Quick Revision Notes: Class 12 Physics

Chapter: Wave Optics (प्रकाशिकी)

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### 1. Wavefront and Huygens' Principle (तरंगाग्र तथा हाइगेन्स का सिद्धान्त)

• Wavefront (तरंगाग्र): The locus of all adjacent points vibrating in the same phase is called a wavefront.
  • Types: Spherical (गोलीय), Cylindrical (बेलनाकार), and Plane (समतल).
• Huygens' Principle (हाइगेन्स का सिद्धान्त):
  1. Every point on a given wavefront acts as a source of secondary wavelets (द्वितीयक तरंगिकाएँ) travelling in all directions with the speed of light.
  2. The envelope of these secondary wavelets at any later time gives the new position of the wavefront.
• Laws of Reflection and Refraction: Huygens' principle successfully explains the laws of reflection ($\angle i = \angle r$) and refraction ($\frac{\sin i}{\sin r} = \frac{v_1}{v_2} = \mu$).

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### 2. Interference of Light Waves (प्रकाश का व्यतिकरण)

The modification in the intensity of light due to the superposition of two or more light waves is called interference.

• Superposition Principle: Resultant displacement $y = y_1 + y_2$.
• Condition for Sustained Interference: The sources must be coherent (कला-संबद्ध) i.e., they must maintain a constant phase difference.

Mathematical Analysis:

Let two waves be:

• $y_1 = a_1 \sin(\omega t)$

• $y_2 = a_2 \sin(\omega t + \phi)$

• Resultant Amplitude ($R$): R = sqrt(a_1^2 + a_2^2 + 2a_1a_2 \cos\phi)

• Resultant Intensity ($I$): Since $I \propto R^2$, I = I_1 + I_2 + 2\sqrt{I_1I_2} \cos\phi

Conditions for Interference:

1. Constructive Interference (संपोषी व्यतिकरण - Maximum Intensity):

   • Path difference ($\Delta x$): \Delta x = n\lambda (where $n = 0, 1, 2, 3, \dots$)
   • Phase difference ($\phi$): \phi = 2n\pi
   • Maximum Intensity: I_max = (\sqrt{I_1} + \sqrt{I_2})^2 \propto (a_1 + a_2)^2

2. Destructive Interference (विनाशी व्यतिकरण - Minimum Intensity):

   • Path difference ($\Delta x$): \Delta x = (2n - 1)\frac{\lambda}{2} (where $n = 1, 2, 3, \dots$)
   • Phase difference ($\phi$): \phi = (2n - 1)\pi
   • Minimum Intensity: I_min = (\sqrt{I_1} - \sqrt{I_2})^2 \propto (a_1 - a_2)^2

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### 3. Young's Double Slit Experiment (YDSE) (यंग का द्वि-स्लिट प्रयोग)

• Fringe Width ($\beta$): The distance between two consecutive bright or dark fringes. \beta = \frac{\lambda D}{d}

  • $\lambda$ = Wavelength of light (प्रकाश की तरंगदैर्ध्य)
  • $D$ = Distance between slits and screen (स्लिट और पर्दे के बीच की दूरी)
  • $d$ = Distance between the two slits (दोनों स्लिटों के बीच की दूरी)

• Angular Fringe Width ($\theta$): \theta = \frac{\beta}{D} = \frac{\lambda}{d}

• Position of $n$-th Bright Fringe: x_n = \frac{n\lambda D}{d}

• Position of $n$-th Dark Fringe: x_n' = (2n - 1)\frac{\lambda D}{2d}

• Effect of Medium: When the entire setup is immersed in a medium of refractive index $\mu$:

  • New wavelength: \lambda' = \frac{\lambda}{\mu}
  • New fringe width: \beta' = \frac{\beta}{\mu}

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### 4. Diffraction of Light (प्रकाश का विवर्तन)

The bending of light around the corners of an obstacle or aperture into the region of geometrical shadow is called diffraction.

• Condition for Minima (Single Slit): a \sin\theta = n\lambda (where $n = 1, 2, 3, \dots$)

  • $a$ = Width of the slit

• Condition for Maxima: a \sin\theta = (2n + 1)\frac{\lambda}{2} (where $n = 1, 2, 3, \dots$)

• Central Maximum Width:

  • Linear Width: \frac{2\lambda D}{a}
  • Angular Width: \frac{2\lambda}{a}

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### 5. Polarization of Light (प्रकाश का ध्रुवण)

The phenomenon of restricting the vibrations of light waves into a single plane is called polarization. (Proves the transverse nature of light waves).

• Malus's Law (मालस का नियम): When completely plane-polarized light is incident on an analyzer, the transmitted intensity $I$ is directly proportional to the square of the cosine of the angle ($\theta$) between the transmission axes of the polarizer and the analyzer. I = I_0 \cos^2\theta (where $I_0$ is the initial intensity)

• Brewster's Law (ब्रूस्टर का नियम): When unpolarized light is incident at a specific polarizing angle ($i_p$ or $i_B$) on a transparent surface, the reflected light is completely plane-polarized, and the reflected and refracted rays are perpendicular to each other. \mu = \tan(i_p)

  • $\mu$ = Refractive index of the medium
  • $i_p$ = Brewster's angle (ध्रुवण कोण)

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### Important Constants & Relations for Quick Numericals

• Speed of light in vacuum: c = 3 \times 10^8 \text{ m/s}
• Relationship: c = \nu\lambda (where $\nu$ is frequency)
• Visible light wavelength range: 4000 \text{ \AA} \text{ to } 7000 \text{ \AA} ($1 \text{ \AA} = 10^{-10} \text{ m}$)