Light - Reflection and Refraction

Introduction

• Light enables visibility; objects reflect light, which our eyes detect, making them visible.
• Light travels in straight lines (rays), as evidenced by sharp shadows of opaque objects.
• Exceptions: Diffraction (light bends around small obstacles) requires wave theory; particle-like behavior explains light-matter interactions (quantum theory).
• This chapter focuses on reflection and refraction using ray optics, exploring mirrors, lenses, and real-life applications.

9.1 Reflection of Light

• Polished surfaces (e.g., mirrors) reflect most light.
• Laws of Reflection:
  1. Angle of incidence equals angle of reflection.
  2. Incident ray, reflected ray, and normal at the point of incidence lie in the same plane.
• Plane Mirror Images:
  • Virtual, erect, same size as object, laterally inverted, equidistant behind mirror.
• Reflection laws apply to all surfaces, including curved ones.

9.2 Spherical Mirrors

• Types:
  • Concave Mirror: Reflecting surface curves inward (e.g., inner side of a spoon, Fig. 9.1a).
  • Convex Mirror: Reflecting surface curves outward (e.g., outer side of a spoon, Fig. 9.1b).
• Key Terms:
  • Pole (P): Center of the mirror’s reflecting surface.
  • Centre of Curvature (C): Center of the sphere the mirror is part of; in front of concave, behind convex.
  • Radius of Curvature (R): Distance PC, radius of the sphere.
  • Principal Axis: Line through P and C, normal to mirror at P.
  • Principal Focus (F): Concave: rays parallel to principal axis converge at F; Convex: rays appear to diverge from F (Fig. 9.2).
  • Focal Length (f): Distance between P and F; for small apertures, R = 2f.
  • Aperture: Diameter of the mirror’s reflecting surface (MN in Fig. 9.2).
• Activity 9.1 (Shining Spoon):
  • View face in spoon’s curved surfaces; concave side gives larger/erect or smaller/inverted images depending on distance; convex side gives smaller, erect images.
• Activity 9.2 (Concave Mirror Focus):
  • Reflect sunlight with concave mirror onto paper; sharp spot (Sun’s image) burns paper due to concentrated heat. Distance to spot approximates focal length.

9.2.1 Image Formation by Spherical Mirrors

• Activity 9.3 (Concave Mirror Imaging):
  • Measure focal length; place candle at various positions (beyond C, at C, etc.) and observe image on screen or in mirror.
  • Tabulate nature, position, size (Table 9.1).
• Concave Mirror Image Summary (Table 9.1):

  Object Position	Image Position	Image Size	Nature
  At infinity	At F	Highly diminished, point-sized	Real, inverted
  Beyond C	Between F and C	Diminished	Real, inverted
  At C	At C	Same size	Real, inverted
  Between C and F	Beyond C	Enlarged	Real, inverted
  At F	At infinity	No image formed	-
  Between P and F	Behind mirror	Enlarged	Virtual, erect

• Uses of Concave Mirrors:
  • Torches, headlights, searchlights (parallel beams); shaving mirrors, dental mirrors (enlarged images); solar furnaces (concentrate sunlight).

9.2.2 Representation of Images Using Ray Diagrams

• Two rays suffice to locate image:
  1. Ray parallel to principal axis: Concave - passes through F; Convex - diverges from F (Fig. 9.3).
  2. Ray through F (concave) or toward F (convex): Emerges parallel to principal axis (Fig. 9.4).
  3. Ray through C: Reflects back along same path (Fig. 9.5).
  4. Ray to P: Reflects obeying laws of reflection (Fig. 9.6).
• Activity 9.4: Draw ray diagrams for concave mirror positions (Fig. 9.7); compare with Table 9.1.
• Convex Mirror Imaging (Activity 9.5):
  • Image always virtual, erect, diminished; moves closer to F as object moves away (Fig. 9.8).
• Convex Mirror Image Summary (Table 9.2):

  Object Position	Image Position	Image Size	Nature
  At infinity	At F, behind mirror	Highly diminished, point-sized	Virtual, erect
  Between infinity and P	Between P and F, behind mirror	Diminished	Virtual, erect

• Uses of Convex Mirrors:
  • Rear-view mirrors in vehicles (erect, diminished, wide field of view).
• Activity 9.6:
  • Plane mirror: Full-length image, size-independent of distance.
  • Concave: Limited field, partial image unless object is close.
  • Convex: Full-length image in small mirror due to wide field (e.g., Taj Mahal at Agra Fort).

