Wave Optics Class 12 Notes

• A wavefront is a surface of constant phase.
• Spherical wavefronts come from a point source; plane wavefronts from a distant source.
• Using Huygens' construction, the laws of reflection and refraction (Snell's law) are derived geometrically.

• Constructive interference (bright fringe): path difference = an integral multiple of wavelength.
• Destructive interference (dark fringe): path difference = a half-integral multiple of wavelength.
• The principle of superposition governs the resultant amplitude or intensity.

• Path difference to a point at angle θ:
• $\Delta =d\sin \theta$
• .
• Bright fringes when
• $d\sin \theta =n\lambda$
• , dark when
• $d\sin \theta =(n+\frac{1}{2})\lambda$
• .
• Intensity at a point:
• $I=I_0{\cos }^2\left(\frac{\pi d\sin \theta }{\lambda }\right)$
• .

• Brewster's law: at the polarising angle
• $\tan i_p=n$
• , the reflected and refracted rays are at right angles.
• A polaroid transmits vibrations along its transmission axis; crossed polaroids block light completely.
• Intensity after a polaroid at angle θ (Malus's law):
• $I=I_0{\cos }^2\theta$
• .

Example: In Young's experiment, slits are 0.2 mm apart and the screen is 1 m away with green light of wavelength 500 nm. Find the fringe width.

Solution: $\beta =\frac{\lambda D}{d}=\frac{500\times 10^{-9}\times 1}{0.2\times 10^{-3}}=2.5\text{mm}$.

Exercises (NCERT Q10.1–Q10.6)

• 10.1) Monochromatic light of wavelength 589 nm is incident from air on a water surface. What are the wavelength, frequency and speed of (a) reflected, and (b) refracted light? Refractive index of water is 1.33.
• 10.2) What is the shape of the wavefront in each of the following cases: (a) Light diverging from a point source. (b) Light emerging out of a convex lens when a point source is placed at its focus. (c) The portion of the wavefront of light from a distant star intercepted by the Earth.
• 10.3) (a) The refractive index of glass is 1.5. What is the speed of light in glass? (Speed of light in vacuum is 3.0 × 10⁸ m s⁻¹) (b) Is the speed of light in glass independent of the colour of light? If not, which of the two colours red and violet travels slower in a glass prism?
• 10.4) In a Young's double-slit experiment, the slits are separated by 0.28 mm and the screen is placed 1.4 m away. The distance between the central bright fringe and the fourth bright fringe is measured to be 1.2 cm. Determine the wavelength of light used in the experiment.
• 10.5) In Young's double-slit experiment using monochromatic light of wavelength λ, the intensity of light at a point on the screen where path difference is λ, is K units. What is the intensity of light at a point where path difference is λ/3?
• 10.6) A beam of light consisting of two wavelengths, 650 nm and 520 nm, is used to obtain interference fringes in a Young's double-slit experiment. (a) Find the distance of the third bright fringe on the screen from the central maximum for wavelength 650 nm. (b) What is the least distance from the central maximum where the bright fringes due to both the wavelengths coincide?

Answers at a glance

1. 1
   Q10.1 → (a) Reflected: λ = 589 nm, ν = 5.09 × 10¹⁴ Hz, v = 3.0 × 10⁸ m s⁻¹. (b) Refracted: λ = 443 nm, ν = 5.09 × 10¹⁴ Hz, v = 2.26 × 10⁸ m s⁻¹.
2. 2
   Q10.2 → (a) Spherical. (b) Plane. (c) Plane (spherical wavefront of very large radius).
3. 3
   Q10.3 → (a) 2.0 × 10⁸ m s⁻¹. (b) No, the speed is not colour-independent; violet travels slower.
4. 4
   Q10.4 → λ = 600 nm.
5. 5
   Q10.5 → Intensity = K/4.
6. 6
   Q10.6 → (a) x₃ = 1950(D/d) nm. (b) Coincidence first occurs at x = 2600(D/d) nm, where D/d is the ratio of screen distance to slit separation (values not stated in the problem).

MCQ Questions

1 Mark Questions

1. 1
   State the condition for constructive interference in terms of path difference.
2. 2
   Define coherence of light sources.
3. 3
   What is the difference between interference and diffraction?
4. 4
   State Brewster's law.

2 Mark Questions

1. 1
   In Young's double slit experiment, λ = 600 nm, d = 1 mm and D = 1 m. Find the fringe width. (Ans: β = λD/d = 0.6 mm)
2. 2
   Light of wavelength 500 nm in air enters glass of refractive index 1.5. Find its wavelength and speed in glass. (Ans: λ = 333 nm, v = 2 × 10⁸ m/s)
3. 3
   Find the Brewster angle for light passing from air into glass of refractive index 1.5. (Ans: iB = tan⁻¹(1.5) ≈ 56.3°)

3 Mark Questions

1. 1
   Derive the expression for the fringe width in Young's double slit experiment. (Ans: β = λD/d)
2. 2
   Explain Huygens' principle and use it to establish the laws of reflection of light.
3. 3
   Explain how polarisation of light by reflection occurs and state Brewster's law with its experimental verification.

5 Mark Questions

1. 1
   Describe Young's double slit experiment with a ray diagram. Derive the positions of bright and dark fringes and the fringe width, and explain how the intensity distribution confirms the wave nature of light.
2. 2
   Distinguish between interference and diffraction of light. Explain diffraction at a single slit, derive the condition for minima, and discuss why the wave nature of light is shown by diffraction but not by the particle picture.