and upright or inverted the image is. You will use a single, consistent sign convention so you never have to memorize separate cases. Ray diagrams give you a quick picture, and the formulas confirm your result. This topic is a scoring area because the logic is clean: define your u and f with the correct signs, apply 1 v + 1 u = 1 f , then compute m = - v u . Edge cases like u or u = f are quick checks for focus and infinity images. For a rotating mirror, the reflected ray rotates by twice the mirror’s angle. With practice, you will translate words into a neat diagram, run the formula with signs, and avoid traps like mixing units or flipping u and v . remember Everyday anchors: bathroom mirror (plane) shows you laterally inverted; car’s rear-view (convex) gives a wide field but smaller images; a shaving/makeup mirror (concave) magnifies your face when you come closer than its focal length. Reflection The bouncing back of light from a surface. Angle of incidence equals angle of reflection, and incident ray, reflected ray, and normal lie in one plane. The incoming ray that strikes the mirror surface. Incident Ray Normal A line perpendicular to the reflecting surface at the point of incidence. Angle between the incident ray and the normal at the point of incidence. Angle of Incidence (i) Angle between the reflected ray and the normal at the point of incidence. Angle of Reflection (r) A flat mirror that forms a virtual, upright image with the same size as the object, with |v| = |u| and lateral inversion. Plane Mirror A mirror whose reflecting surface is part of a sphere. Concave mirrors reflect from the inner surface; convex mirrors reflect from the outer surface. Spherical Mirror A converging spherical mirror that can form real, inverted images for distant objects and virtual, upright, magnified images when the object is inside the focal length. Concave Mirror Convex Mirror A diverging spherical mirror that always forms virtual, upright, and diminished images behind the mirror. The geometric center of the mirror surface. Pole (P) Principal Axis The straight line passing through the pole and the center of curvature. Centre of Curvature (C) The center of the sphere of which the mirror is a part. Its distance from the pole is the radius of curvature R . Radius of Curvature (R) Distance PC ; related to focal length by f = R/2 for spherical mirrors (paraxial approximation). Focus (F) The point on the principal axis where paraxial rays parallel to the axis converge (concave) or appear to diverge from (convex). Distance PF . For spherical mirrors in the paraxial limit, f = R/2 . Focal Length (f) Real Image Formed by actual convergence of reflected rays; inverted and can be obtained on a screen. Virtual Image Formed by the apparent divergence of rays; upright and cannot be obtained on a screen. Ratio of image height to object height: m = h i/h o = -v/u for mirrors. Magnification (m) Distances measured from the pole P along the principal axis. Positive to the right. For mirrors: object to the left gives u < 0 ; concave mirror has f < 0 ; convex has f > 0 . Real image in front of the mirror has v < 0 ; virtual image behind has v > 0 . Cartesian Sign Convention We will use the Cartesian sign convention consistently: take the pole P as origin, principal axis to the right as positive. Light for mirror problems usually travels left to right. Real objects are typically on the left, so u is negative. Concave mirrors have the focus on the left, hence f < 0 . Convex mirrors have focus on the right (behind the mirror), so f > 0 . A negative v means the image is in front of the mirror (real), while a positive v means the image is behind the mirror (virtual). Laws of Reflection Core geometry of reflection. Equal angles immediately give plane mirror results: draw a normal, mark equal i and r , and use congruent triangles to show the image is the same distance behind the mirror as the object is in front, with lateral inversion. Mirror Formula Valid for spherical mirrors under paraxial approximation with Cartesian sign convention. Calculates the position of the image by relating the object distance, image distance, and focal length of the mirror. Magnification Negative m means inverted image; positive m means