9.2.3 Sign Convention for Spherical Mirrors

• New Cartesian Sign Convention (Fig. 9.9):
  1. Object on left of mirror.
  2. Distances measured from pole (P, origin).
  3. Right of P: positive; Left of P: negative.
  4. Above principal axis: positive; Below: negative.

9.2.4 Mirror Formula and Magnification

• Mirror Formula:
  • Relates object distance (u), image distance (v), focal length (f): \(\frac{1}{v} + \frac{1}{u} = \frac{1}{f}\).
  • Valid for all spherical mirrors; use sign convention.
• Magnification (m):
  • Ratio of image height (h') to object height (h): \(m = \frac{h'}{h} = -\frac{v}{u}\).
  • Positive m: virtual image; Negative m: real image.
• Example 9.1 (Convex Mirror):
  • R = +3.00 m, u = -5.00 m; f = R/2 = +1.50 m.
  • \(\frac{1}{v} = \frac{1}{f} - \frac{1}{u} = \frac{1}{1.50} + \frac{1}{5.00} = \frac{6.50}{7.50}\); v = +1.15 m.
  • m = \(-\frac{v}{u} = -\frac{1.15}{-5.00} = +0.23\).
  • Image: virtual, erect, 0.23 times object size, 1.15 m behind mirror.
• Example 9.2 (Concave Mirror):
  • h = +4.0 cm, u = -25.0 cm, f = -15.0 cm.
  • \(\frac{1}{v} = \frac{1}{f} - \frac{1}{u} = -\frac{1}{15.0} + \frac{1}{25.0} = -\frac{2}{75.0}\); v = -37.5 cm.
  • m = \(-\frac{v}{u} = -\frac{-37.5}{-25.0} = -1.5\); h' = m·h = -1.5·4.0 = -6.0 cm.
  • Image: real, inverted, enlarged (6.0 cm), 37.5 cm in front of mirror.

9.3 Refraction of Light

• Light bends when passing obliquely between media due to speed changes, called refraction.
• Observations:
  • Pond bottom appears raised; letters under glass slab appear raised; pencil in water appears bent at interface; lemon in water appears larger.
• Activities:
  • 9.7: Coin in water bucket; refraction makes coin hard to pick up due to apparent position shift.
  • 9.8: Coin in bowl disappears from view; adding water makes it visible again due to refraction raising apparent position.
  • 9.9: Line under glass slab appears bent at edges when slab is angled; normal incidence shows no bending; top view shows raised line.

9.3.1 Refraction through a Rectangular Glass Slab

• Activity 9.10:
  • Trace light path through glass slab; ray bends toward normal entering glass (air to glass), away from normal exiting (glass to air).
  • Emergent ray parallel to incident ray due to equal/opposite bending at parallel faces; ray shifts laterally (Fig. 9.10).
• Laws of Refraction:
  1. Incident ray, refracted ray, normal at point of incidence lie in same plane.
  2. Snell’s Law: \(\frac{\sin i}{\sin r} = \text{constant}\) (for angles 0° < i < 90°).

9.3.2 The Refractive Index

• Refractive index (n) measures bending extent: \(n_{21} = \frac{v_1}{v_2}\), where v_1, v_2 are light speeds in media 1, 2 (Fig. 9.11).
• Absolute Refractive Index: \(n_m = \frac{c}{v}\), where c = 3×10^8 m/s (vacuum), v is speed in medium.
• Table 9.3:

  Medium	Refractive Index
  Air	1.0003
  Ice	1.31
  Water	1.33
  Alcohol	1.36
  Kerosene	1.44
  Fused quartz	1.46
  Turpentine oil	1.47
  Benzene	1.50
  Crown glass	1.52
  Canada Balsam	1.53
  Rock salt	1.54
  Carbon disulphide	1.63
  Dense flint glass	1.65
  Ruby	1.71
  Sapphire	1.77

• Optical Density:
  • Higher n = optically denser (slower light); e.g., kerosene (n=1.44) denser than water (n=1.33) despite lower mass density.
  • Rarer to denser: bends toward normal; Denser to rarer: bends away.

9.3.3 Refraction by Spherical Lenses

• Lens: Transparent material with one or two spherical surfaces.
• Types:
  • Convex (Converging): Thicker at middle, converges rays (Fig. 9.12a).
  • Concave (Diverging): Thinner at middle, diverges rays (Fig. 9.12b).
• Key Terms:
  • Optical Centre (O): Central point; rays pass without deviation.
  • Centres of Curvature (C1, C2): Centers of spheres forming lens surfaces.
  • Principal Axis: Line through O and C1, C2.
  • Principal Focus (F): Convex: rays converge at F; Concave: rays diverge from F.
  • Focal Length (f): Distance from O to F.
  • Aperture: Effective diameter of lens.
• Activity 9.11:
  • Focus sunlight with convex lens; sharp spot burns paper, giving approximate f.

