Lernzettel: Apparent Depth and Total Internal Reflection

📋 Course Outline

1. Real and Apparent Depth
2. Apparent Depth Calculations
3. Apparent Displacement
4. Critical Angle and TIR
5. Critical-Angle Applications

📖 1. Real and Apparent Depth

★ Must-know

📐 Formula — For perpendicular observation at a plane interface, the refractive-index ratio equals the ratio of real depth to apparent depth: $\frac{n_i}{n_r}=\frac{\text{real depth}}{\text{apparent depth}}$.

• The real-depth/apparent-depth relation applies to plane surfaces, requires perpendicular observation, and in the stated simplified form requires the observing eye to be in air.

• When an object in a denser medium is viewed from a rarer medium, its apparent depth is smaller and the virtual image appears closer to the interface; viewing from a denser medium toward a rarer medium gives a larger apparent depth.

Further detail

• A ray from the real object O refracts at the interface and is traced backward by the eye to form a virtual image I at the apparent depth.

📖 2. Apparent Depth Calculations

★ Must-know

📐 Formula — When the rarer medium is air, the refractive index of the denser medium satisfies $n=\frac{\text{real depth}}{\text{apparent depth}}$.

• To make a 24 cm container appear half-filled from air, the water must have a real depth of approximately 13.72 cm, because $\frac{4}{3}=\frac{x}{24-x}$.

Further detail

📐 Formula — The apparent depth calculation follows from Snell’s law and the small-angle relation $\sin\theta\approx\tan\theta$, giving $\frac{n}{1}=\frac{AO}{AI}$.

• For water with refractive index $\frac{4}{3}$ in a 24 cm container, a truly half-filled level has real depth 12 cm and apparent depth 9 cm.

💡 Memory Hook

Real depth → apparent depth → apparent filling level

📖 3. Apparent Displacement

🔑 Key Concepts & Definitions

• Apparent displacement : The shift between a real object and its virtual image caused by refraction at an interface.

★ Must-know

📐 Formula — For a layer of thickness $t$ and refractive index $n$ viewed from air, the apparent displacement is $d=t\left(1-\frac{1}{n}\right)$.

Further detail

• Objects at different real positions, such as O₁ and O₂, form virtual images I₁ and I₂, and the corresponding displacement segments O₁I₁ and O₂I₂ are equal in the stated parallel-interface arrangement.

📖 4. Critical Angle and TIR

🔑 Key Concepts & Definitions

• Critical angle : The angle of incidence in the denser medium for which the refracted ray in the rarer medium travels along the interface at 90°.
• Total internal reflection : Occurs when light travels from a denser medium to a rarer medium with an incidence angle greater than the critical angle, causing the ray to return entirely into the denser medium.

📝 Essential Points

• As the incidence angle increases from below the critical angle, the refracted angle increases until the refracted ray reaches the interface at 90°; any further increase produces total internal reflection.

💡 Memory Hook

Incident angle greater than critical angle → total internal reflection.

📖 5. Critical-Angle Applications

★ Must-know

📐 Formula — For light passing from a medium of refractive index $n_1$ to a rarer medium of refractive index $n_2$, the critical angle satisfies $\sin C=\frac{n_2}{n_1}$.

📐 Formula — When the rarer medium is air, the critical angle satisfies $\sin C=\frac{1}{n}$.

• For yellow light, the glass–air critical angle is approximately 42° because $\sin C=\frac{1}{3/2}=\frac{2}{3}$, while the water–air critical angle is approximately **48°**35′ because $\sin C=\frac{1}{4/3}=\frac{3}{4}$.

Further detail

• The critical angle depends on the denser medium, the rarer medium, and the colour of the light ray.

• For the glass–water pair, the critical angle is approximately **62°**44′ because $\sin C=\frac{4/3}{3/2}=\frac{8}{9}$.

📊 Synthesis Tables

Critical Angles for Yellow Light

Medium pair	Sine of critical angle	Critical angle
Glass → air	2/3	Approximately 42°
Water → air	3/4 = 0.75	48°35′
Glass → water	8/9 = 0.8889	62°44′