Refraction of light at a Plane Interface: Total Internal Reflection: Optical Fibre

TIR at each 45° face; ray emerges anti-parallel to the incident ray (turned 180°)

Step 1: When light enters through the hypotenuse face normally (perpendicular), it travels straight inside and hits one of the 45° faces. Step 2: At this face, the angle of incidence = 45° > critical angle (42°), so TIR occurs. Step 3: The ray reflects and travels to the second 45° face, again at 45° > 42°, so TIR occurs again. Step 4: After two TIRs at the 45° faces, the ray emerges through the hypotenuse anti-parallel (180° reversed) to the original direction. Step 5: This prism is used as a 'reversing prism' in binoculars. Option B describes a simple reflecting prism (entry through 45° face

85.0 cm

Step 1: sin C = 1/n = 3/4. Step 2: cos C = √(1 – 9/16) = √(7/16) = √7/4. Step 3: tan C = sin C / cos C = (3/4) / (√7/4) = 3/√7. Step 4: Radius r = h × tan C = 75 × (3/√7) = 225/√7 = 225/2.6458 ≈ 85.0 cm. Step 5: The area through which light can escape is πr². Here the question asks for radius = 85.0 cm. Option A uses sin C instead of tan C; Option C uses tan C = 1; Option D uses an incorrect value of tan C.

62.7°

Step 1: For the glass-water interface, the critical angle C is where the refracted ray in water makes 90° with the normal. Step 2: sin C = n_water / n_glass = (4/3) / (3/2) = (4/3) × (2/3) = 8/9. Step 3: sin C = 8/9 = 0.8889. Step 4: C = sin⁻¹(0.8889) ≈ 62.7°. Step 5: This is larger than the glass-air critical angle (≈41.8°) because water is denser than air, so a larger angle is needed for TIR at glass-water interface. Option A (48.6°) is the water-air critical angle; Option B (41.8°) is the glass-air critical angle; Option D is an incorrect calculation.

9.6 cm

Step 1: When the microscope is raised by 2.4 cm, the normal displacement of the image must equal 2.4 cm to refocus the mark. Step 2: The normal displacement formula is: d = t(1 – 1/n), where t is the height of water. Step 3: 2.4 = t(1 – 1/(4/3)) = t(1 – 3/4) = t × (1/4). Step 4: t = 2.4 × 4 = 9.6 cm. Step 5: So water must be poured to a height of 9.6 cm. Option A (7.2 cm) uses d = t × (1/3) incorrectly; Option B (3.2 cm) uses t = d × (4/3); Option D (6.4 cm) uses d = t × (3/8).