What is Snell's Law?
Snell's Law (also called the Law of Refraction) states that when light passes from one medium to another, the product of the refractive index and the sine of the angle of incidence remains constant: n₁ sin(θ₁) = n₂ sin(θ₂). Light bends towards the normal when entering a denser medium and away from the normal when entering a rarer medium.
Core Optics Formulas
| Symbol | Quantity | Unit | Notes |
|---|
| n₁, n₂ | Refractive indices | Dimensionless | Always ≥ 1; vacuum = 1.000 |
| θ₁ | Angle of incidence | degrees / radians | Measured from the normal |
| θ₂ | Angle of refraction | degrees / radians | Measured from the normal |
| C | Critical angle | degrees | Angle beyond which TIR occurs |
| R₁, R₂ | Radii of curvature | cm or m | Sign by Cartesian convention |
| f | Focal length | cm or m | Positive for convex; negative for concave |
| P | Power of lens | Dioptre (D) | P = 1/f (f in metres) |
| u, v | Object, Image distance | cm or m | Measured from optical centre |
Key Concepts in Refraction & Lenses
Total Internal Reflection (TIR)
Total Internal Reflection occurs when light travels from a denser medium to a rarer medium and the angle of incidence exceeds the critical angle (C). At the critical angle, the refracted ray grazes the interface (emerges at 90°). Beyond it, no refraction occurs — all light is reflected back into the denser medium. TIR is the working principle of optical fibres, diamond brilliance, mirages, and periscopes using prisms.
Optical Fibre
Light signals travel long distances by successive TIR inside the glass/silica core. The cladding has a lower refractive index than the core, ensuring TIR at every internal reflection.
Diamond Sparkle
Diamond has n = 2.417 giving a critical angle of only ~24.4°. Light entering a well-cut diamond undergoes multiple TIRs before exiting, creating the characteristic brilliance and fire.
Mirage Effect
Hot air near the ground has a lower refractive index than cooler air above. Light from the sky undergoes TIR at the hot-air layer, appearing as a reflected water surface to a distant observer.
Medical Endoscopy
Flexible endoscopes use bundles of optical fibres to transmit light into the body and images out, all via TIR. The same principle is used in laparoscopic surgery instruments.
Lens Maker's Equation — Sign Convention
The New Cartesian Sign Convention (NCERT standard) places the optical centre at the origin. All distances are measured from the optical centre. Distances in the direction of incident light (left to right) are positive; opposite direction is negative.
Convex Lens
R₁ > 0, R₂ < 0. Focal length f > 0 (converging). Power P > 0 (positive Dioptres). Also called a converging lens.
Concave Lens
R₁ < 0, R₂ > 0. Focal length f < 0 (diverging). Power P < 0 (negative Dioptres). Also called a diverging lens.
Dispersion and the Prism
The refractive index of a material varies with the wavelength of light (a property called dispersion). Violet light (shorter wavelength) bends more than red light (longer wavelength) as it passes through a glass prism. This causes white light to split into the visible spectrum (VIBGYOR). The angle of deviation in a prism is given by:
Solved Examples (NCERT / JEE Level)
Example 01 — Snell's Law (Air to Glass)
A ray of light passes from air (n₁ = 1.000) into a glass slab (n₂ = 1.500) at an angle of incidence of 45°. Find the angle of refraction.
Given: n₁ = 1.000, n₂ = 1.500, θ₁ = 45°
Snell's Law: n₁ × sin(θ₁) = n₂ × sin(θ₂)
sin(θ₂) = n₁ × sin(θ₁) ÷ n₂
sin(θ₂) = 1.000 × sin(45°) ÷ 1.500
sin(θ₂) = 1.000 × 0.7071 ÷ 1.500
sin(θ₂) = 0.4714
θ₂ = arcsin(0.4714)
θ₂ = 28.13° | Light bends towards the normal (denser medium)
Example 02 — Critical Angle for Diamond
Calculate the critical angle for total internal reflection at the diamond-air interface. Refractive index of diamond = 2.417.
Given: n₁ = 2.417 (diamond), n₂ = 1.000 (air)
Formula: sin(C) = n₂ ÷ n₁
sin(C) = 1.000 ÷ 2.417
sin(C) = 0.41374
C = arcsin(0.41374)
C = 24.43°
Any ray in diamond hitting the surface at θ > 24.43° undergoes TIR.
Critical Angle C = 24.43° | This explains diamond's extraordinary brilliance
Example 03 — Lens Maker's Equation
A biconvex lens has both radii of curvature equal to 20 cm. The lens is made of glass with refractive index 1.5. Find the focal length and power of the lens in air.
Given: n₂ = 1.5 (glass), n₁ = 1.0 (air)
R₁ = +20 cm, R₂ = −20 cm (biconvex, Cartesian convention)
Lens Maker's Equation:
1/f = (n₂/n₁ − 1) × (1/R₁ − 1/R₂)
1/f = (1.5/1.0 − 1) × (1/20 − 1/(−20))
1/f = (0.5) × (1/20 + 1/20)
1/f = (0.5) × (2/20)
1/f = (0.5) × (0.1)
1/f = 0.05 cm−1
f = 20 cm = 0.20 m
P = 1/f = 1/0.20
f = 20 cm | Power P = +5.0 Dioptres | Converging lens
Frequently Asked Questions (FAQs)
Why does light bend when it enters a different medium?
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Light bends at an interface because its speed changes as it enters a new medium. In a denser medium (higher refractive index), light travels more slowly. The bending is a wave effect: one side of the wavefront slows down before the other, causing the wave to change direction. The mathematical relationship between the angles and refractive indices is described by Snell's Law: n₁ sin(θ₁) = n₂ sin(θ₂).
What is the difference between reflection and refraction?
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Reflection occurs when a light ray bounces back into the same medium it came from (angle of incidence = angle of reflection). Refraction occurs when light passes into a second medium and bends due to a change in speed. Both reflection and refraction can occur simultaneously at an interface — some light is reflected, some is refracted. At the critical angle (TIR condition), refraction ceases entirely and all light is reflected.
What is the condition for Total Internal Reflection (TIR)?
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TIR occurs when two conditions are both satisfied: (1) Light must be traveling from a denser medium to a rarer medium (e.g., glass to air, water to air). (2) The angle of incidence must be greater than the critical angle (C), where sin(C) = n₂ / n₁. If either condition is not met, TIR cannot occur. For glass-air, C ≈ 41.8°; for water-air, C ≈ 48.6°; for diamond-air, C ≈ 24.4°.
What is the sign convention used in the Lens Maker's Equation?
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NCERT uses the New Cartesian Sign Convention: (1) The optical centre of the lens is the origin. (2) All distances are measured from the optical centre. (3) Distances in the direction of incident light (left to right) are positive. (4) Distances opposite to incident light are negative. For a biconvex lens: R₁ is positive (centre of curvature on the right) and R₂ is negative (centre of curvature on the left). For a biconcave lens, it is the reverse.
What is the power of a lens and what are Dioptres?
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The power (P) of a lens is defined as the reciprocal of its focal length in metres: P = 1/f. The unit is Dioptre (D). A convex (converging) lens has positive power and a concave (diverging) lens has negative power. When two lenses are placed in contact, their powers add: P₋ₒₜ₊ⁿ = P₁ + P₂. Optometrists prescribe corrective lenses using Dioptres (e.g., −2.5 D for myopia correction).