Reflection and Mirror Formula

• Apply the laws of reflection to plane and curved mirrors
• Distinguish concave and convex mirrors by their focal properties
• Use the mirror formula 1/v + 1/u = 1/f to locate images precisely
• Calculate linear magnification and determine image nature (real/virtual, erect/inverted)
• Master the Cartesian sign convention required for JEE numericals
• Construct image formation tables for concave mirrors across all object positions

Level 1 — Foundations

Laws of reflection

1. Angle of incidence = angle of reflection (both measured from the normal at the point of incidence)
2. Incident ray, reflected ray, and the normal all lie in the same plane

Types of spherical mirrors

• Concave mirror: reflecting surface on the inner (cave) side; converges parallel rays to the principal focus F (in front of mirror)
• Convex mirror: reflecting surface on the outer side; diverges parallel rays; F is behind the mirror

Key terms

• Centre of curvature C: centre of the sphere of which the mirror is a part
• Radius of curvature R: distance PC
• Focal length f = R/2 (concave: f is negative by sign convention; convex: f is positive)
• Aperture: diameter of the mirror's reflecting surface

Mirror formula $\frac{1}{v} + \frac{1}{u} = \frac{1}{f} = \frac{2}{R}$

Magnification $m = -\frac{v}{u}$

• m < 0: image is inverted (real image for concave mirror, u < 2f excluded)
• m > 0: image is erect (virtual)
• |m| > 1: magnified; |m| < 1: diminished

Level 2 — JEE depth

Cartesian sign convention (critical for JEE)

• Incident light travels left to right
• All distances measured from the pole P
• Distances in the direction of incident light: positive
• Distances opposite to incident light: negative
• For a concave mirror: f < 0, u < 0 always (object in front)
• For a convex mirror: f > 0, v > 0 always (image behind mirror)

Derivation of mirror formula (similar triangles) Consider object AB at distance u from pole P, image A'B' at v. Triangles ABF and MPF are similar (where M is point of incidence on mirror): $\frac{A'B'}{AB} = \frac{B'F}{PF} \quad \Rightarrow \quad \frac{-v/u \cdot h}{h} = \frac{v-f}{f}$ After substitution and simplification using f = R/2: $\frac{1}{v} + \frac{1}{u} = \frac{1}{f}$

Power of a mirror $P = \frac{1}{f(\text{in metres})} \quad \text{(in dioptres)}$ Note: Sign of P follows sign convention; concave mirror has negative f → negative P.

Image formation table for concave mirror

Convex mirror: always forms virtual, erect, diminished image behind mirror regardless of object position. Used as rear-view mirrors.

JEE trap: Students forget that for a concave mirror f is negative. Always substitute f = −|f| in the formula and let algebra determine the sign of v.

Concave mirror: find image position and magnification

Given: Concave mirror, f = 15 cm → f = −15 cm (sign convention)
       Object distance u = −45 cm (object in front of mirror)

Mirror formula: 1/v + 1/u = 1/f
→ 1/v + 1/(−45) = 1/(−15)
→ 1/v = −1/15 + 1/45 = −3/45 + 1/45 = −2/45
→ v = −22.5 cm

Magnification: m = −v/u = −(−22.5)/(−45) = −0.5

Answer: Image is 22.5 cm in front of mirror (real), inverted, and half the object size.

Convex mirror: find image position and nature

Given: Convex mirror, f = +20 cm
       Object distance u = −30 cm (in front of mirror)

Mirror formula: 1/v + 1/(−30) = 1/20
→ 1/v = 1/20 + 1/30 = 3/60 + 2/60 = 5/60
→ v = +12 cm

Magnification: m = −v/u = −(12)/(−30) = +0.4

Answer: Image is 12 cm behind the mirror (virtual), erect, and 0.4× (diminished).
This confirms the convex mirror's use as a rear-view mirror — wider field of view, diminished images.

• Q1: A concave mirror has radius of curvature 30 cm. An object is placed 20 cm in front. Find the image distance.
• Q2: A convex mirror produces an image at 10 cm behind it when the object is 40 cm in front. Find the focal length.
• Q3: The magnification produced by a concave mirror is +3. Is the image real or virtual? Erect or inverted?
• Q4: An object 4 cm tall is placed 15 cm in front of a concave mirror of focal length 10 cm. Find the height of the image.
• Stretch: Two concave mirrors face each other with their principal axes coincident. Object is placed at the centre of curvature of mirror 1. If both have f = 20 cm and are 80 cm apart, trace the final image after two reflections.

NCERT Chapter 9 link: Chapter 9 "Ray Optics and Optical Instruments" (Class 12) covers this in Section 9.2 (reflection by spherical mirrors). The derivation using paraxial approximation and all image formation cases are given. NCERT examples 9.1–9.3 are direct JEE-pattern questions.

Exam connections: JEE Main asks 1–2 questions on mirrors per year — typically a numerical on mirror formula with sign convention, or a conceptual on image nature vs object position. JEE Advanced may give a problem involving two mirrors or a mirror-lens combination. Focal point at infinity (u = f) is a frequent MCQ trap. Magnification sign determines real/virtual — examiners specifically test this.

Study strategy: Memorise the sign convention as a rule, not a formula. Draw a diagram for every problem — mark pole P, focus F, centre C, then mark u (negative arrow to left). Solve mirror formula algebraically; the sign of v tells you whether image is real or virtual without any guesswork. Practice 20 mixed problems before attempting past JEE papers.

• Use the platform-native live simulation or PhET-style tool for this topic (Geometric Optics simulation — move object across concave mirror and observe image flip at f).
• Mirror / body / home activity: use a shiny steel spoon — concave side gives inverted image (like concave mirror), convex side gives erect diminished image. Measure approximate focal length by focusing sunlight on paper.
• Voice or text reflection with AI Mentor: explain to a younger student why rear-view mirrors make cars appear farther away than they are.

• "Explain why a concave mirror is used in torches and headlights, using one real example from an Indian road or festival."
• "What is one common mistake students make when applying the mirror formula, and how would you catch yourself making it?"
• Stretch: "How does the concept of focal length connect to telescope design, satellite dishes, or a doctor's ophthalmoscope?"

• Complete the core practice + one extension activity (photo, table, short reflection, or mini-project) for base XP + topic badge.
• 5-7 day streak or family discussion note = multiplier + visible artifact in parent/principal dashboard.
• Best real-world application stories (anonymised) featured on class or national leaderboard.

• One hands-on project: build a simple periscope using two plane mirrors at 45° and verify the image using a ruler and protractor; extend by replacing one mirror with a concave mirror.
• Direct link to Future Skill track: AI Mastery (optical sensors in cameras use mirror/lens geometry) and Green Tech (solar concentrator dishes use concave mirrors to focus solar energy).
• Coding extension: write a Python script that, given u and f, calculates v, m, and prints image nature; plot image distance vs object distance for a concave mirror.

This material emphasises experiential "learning by doing", competency (apply/create/analyse), vocational exposure, critical thinking, and multidisciplinary connections. Designed to feed live worlds, AI Mentor (with memory), gamification, robotics, parent analytics, and future skills — not just exam prep.

Portfolio Evidence Idea: Your photo/table/reflection/project + one sentence on "How this helps me in real life or a possible future path."

Open the Practice tab for aligned questions (easy/medium/hard + case-based) with full AI scaffolding.

See curriculum for cross-links and the full future-skills/robotics chapters.