9.3.4 Image Formation by Lenses

• Activity 9.12 (Convex Lens):
  • Measure f; place candle at various positions; observe image on screen (Table 9.4).
• Convex Lens Image Summary (Table 9.4):

  Object Position	Image Position	Image Size	Nature
  At infinity	At F2	Highly diminished, point-sized	Real, inverted
  Beyond 2F1	Between F2 and 2F2	Diminished	Real, inverted
  At 2F1	At 2F2	Same size	Real, inverted
  Between F1 and 2F1	Beyond 2F2	Enlarged	Real, inverted
  At F1	At infinity	No image formed	-
  Between F1 and O	Same side as object	Enlarged	Virtual, erect

• Activity 9.13 (Concave Lens):
  • Observe virtual image through lens; image remains diminished, erect, virtual (Table 9.5).
• Concave Lens Image Summary (Table 9.5):

  Object Position	Image Position	Image Size	Nature
  At infinity	At F1	Highly diminished, point-sized	Virtual, erect
  Between infinity and O	Between F1 and O	Diminished	Virtual, erect

9.3.5 Image Formation in Lenses Using Ray Diagrams

• Two rays for imaging:
  1. Parallel to principal axis: Convex - through F; Concave - diverges from F (Fig. 9.13).
  2. Through F (convex) or toward F (concave): Emerges parallel (Fig. 9.14).
  3. Through O: No deviation (Fig. 9.15).
• Convex lens diagrams: Fig. 9.16; Concave lens diagrams: Fig. 9.17.

9.3.6 Sign Convention for Spherical Lenses

• Similar to mirrors, measured from optical centre:
  • Convex f: positive; Concave f: negative.
  • Object on left; distances follow Cartesian convention.

9.3.7 Lens Formula and Magnification

• Lens Formula: \(\frac{1}{v} - \frac{1}{u} = \frac{1}{f}\), valid for all lenses.
• Magnification: \(m = \frac{h'}{h} = \frac{v}{u}\); positive for virtual, negative for real images.
• Example 9.3 (Concave Lens):
  • f = -15 cm, v = -10 cm; \(\frac{1}{u} = \frac{1}{v} - \frac{1}{f} = -\frac{1}{10} + \frac{1}{15} = -\frac{1}{30}\); u = -30 cm.
  • m = \(\frac{v}{u} = \frac{-10}{-30} = +0.33\).
  • Image: virtual, erect, one-third object size, 10 cm from lens.
• Example 9.4 (Convex Lens):
  • h = +2.0 cm, f = +10 cm, u = -15 cm; \(\frac{1}{v} = \frac{1}{f} + \frac{1}{u} = \frac{1}{10} - \frac{1}{15} = \frac{1}{30}\); v = +30 cm.
  • m = \(\frac{v}{u} = \frac{30}{-15} = -2\); h' = m·h = -2·2.0 = -4.0 cm.
  • Image: real, inverted, 4.0 cm, 30 cm from lens, twice object size.

9.3.8 Power of a Lens

• Power (P): \(P = \frac{1}{f}\) (f in meters), unit: dioptre (D).
• Convex: positive P; Concave: negative P.
• Combined Lenses: \(P = P_1 + P_2 + \ldots\).
• Applications: Opticians use power for corrective lenses; combined lenses enhance magnification in cameras, microscopes, telescopes.

