Light – Reflection & Refraction
Class 10 (CBSE) – Science
(Advanced MIT‑style treatment – first principles, rigorous reasoning, edge cases, and a challenging problem.)
📚 Table of Contents
| # | Topic | Page |
|---|---|---|
| 1 | Introduction & Basic Definitions | 1 |
| 2 | Reflection of Light | 3 |
| 3 | Refraction of Light | 8 |
| 4 | Advanced Theory – Fermat’s Principle & Wave Optics | 13 |
| 5 | Edge Cases & Exotic Phenomena | 18 |
| 6 | Applications (lenses, prisms, optical fibres) | 22 |
| 7 | Practice Problems | 26 |
| 8 | Challenging Problem | 29 |
| 9 | References & Further Reading | 30 |
Note: For brevity, page numbers are illustrative. Use these as anchors in your notes.
1. Introduction & Basic Definitions (Page 1)
| Concept | Definition |
|---|---|
| Light | Electromagnetic radiation in the visible spectrum (≈ 400–700 nm). |
| Ray | Idealised straight line along which energy propagates. |
| Incident Ray | Ray before interacting with a surface. |
| Reflected Ray | Ray after bouncing off a surface. |
| Refracted Ray | Ray after passing into a second medium. |
| Plane of Incidence | Plane containing incident, reflected, and refracted rays. |
| Normal | Perpendicular to the interface at the point of interaction. |
| Angle of Incidence (θ₁) | Angle between incident ray and normal. |
| Angle of Reflection (θᵣ) | Angle between reflected ray and normal. |
| Angle of Refraction (θ₂) | Angle between refracted ray and normal. |
| Refractive Index (n) | Ratio of speed of light in vacuum to that in the medium: |
| ( n = \dfrac{c}{v} ) (dimensionless). |
Key Law – Law of Reflection
The angle of incidence equals the angle of reflection:
[ \theta_1 = \theta_r ]
Key Law – Snell’s Law (Law of Refraction)
[ n_1 \sin \theta_1 = n_2 \sin \theta_2 ]
2. Reflection of Light (Page 3)
2.1 Geometric Derivation
-
Plane Mirror
- Incident and reflected rays lie on the same side of the normal.
- Using vector reflection:
[ \mathbf{r} = \mathbf{i} - 2(\mathbf{i}!\cdot!\mathbf{n})\mathbf{n} ] where (\mathbf{i}) is incident unit vector, (\mathbf{n}) is normal unit vector.
-
Diffuse (Lambertian) Surface
- Roughness causes multiple scattering.
- Intensity obeys Lambert’s cosine law:
[ I = I_0 \cos \theta ] where (\theta) is angle between surface normal and observation direction.
2.2 Polarization by Reflection
- Brewster’s Angle ((\theta_B)) – angle at which reflected light is perfectly polarized transverse to the plane of incidence.
[ \tan \theta_B = \dfrac{n_2}{n_1} ]
Edge Case: At (\theta_B), the reflected intensity of the p‑polarized component goes to zero.
2.3 Total Internal Reflection (TIR)
- Occurs when light travels from a medium of higher (n) to lower (n) and (\theta_1) exceeds the critical angle (\theta_c).
[ \sin \theta_c = \dfrac{n_2}{n_1} ] - All incident light is reflected; no refraction occurs.
- Evanescent Wave: Exponential decay into the second medium.
3. Refraction of Light (Page 8)
3.1 Snell’s Law from Wavefront Geometry
- Consider a wavefront crossing a boundary at an angle.
- The component of velocity parallel to the interface remains unchanged, leading to: [ \frac{\sin \theta_1}{\sin \theta_2} = \frac{v_1}{v_2} = \frac{n_2}{n_1} ]
3.2 Dispersion
- Refractive index varies with wavelength (\lambda): (n(\lambda)).
- Normal Dispersion: (dn/d\lambda > 0) (red light refracts less than blue).
- Anomalous Dispersion: (dn/d\lambda < 0) near absorption lines.
3.3 Wave Optics View
- Phase Matching: Continuity of the tangential component of the wavevector across interface:
[ k_{1,\parallel} = k_{2,\parallel} ] - Leads to Snell’s law via (k = \dfrac{2\pi n}{\lambda_0}).
4. Advanced Theory – Fermat’s Principle & Wave Optics (Page 13)
4.1 Fermat’s Principle (Principle of Least Time)
- Light follows the path that minimises the travel time between two points.
- Formally:
[ \delta \int_{A}^{B} n(\mathbf{r}) , ds = 0 ] - Applying Euler–Lagrange yields Snell’s law.
**Rigorous derivation (for constant (n_i))