Wave Optics

Wave theory of light was first proposed by Christian Huygens.

Step 1: Fringe width. β = (λ D)/(d)
Step 2: Distance relation. Distance between 1st maxima and 5th minima: x = (9)/(2)β
Step 3: Solving for wavelength. λ = (2xd)/(9D) = 600 nm

Step 1: Angular width of fringe.
The angular width of the fringe in Young’s double slit experiment is given by: where is the wavelength of the light and is the distance between the two slits.

Step 2: Solve for .
Given that , , and converting the angle to radians, we get: Thus, the correct answer is (1).

Step 1: Fringe width:
Step 2: Using lens formula:
Step 3: From data, slit separation:
Step 4: Substitute values:
Step 5: Convert:

Step 1: Extra path introduced by sheet:
Step 2: Condition for central maxima:
Step 3: Substituting values:
Step 4: Hence:
Step 5: Linear shift on screen:

Step 1: Central bright fringe is order zero.
Step 2: Counting outward, the third bright fringe corresponds to marking 4.

Step 1: For half maximum intensity: cos² β = (1)/(2) ⟹ β = (π)/(4)
Step 2: Path difference: β = (π d x)/(λ D)
Step 3: x = (λ D)/(4d)
Step 4: Substitute values: x = dfrac500 × 10⁻94 × 10⁻3 = 0.3125 × 10⁻4m

Step 1: Optical path difference introduced by the sheet: Δ = (μ-1)t
Step 2: Condition for central maxima shift: dsinθ = (μ-1)t
Step 3: Substituting values: sinθ = frac(1.17-1)×1.5×10⁻73×10⁻7 ≈ 0.085 ⟹ θ ≈ 4.9^∘
Step 4: Linear shift on screen: y = (D(μ-1)t)/(2d)

Step 1: Intensity in YDSE: I = 4I₀cos²((πΔ)/(λ))
Step 2: At centre (Δ=0): Imax=4I₀=K
Step 3: For Δ=λ/4: I=4I₀cos²((π)/(4)) =4I₀((1)/(2))=2I₀
Step 4: Hence, I=(K)/(2)

This question involves de Broglie's hypothesis, which describes the wave-particle duality of matter.
1. de Broglie Wavelength Formula: - According to de Broglie, any moving particle, including electrons, has an associated wave. The de Broglie wavelength is given by the formula: Where: - is Planck's constant, - is the mass of the particle (electron in this case), - is the velocity of the particle. 2. Understanding the Formula: - This equation tells us that the wavelength associated with a particle is inversely proportional to its momentum (mass velocity). - The higher the velocity, the shorter the wavelength. 3. Analysis of Options: - Option (1) is the correct de Broglie wavelength formula. - Option (2) is incorrect because it introduces an unnecessary factor of 2. - Option (3) is incorrect because it reverses the relationship. - Option (4) is incorrect for the same reason as Option (3).

When light enters a medium with refractive index , its speed and wavelength decrease, but frequency remains constant. The wavelength in the medium is given by: Given: