Govt. Exams
Entrance Exams
For total internal reflection from denser to rarer medium, sin(θc) = n₂/n₁ where n₁ is refractive index of denser medium and n₂ of rarer medium
Fringe width β = λD/d. If d is doubled, β becomes β/2, i.e., half of the original value
Using Snell's law: n₁sin(θ₁) = n₂sin(θ₂). 1.5 × sin(30°) = 1.0 × sin(θ₂). 1.5 × 0.5 = sin(θ₂). sin(θ₂) = 0.75, θ₂ = 48.6°
Frequency f = c/λ = (3 × 10⁸)/(632.8 × 10⁻⁹) = (3 × 10⁸)/(6.328 × 10⁻⁷) ≈ 4.74 × 10¹⁴ Hz.
Grating constant d = 1/(number of lines per unit length) = 1/5000 cm⁻¹ = 2.0 × 10⁻⁴ cm or 2 μm.
At critical angle: sin(θc) = 1/n. So sin(45°) = 1/n gives 1/√2 = 1/n, therefore n = √2 ≈ 1.41.
Using Snell's law: n₁sin(θ₁) = n₂sin(θ₂). So 1.33 × sin(30°) = 1.5 × sin(θ₂). Therefore 1.33 × 0.5 = 1.5 × sin(θ₂), which gives sin(θ₂) = 0.443, θ₂ ≈ 27.8°.
Fringe width β = λD/d = (500 × 10⁻⁹ × 1)/(0.5 × 10⁻³) = 500 × 10⁻⁶/0.5 × 10⁻³ = 1.0 mm.
Using lens formula: 1/f = 1/v + 1/u. For concave lens, f = -20 cm, v = -10 cm (virtual image). So 1/-20 = 1/-10 + 1/u gives 1/u = -1/20 + 1/10 = 1/20, therefore u = 20 cm.
When a plane mirror moves towards an object with velocity v, the image also moves towards the mirror with velocity v. The relative velocity of approach between object and image is 2v. Therefore, velocity of image = 2 × 2 = 4 m/s.