upright image. This ratio determines the relative size of the image formed compared to the original object size. True for spherical mirrors for paraxial rays. Focal Length and Radius If the mirror rotates by , the reflected ray rotates by 2 . Rotation of Mirror Paraxial condition: rays must make small angles with the principal axis and strike near the pole. Far-off-axis rays suffer spherical aberration; the simple f = R/2 and the mirror formula are then approximate. tip 1 v + 1 u = 1 f Mirror formula for spherical mirrors Small-angle (paraxial) approximation: Object, image, and focus lie close to the principal axis Cartesian sign convention Use the mirror formula with sign care. For a concave mirror, f<0 , so if a real object is placed beyond F ( u<0 and |u|>|f| ), you expect a real image ( v<0 ). Magnification m = -v/u will be negative, confirming inversion. Use i=r , congruent triangles, and plane-mirror property |v|=|u| . Image distance from mirror; how the label appears Plane mirror distance and lateral inversion NCERT exemplar: plane mirror basics easy An object is 0.50 m in front of a plane mirror A label on the object reads ‘NEET’ Lateral inversion is not a front-back flip; it is a left-right reversal across the mirror plane. Your right hand appears as the left hand in the mirror because the image is a symmetric counterpart behind the mirror. cm Use 1 v + 1 u = 1 f and m = - v u = h i h o . Concave mirror image location and size Image distance v , magnification m , and image height h i medium Mirror formula direct application Concave mirror with focal length f = -15 cm Object distance u = -30 cm Object height h o = 2.0 cm Linear relation from the mirror formula in 1/u–1/v space. 1/v 1/m For a fixed mirror, points (1/u, 1/v) lie on a straight line with slope −1 and intercept 1/f. 1/f derived custom 1/u 1/m Object at infinity → 1/u = 0, image at focus: 1/v = 1/f 1/f −1/f Object at focus → v at infinity Rewriting 1 v = 1 f - 1 u shows a straight-line plot with intercept 1/f and slope -1 . This gives a quick experimental route for determining f from data. Mirror / Object Position Image Position Nature Size Sign of m Concave, object beyond C Between C and F (in front) Real, inverted Diminished m < 0 Concave, object at C At C (in front) Real, inverted Same size m = −1 Concave, object between C and F Beyond C (in front) Real, inverted Magnified m < 0 Concave, object at F At infinity Real, inverted Highly magnified m < 0 Concave, object between F and P Behind mirror Virtual, upright Magnified m > 0 Convex, object anywhere Between P and F (behind mirror) Virtual, upright Diminished m > 0 Plane, object anywhere Equal distance behind mirror Virtual, upright Same size m = +1 Draw the principal axis and mark P, F, C with f = R/2 . From the top of the object, draw a ray parallel to the axis; reflect it through F. Draw a second ray through C; it retraces back on itself after reflection. Their intersection gives the image top; drop to the axis for image base. Read nature and size. If the rays do not actually meet (object inside F), extend reflected rays backward to locate a virtual image behind the mirror. Steps to draw a concave mirror ray diagram (quick method) Ray parallel to axis appears to diverge from F after reflection. Ray directed toward C reflects back on itself (appears to). Ray aimed toward F emerges parallel to the axis. Principal rays for convex mirrors A mirror is rotated by 10 about an axis in its plane An incident ray hits at a fixed point easy Standard rotation-of-mirror result Rotation of the reflected ray Rotating mirror: deflection of reflected ray degree Use = 2 , for small rotations. Sign trap: m = -v/u . If you drop the minus or swap u,v , you will flip ‘upright’ vs ‘inverted’. Always write u<0 for a real object in front of a mirror before substituting. neet-alert In Cartesian sign convention for mirrors, u<0 for a real object to the left of the mirror. The sign encodes geometry; dropping it breaks the formula. Object distance is always positive because distance is positive in math. A convex mirror can form a real image if the object is very far. Ideal convex mirrors always form virtual, upright, diminished images behind the mirror for real objects. RIVU: Real → Inverted; Virtual → Upright. Also remember F=R/2 for spherical mirrors. Quick recall for image orientation and focal length relation For a thin prism, angular dispersion = (n v - n r)A (difference between deviations of violet and red). Angular dispersion for a thin prism: = (n v - n r)A Thin prism, small apex angle A Small-angle approximation: in radians Deviations add linearly: (n-1)A = (n v - n r)A Dispersive power = n v - n r n y - 1 ; material property, independent of prism angle. Dispersive power Thin prism with small apex angle Mean deviation uses yellow (or mean) refractive index n y = n v - n r n y - 1 Apparent depth when viewed from a rarer medium at near-normal incidence: d app = d real . d app = d real Observer in rarer medium (air) Small angles (paraxial), planar interface Refractive index is of denser medium relative to air Apparent depth d app = d real Equivalent form: n = h h' (real depth over apparent depth). n = h h' Relation n = h h' Same as apparent depth derivation Rename d real =h , d app =h' Critical angle for light from denser to rarer medium: C = 1 n (when rarer is air). C = 1 n Critical angle condition C = 1 n Light goes from denser medium n to air Snell’s law applies at the boundary For thin lenses in contact: 1 F = 1 f 1 + 1 f 2 (powers add). Thin lenses, negligible separation Common principal axis, same medium Equivalent focal length of two thin lenses in contact 1 F = 1 f 1 + 1 f 2 Same result: 1 F eq = 1 f 1 + 1 f 2 . Powers add: P eq = P 1 + P 2 . Equivalent focal length F eq for thin lenses in contact Thin, coaxial lenses in contact Paraxial approximation 1 F eq = 1 f 1 + 1 f 2 Two plane mirrors make an angle = 60 Object on the angle bisector Repeated in competitive exams hard Two plane mirrors inclined: number of images Number of images Use n = 360 - 1 when 360 is an integer; else n = 360 if the object is on the bisector. Two-mirror images: If 360°/θ is even integer and object on bisector → n = 360/θ − 1. If not an integer, use floor(360/θ). Off-bisector placement can reduce the count by 1. remember neet-alert Units trap: If f is in cm, keep u and v in cm in the same calculation. Mixing cm and m is a silent mistake that shifts answers by 100×. Boundary checks help catch errors fast: as u - , v f (image at focus). If u = f , then v - (image at infinity). If u is between P and F for a concave mirror, v>0 and m>0 (virtual, upright, magnified). For a convex mirror, v>0 and 0<m<1 for any real object distance. Clean line drawing: concave mirror with P, F, C marked; object beyond C; two principal rays showing image between C and F. Clean line drawing: concave mirror with P, F, C marked; object beyond C; two principal rays showing image between C and F. Ray diagram for a concave mirror forming a real, inverted image between F and C. Practical tip: When unsure, sketch two principal rays and read off the image’s qualitative features (location relative to F and C, upright or inverted, big or small). Then use the mirror formula to compute exact v and m . cm Apply mirror formula and magnification. Convex mirror: image distance and size Image distance v , magnification m , image height h i medium Convex mirror application Convex mirror with focal length f = +20 cm Object distance u = -40 cm Object height h o = 3.0 cm Multiple reflections between two plane mirrors produce many images. When 360 is an integer, n = 360 - 1 . If not, n = 360 for an object on the bisector; shifting the object off the bisector may reduce the count by one. tip Quick sanity checks: For a convex mirror, v is always positive and less than f in magnitude. For a concave mirror with a real object, v is negative unless the object is inside the focal length (then v>0 ). Plane Mirror Flat mirror; forms virtual, upright, same-size images at |v|=|u| . Concave Mirror Converging; can form real or virtual images depending on object position. Convex Mirror Diverging; always forms virtual, upright, diminished images. Mirror Formula 1 v + 1 u = 1 f under paraxial approximation. Magnification m = - v u = h i h o ; sign indicates orientation. Rotation Rule Reflected ray rotates by twice the mirror’s rotation: = 2 . Sign Convention Cartesian: right positive; real object u<0 ; concave f<0 ; convex f>0 . Glossary Recap