Refraction, TIR, Prism and Lens Formula

• Apply Snell's law to calculate angles and refractive indices at interfaces
• Derive and use the condition for total internal reflection (TIR)
• Use the lens formula and lensmaker's equation to locate images
• Analyse prism geometry at minimum deviation to find refractive index
• Combine lenses in contact using power addition
• Identify refraction at a spherical surface as the bridge between mirrors and lenses

Level 1 — Foundations

Snell's Law $n_1 \sin\theta_1 = n_2 \sin\theta_2$

• n = c/v = refractive index (dimensionless, always ≥ 1 for real media)
• Light bends towards normal when entering a denser medium (n₂ > n₁)

Total Internal Reflection (TIR) Occurs only when light travels from denser to rarer medium and angle of incidence exceeds the critical angle θ_c: $\sin\theta_c = \frac{n_2}{n_1} \quad (n_1 > n_2)$ At θ > θ_c, no refracted ray — all light reflects internally.

Lens Formula (Cartesian sign convention, same as mirrors but light travels left to right and distances measured from optical centre) $\frac{1}{v} - \frac{1}{u} = \frac{1}{f}$ Magnification: m = v/u (note: no negative sign unlike mirror formula)

Lensmaker's Equation $\frac{1}{f} = (n-1)\left(\frac{1}{R_1} - \frac{1}{R_2}\right)$

• R₁: radius of first surface (positive if centre of curvature is to the right)
• R₂: radius of second surface
• Convex lens: f > 0; concave lens: f < 0

Level 2 — JEE depth

Refraction at a single spherical surface $\frac{n_2}{v} - \frac{n_1}{u} = \frac{n_2 - n_1}{R}$ This is the fundamental formula — lens formula is derived by applying this twice (two surfaces).

Prism — minimum deviation At minimum deviation δ_m, the ray inside the prism is parallel to the base (symmetrical path):

• Angle of incidence at entry = angle of emergence at exit = i
• Refraction condition at minimum deviation: $n = \frac{\sin\left(\frac{A + \delta_m}{2}\right)}{\sin\left(\frac{A}{2}\right)}$ where A = apex angle of prism.

Deviation by a prism (general) $\delta = (i_1 + i_2) - A$ For thin prism (small A): δ ≈ (n − 1)A (independent of angle of incidence)

Combination of lenses in contact $\frac{1}{f} = \frac{1}{f_1} + \frac{1}{f_2}$ $P = P_1 + P_2 \quad (\text{in dioptres, where } P = 1/f \text{ in metres})$

TIR applications

• Optical fibre: light undergoes repeated TIR at glass-air interface; θ_c(glass) ≈ 42°
• Diamond: high n ≈ 2.42 → θ_c ≈ 24° → multiple TIR inside → brilliance
• Periscopes, endoscopes, rain sensors

JEE traps

• Lens formula: 1/v − 1/u = 1/f (not 1/v + 1/u like mirrors!)
• Magnification for lens: m = v/u (no minus sign)
• Lensmaker's: R₁ and R₂ signs depend on which surface faces the object — always draw diagram and apply Cartesian convention carefully
• Power in dioptres: convert f to metres first
• Prism formula: valid only at minimum deviation — if problem doesn't say "minimum deviation," use Snell's law at each surface separately

Glass prism: find refractive index at minimum deviation

Given: Prism apex angle A = 60°, minimum deviation δ_m = 40°

Prism formula at minimum deviation:
n = sin[(A + δ_m)/2] / sin[A/2]
  = sin[(60° + 40°)/2] / sin[60°/2]
  = sin(50°) / sin(30°)
  = 0.766 / 0.500
  = 1.532

Answer: Refractive index of glass = 1.53 (typical crown glass value)

Verification: This is a real material, confirming calculation is correct.

Convex lens: image position and size

Given: Convex lens f = +20 cm
       Object distance u = −30 cm (object to the left of lens)
       Object height h = 3 cm

Lens formula: 1/v − 1/u = 1/f
→ 1/v − 1/(−30) = 1/20
→ 1/v + 1/30 = 1/20
→ 1/v = 1/20 − 1/30 = 3/60 − 2/60 = 1/60
→ v = +60 cm (positive → image on right side of lens → real)

Magnification: m = v/u = 60/(−30) = −2

Image height = m × h = −2 × 3 = −6 cm

Answer: Real image 60 cm to the right of lens, inverted, 6 cm tall (magnified 2×).
This is the geometry used in slide projectors and camera lenses.

• Q1: Light travels from glass (n=1.5) to water (n=1.33). Find the critical angle for TIR.
• Q2: A concave lens of focal length 25 cm has an object at 15 cm in front. Find the image position and state its nature.
• Q3: Two thin lenses of focal length +20 cm and −30 cm are in contact. Find the equivalent focal length and power.
• Q4: A prism of apex angle 45° gives minimum deviation of 30°. Find the refractive index.
• Stretch: An optical fibre of core refractive index 1.62 and cladding index 1.52 — find the critical angle and the maximum acceptance angle (numerical aperture) for light to remain confined.

NCERT Chapter 9 link: Sections 9.3–9.6 cover refraction at plane surfaces, TIR, refraction at spherical surfaces, lenses, and prisms. Table 9.1 (image formation by lenses) and the optical instrument section (9.9–9.11) extend these concepts. NCERT examples 9.4–9.12 directly prepare for JEE Main.

Exam connections: JEE Main: lens formula numerical (1–2 questions per paper), TIR-based MCQ (optical fibre or diamond), prism at minimum deviation. JEE Advanced: combination of lens + mirror, prism with two refractions through different media, image tracing through multiple surfaces. Power of lens combination appears in almost every paper. Common JEE trick: image formed by lens acts as object for mirror behind it.

Study strategy: Keep a formula card with mirror and lens formulas side by side — the difference in sign is the #1 source of errors. For prism problems, always draw the ray path through the prism and label all angles. Practice the refraction-at-spherical-surface formula as it directly leads to the lensmaker's equation — understanding the derivation prevents formula confusion.

• Use the platform-native live simulation or PhET-style tool for this topic (Bending Light simulation — observe TIR angle, change n₁ and n₂ in real time).
• Mirror / body / home activity: Fill a glass with water and shine a torch from below at various angles to observe TIR at the water-air interface; find the approximate critical angle visually.
• Voice or text reflection with AI Mentor: Explain to a family member why optical fibres are used for internet and how TIR makes data travel without loss.

• "Explain why a diamond sparkles more than glass using one concept from TIR, as if talking to a Class 7 student at a jewellery shop."
• "What is one common mistake students make when applying the lens formula vs the mirror formula, and how would you avoid it?"
• Stretch: "How does refraction of light at spherical surfaces connect to the design of camera lenses, microscopes, or corrective eyeglasses for presbyopia?"

• 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 optical fibre model using a water stream from a torch; photograph the light bending inside the stream and measure the approximate angle.
• Direct link to Future Skill track: AI Mastery (camera and LiDAR sensors in autonomous vehicles use lens refraction), Cyber Defenders (optical fibre is the backbone of secure internet infrastructure).
• Coding extension: write a Python ray-tracer for a single convex lens — given u and f, plot the image position as u varies from −∞ to −f.

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.