In two separate Young's double-slit experimental set-ups, two monochromatic light sources of different wavelengths are used to get fringes of equal width. The ratios of the slit separations and that of the wavelengths of light used are 2:1 and 1:2 respectively. The corresponding ratio of the distances between the slits and the respective screens (D₁ / D₂$D_{1} / D_{2}$) is ____.

Related Formula

Brewster's Law: iₚ = (μ₂)/(μ₁)$\tan i_p = \frac{\mu_2}{\mu_1}$ r = 90° - iₚ$r = 90^{\circ} - i_p$

Core Logic

Linearly polarized reflected light implies the light was incident at Brewster's angle iₚ$i_p$. At Brewster's angle, the reflected and refracted rays are perpendicular to each other (iₚ + r = 90°$i_p + r = 90^{\circ}$).

Step 1: Determine Incident Angle

Step 2: Calculate Angle of Refraction

Pattern Recognition

Brewster's angle immediately triggers the relationship r = 90° - i$r = 90^{\circ} - i$. No complex Snell's law calculation is needed if the Brewster condition is identified from "reflected beam is linearly polarized".

Chapter Mix

Class 12 Physics: Wave Optics

Related Formula

Core Logic

For the reflected and refracted rays to be mutually perpendicular, the angle of incidence must equal Brewster's angle, θₚ$\theta_p$. Using Brewster's law:

Step 1: Calculate Angle

Given incident medium index μ₁ = 2$\mu_1 = 2$ and refracting medium index μ₂ = 2√(3)$\mu_2 = 2\sqrt{3}$.

Therefore, θ = 60°$\theta = 60^{\circ}$.

Pattern Recognition

The condition 'reflected and refracted rays perpendicular' is the strict definitional trigger for Brewster's law.

Chapter Mix

Class 12 Physics: Wave Optics

Related Formula

where path difference phase Δ φ = (2π)/(λ) · Δ x = (2π)/(λ) · (yd)/(D)$\Delta \phi = \frac{2\pi}{\lambda} \cdot \Delta x = \frac{2\pi}{\lambda} \cdot \frac{yd}{D}$

Core Logic

Given values: d = 2 mm = 2 × 10⁻³ m$d = 2 \text{ mm} = 2 \times 10^{-3} \text{ m}$ D = 10 m$D = 10 \text{ m}$ λ = 6000 AA = 6 × 10⁻⁷ m$\lambda = 6000 \text{ \AA} = 6 \times 10^{-7} \text{ m}$ Position y = (d)/(2)$y = \frac{d}{2}$ (since it is directly in front of one of the slits).

Step 1: Calculate Phase Difference

The path difference at this position is:

Step 2: Substitution

Substitute the values:

Step 3: Intensity Output

Using the intensity formula:

Pattern Recognition

The point "in front of one slit" always means y = d/2$y = d/2$. Calculate phase directly using π d² / (2 λ D)$\pi d^2 / (2 \lambda D)$.

Chapter Mix

Class 12 Physics: Wave Optics

Related Formula

θ = (λ)/(a)$$\theta = \frac{\lambda}{a}$$ (Diffraction angle)

Core Logic

Statement-I is correct because a prism changes the direction of a plane wave without changing its planar wavefront nature, whereas a small pinhole acts as a point source (Huygens' principle), generating spherical wavefronts. Statement-II is incorrect. Increasing the slit width a$a$ decreases the diffraction angle θ$\theta$ and reduces the spreading of the wave. A narrower slit produces a more pronounced spherical wave (high curvature) while a wider slit leads to a flatter, less curved (more planar) wave.

Step 1: Final Conclusion

Therefore, Statement I is true, and Statement II is false.

Pattern Recognition

Curvature of a wavefront coming out of an aperture is inversely related to aperture size relative to wavelength. Small aperture = high diffraction = highly curved spherical wave.

Chapter Mix

Class 12 Physics: Wave Optics