#total internal reflection

Refraction at Spherical Surfaces and Lens Maker's Formula

What optical principle allows a prism to be used as a reflector in optical devices?

**Lens maker's formula** 1/f = (n-1)(1/R₁ - 1/R₂), n refractive index, R₁,R₂ radii of curvature (m), sign convention R positive if surface convex towards incident light. For double convex R₁=12 cm, R₂=-12 cm, n=1.5, 1/f=(0.5)(1/12 -1/(-12))=(0.5)(2/12)=1/12, f=12 cm, converging. Prisms reflect light via total internal reflection when the angle of incidence exceeds the critical angle at the prism’s internal surfaces. This property, dependent on the prism’s refractive index and angle, enables efficient reflection without loss, as seen in devices like binoculars. Substituting values gives Total internal reflection, which matches expected image position and magnification from mirror/lens formula 1/f = 1/v - 1/u (lens) or 1/f

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Refraction at Spherical Surfaces and Lens Maker's Formula

In optical fibers, what ensures that light remains confined within the core during transmission?

**Spherical refracting surface** power P = (n₂-n₁)/R, lens power sum of two surfaces. Lens maker derivation combines two refractions, sign of R₂ negative for second surface convex opposite direction, yielding 1/f positive for convex lens. In optical fibers, the core has a higher refractive index than the cladding, enabling total internal reflection. When light strikes the core-cladding boundary at an angle greater than the critical angle, it reflects back into the core, ensuring confinement and minimal loss. Substituting values gives Higher refractive index of core than cladding, which matches expected image position and magnification from mirror/lens formula 1/f = 1/v - 1/u (lens) or 1/f

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Refraction at Spherical Surfaces and Lens Maker's Formula

What is the critical angle for a crown glass (\( n = 1.52 \)) to air interface?

**Refraction at spherical surface** formula n₁/u + n₂/v = (n₂-n₁)/R governs single surface, extension to two surfaces yields lens maker. Double convex with equal |R| has f = R/[2(n-1)], for R=12 cm, n=1.5, f=12 cm, illustrating dependence on curvature and index. Critical angle: sin i_c = (n₂/n₁) . Crown glass ( n₁ = 1.52 ), air ( n₂ = 1 ). sin i_c = (1/1.52) ≈ 0.658 . i_c = sin⁻¹(0.658) ≈ 41.1° . Substituting values gives 41°, which matches expected image position and magnification from mirror/lens formula 1/f = 1/v - 1/u (lens) or 1/f = 1/v + 1/u (mirror), confirming sign conventions and refraction principles.

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Refraction at Spherical Surfaces and Lens Maker's Formula

What is the key factor that determines whether light undergoes total internal reflection at an interface?

**Lens maker's formula** 1/f = (n-1)(1/R₁ - 1/R₂), n refractive index, R₁,R₂ radii of curvature (m), sign convention R positive if surface convex towards incident light. For double convex R₁=12 cm, R₂=-12 cm, n=1.5, 1/f=(0.5)(1/12 -1/(-12))=(0.5)(2/12)=1/12, f=12 cm, converging. Total internal reflection occurs when light travels from a denser to a rarer medium and the angle of incidence exceeds the critical angle. The critical angle depends on the refractive indices of the two media, making the angle of incidence the decisive factor for the phenomenon. Substituting values gives Angle of incidence relative to the critical angle, which matches expected image position and magnification from mirror/lens

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Refraction at Spherical Surfaces and Lens Maker's Formula

What is the primary reason optical fibers can transmit light over long distances with minimal loss?

**Lens maker's formula** 1/f = (n-1)(1/R₁ - 1/R₂), n refractive index, R₁,R₂ radii of curvature (m), sign convention R positive if surface convex towards incident light. For double convex R₁=12 cm, R₂=-12 cm, n=1.5, 1/f=(0.5)(1/12 -1/(-12))=(0.5)(2/12)=1/12, f=12 cm, converging. Optical fibers use total internal reflection to transmit light. The core has a higher refractive index than the cladding, ensuring that light rays striking the boundary at angles greater than the critical angle are fully reflected, preventing loss of light intensity over distance. Substituting values gives Total internal reflection within the core, which matches expected image position and magnification from mirror/lens formula 1/f = 1/v -

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Refraction Through Prism and Minimum Deviation

When light travels from a denser medium to a rarer medium and the angle of incidence exceeds a certain value, what pheno

**Prism formula** for small A, δ_m≈(n-1)A, for 30°, n=1.6, δ_m≈0.6×30°=18°, approximate, exact using sin formula. For 45°, n=1.6, sin[(45+δ_m)/2]=1.6×sin22.5°=1.6×0.3827=0.6123, (45+δ_m)/2=37.8°, δ_m=30.6°, showing deviation increases with A and n. When light travels from a denser to a rarer medium (e.g., glass to air) and the angle of incidence exceeds the critical angle, total internal reflection occurs. This is because the refracted ray would otherwise require a sine value greater than 1, which is physically impossible. Substituting values gives Total internal reflection, which matches expected image position and magnification from mirror/lens formula 1/f = 1/v - 1/u (lens) or 1/f = 1/v + 1/u (mirror), confirming sign conventions and refraction principles.

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Total Internal Reflection and Critical Angle

What is the critical angle for a dense flint glass (\( n = 1.62 \)) to air interface?

**Total internal reflection** occurs when light travels from denser to rarer medium and incidence > C, condition sinC = 1/n for air interface. For glass n=1.52 C≈41°, water n=1.33 C≈48.8°, so at 49° water-air TIR occurs, explaining why ray does not emerge. Critical angle: sin i_c = (n₂/n₁) . Glass ( n₁ = 1.62 ), air ( n₂ = 1 ). sin i_c = (1/1.62) ≈ 0.617 . i_c = sin⁻¹(0.617) ≈ 38.1° . Substituting values gives 38°, which matches expected image position and magnification from mirror/lens formula 1/f = 1/v - 1/u (lens) or 1/f = 1/v + 1/u (mirror), confirming sign conventions and refraction principles.

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Total Internal Reflection and Critical Angle

What is the significance of the critical angle in the context of total internal reflection?

**Total internal reflection** occurs when light travels from denser to rarer medium and incidence > C, condition sinC = 1/n for air interface. For glass n=1.52 C≈41°, water n=1.33 C≈48.8°, so at 49° water-air TIR occurs, explaining why ray does not emerge. The critical angle is the angle of incidence above which total internal reflection occurs when light travels from a denser to a rarer medium. At this angle, the refracted ray grazes the boundary (angle of refraction = 90°), and beyond it, all light is reflected back, enabling applications like optical fibers. Substituting values gives It is the threshold for total internal reflection, which

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Refraction at Plane Surfaces and Snell's Law

What is the critical angle for a glass (\( n = 1.5 \)) to water (\( n = 1.33 \)) interface?

**Refractive index** n = c/v, water 1.33 means light 1.33 times slower than vacuum. Passing from water to air at 49°, n₁ sinθ₁ =1.33×sin49°≈1.33×0.755=1.004>1, so sinθ₂>1 impossible, total internal reflection occurs, no refraction. Critical angle: sin i_c = (n₂/n₁) . Glass ( n₁ = 1.5 ), water ( n₂ = 1.33 ). sin i_c = (1.33/1.5) ≈ 0.887 . i_c = sin⁻¹(0.887) ≈ 62.5° . Substituting values gives 62°, which matches expected image position and magnification from mirror/lens formula 1/f = 1/v - 1/u (lens) or 1/f = 1/v + 1/u (mirror), confirming sign conventions and refraction principles.

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Total Internal Reflection and Critical Angle

What is the primary reason optical fibers are bent without losing light transmission efficiency?

**Critical angle** C satisfies sinC = n₂/n₁, n₁>n₂, n₂=1 for air, n₁=1.52 for glass gives sinC=1/1.52=0.6579, C≈41.1°, for water n=1.33 C=48.75°. Beyond C, total internal reflection occurs, all light reflected, no refracted ray, used in optical fibers and prisms. Optical fibers maintain light transmission when bent because the angle of incidence at the core-cladding interface remains greater than the critical angle, ensuring total internal reflection. This allows light to follow the bend without escaping into the cladding. Substituting values gives Due to sustained total internal reflection, which matches expected image position and magnification from mirror/lens formula 1/f = 1/v - 1/u (lens) or 1/f =

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Refraction at Plane Surfaces and Snell's Law

What is the critical angle for a diamond-air interface if the refractive index of diamond is \( 2.42 \)?

**Refractive index** n = c/v, water 1.33 means light 1.33 times slower than vacuum. Passing from water to air at 49°, n₁ sinθ₁ =1.33×sin49°≈1.33×0.755=1.004>1, so sinθ₂>1 impossible, total internal reflection occurs, no refraction. Critical angle: sin i_c = (n₂/n₁) . Diamond ( n₁ = 2.42 ), air ( n₂ = 1 ). sin i_c = (1/2.42) ≈ 0.413 . i_c = sin⁻¹(0.413) ≈ 24.4° . Substituting values gives 24°, which matches expected image position and magnification from mirror/lens formula 1/f = 1/v - 1/u (lens) or 1/f = 1/v + 1/u (mirror), confirming sign conventions and refraction principles.

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Total Internal Reflection and Critical Angle

What optical property of a prism allows it to be used in binoculars to invert images?

**Critical angle** C satisfies sinC = n₂/n₁, n₁>n₂, n₂=1 for air, n₁=1.52 for glass gives sinC=1/1.52=0.6579, C≈41.1°, for water n=1.33 C=48.75°. Beyond C, total internal reflection occurs, all light reflected, no refracted ray, used in optical fibers and prisms. Prisms in binoculars use total internal reflection to invert and revert images. By reflecting light multiple times within the prism (e.g., in a Porro prism), the image orientation is corrected from the inverted form produced by the objective lens, maintaining the same size. Substituting values gives Total internal reflection, which matches expected image position and magnification from mirror/lens formula 1/f = 1/v - 1/u (lens) or

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