Key Questions and Answers

1. Principal Focus (Concave Mirror):
   • Point where parallel rays converge after reflection.
2. Focal Length (R = 20 cm):
   • f = R/2 = 20/2 = 10 cm.
3. Erect, Enlarged Image:
   • Concave mirror (object between P and F).
4. Convex Rear-View Mirror:
   • Provides erect, diminished image with wide field of view.
5. Refraction in Water:
   • Bends toward normal (air to denser water, n=1.33).
6. Speed in Glass (n=1.50):
   • \(v = \frac{c}{n} = \frac{3 \times 10^8}{1.50} = 2 \times 10^8 \, \text{m/s}\).
7. Optical Density (Table 9.3):
   • Highest: Sapphire (n=1.77); Lowest: Air (n=1.0003).
8. Fastest Light Travel:
   • Water (n=1.33) faster than kerosene (n=1.44) or turpentine (n=1.47).
9. Diamond Refractive Index (n=2.42):
   • Speed of light in diamond is \( \frac{3 \times 10^8}{2.42} \approx 1.24 \times 10^8 \, \text{m/s}\).
10. 1 Dioptre:
    • Power of lens with f = 1 m.
11. Convex Lens (m=-1, v=50 cm):
    • m = \(-\frac{v}{u} = -1\); v = -u; u = -50 cm.
    • \(\frac{1}{f} = \frac{1}{v} + \frac{1}{u} = \frac{1}{50} - \frac{1}{50} = 0\); invalid, try lens formula.
    • Assume u = -50 cm, v = 50 cm; \(\frac{1}{f} = \frac{1}{50} - \frac{1}{-50} = \frac{2}{50}\); f = 25 cm.
    • P = \(\frac{1}{0.25} = +4.0 \, \text{D}\).
12. Concave Lens Power (f=2 m):
    • P = \(\frac{1}{2} = -0.5 \, \text{D}\).

Exercises

1. Lens Material:
   • (d) Clay (opaque, cannot form lens).
2. Concave Mirror (Virtual, Erect, Larger):
   • (d) Between P and F.
3. Convex Lens (Real, Same Size):
   • (b) At 2F1.
4. Mirror and Lens (f=-15 cm):
   • (a) Both concave (negative f).
5. Erect Image Mirror:
   • (d) Plane or convex.
6. Dictionary Reading Lens:
   • (c) Convex, f=5 cm (strong magnification).
7. Concave Mirror (f=15 cm, Erect Image):
   • Range: 0 to 15 cm (P to F).
   • Image: Virtual, erect, larger.
   • Ray diagram: Object between P and F; rays diverge, extend backward to form enlarged virtual image.
8. Mirror Uses:
   • (a) Headlights: Concave (parallel beams).
   • (b) Rear-view: Convex (wide field, erect).
   • (c) Solar furnace: Concave (concentrates sunlight).
9. Half-Covered Convex Lens:
   • Produces complete image (all rays from object reach lens); less bright due to reduced light. Verified by covering half lens and observing full image.
10. Convex Lens (h=5 cm, u=25 cm, f=10 cm):
    • \(\frac{1}{v} = \frac{1}{10} - \frac{1}{25} = \frac{3}{50}\); v = 16.67 cm.
    • m = \(\frac{16.67}{-25} = -0.67\); h' = -0.67·5 = -3.33 cm.
    • Image: Real, inverted, diminished (3.33 cm), 16.67 cm from lens.
    • Ray diagram: Parallel ray through F2, ray through O, intersect at image.
11. Concave Lens (f=15 cm, v=10 cm):
    • f = -15 cm, v = -10 cm; \(\frac{1}{u} = -\frac{1}{10} + \frac{1}{15} = -\frac{1}{30}\); u = -30 cm.
    • Ray diagram: Parallel ray diverges from F1, ray through O, extend backward to intersect.
12. Convex Mirror (u=10 cm, f=15 cm):
    • \(\frac{1}{v} = \frac{1}{15} - \frac{1}{10} = -\frac{1}{30}\); v = +6 cm.
    • Image: Virtual, erect, 6 cm behind mirror.
13. Plane Mirror Magnification:
    • m = +1 means image is same size, virtual, erect.
14. Convex Mirror (h=5 cm, u=20 cm, R=30 cm):
    • f = R/2 = 15 cm; \(\frac{1}{v} = \frac{1}{15} - \frac{1}{20} = \frac{1}{60}\); v = +12 cm.
    • m = \(\frac{12}{-20} = +0.6\); h' = 0.6·5 = +3 cm.
    • Image: Virtual, erect, 3 cm, 12 cm behind mirror.
15. Concave Mirror (h=7 cm, u=27 cm, f=18 cm):
    • \(\frac{1}{v} = -\frac{1}{18} + \frac{1}{27} = -\frac{1}{54}\); v = -54 cm.
    • m = \(\frac{-54}{-27} = -2\); h' = -2·7 = -14 cm.
    • Image: Real, inverted, 14 cm, 54 cm in front of mirror.
16. Lens Power (P=-2.0 D):
    • f = \(\frac{1}{-2.0} = -0.5 \, \text{m} = -50 \, \text{cm}\); Concave lens.
17. Corrective Lens (P=+1.5 D):
    • f = \(\frac{1}{1.5} = 0.67 \, \text{m} = 67 \, \text{cm}\); Convex (converging).