ICSE 10 Physics Refraction in Plane Surface Short Notes

⚡ Fast Revision: Refraction at Plane Surfaces  - Phenomenon & Laws

1. Refraction & Its Cause

• Definition: The change in the direction of the path of light when it passes obliquely from one transparent medium to another.
• The Root Cause: Refraction occurs because the **speed of light changes** when transitioning from one optical medium to another.
• Frequency Invariance: When light enters a new medium, its speed and wavelength change, but its frequency remains completely unchanged because it depends strictly on the source of light.

2. Deviation Governing Rules

• Rarer to Denser: When light travels from an optically rarer to a denser medium, its speed decreases and it bends **toward the normal** ($i > r$).
• Denser to Rarer: When light travels from an optically denser to a rarer medium, its speed increases and it bends **away from the normal** ($i < r$).
• No Deviation Conditions: No bending occurs if the ray strikes the interface normally ($\angle i = 0^\circ \implies \angle r = 0^\circ$), or if both media share the exact same refractive index.

3. Laws of Refraction

• First Law: The incident ray, the refracted ray, and the normal at the point of incidence all lie in the same plane.
• Snell's Law: For a given pair of media and light of a specific color, the ratio of the sine of the angle of incidence ($\sin i$) to the sine of the angle of refraction ($\sin r$) is a constant.
• Refractive Index ($\mu$): This constant ratio is equal to the refractive index of the second medium with respect to the first medium ($_1\mu_2$).

Snell's Law Formula:

$_1\mu_2 = \frac{\sin i}{\sin r}$

Angle of Deviation: $\delta = |i - r|$

Unit Alert

Refractive Index ($\mu$): Being a ratio of identical trigonometric values (or speeds), it is a **pure number**.

SI Unit: It has no unit.

❌ Common Error:

Measuring the angle of incidence ($i$) or refraction ($r$) from the glass surface instead of the normal.
Fix: Always measure angles **between the light ray and the perpendicular normal line**. If the ray makes $30^\circ$ with the surface, $i = 90^\circ - 30^\circ = 60^\circ$.

Incident Ray ➘ │ Normal (N)
\ │
\ i │ (Medium 1: Rarer)
════════════════\════┿════════════════ Interface
\ r │ (Medium 2: Denser)
\ │
▼ │ Refracted Ray (Bends toward Normal)

Important Exam Diagram: Refraction from Rarer to Denser Medium

⚡ Fast Revision: Refraction at Plane Surfaces - Refractive Index Relations

1. Types of Refractive Index

• Absolute Refractive Index ($\mu$): The ratio of the speed of light in vacuum or air ($c$) to the speed of light in that specific medium ($v$). It is always greater than or equal to 1 ($\mu \ge 1$).
• Relative Refractive Index ($_1\mu_2$): The ratio of the speed of light in medium 1 to the speed of light in medium 2 when light transitions between two distinct materials.
• Wavelength Dependence: Since frequency is constant, the refractive index can also be expressed as the ratio of the wavelength in air ($\lambda_0$) to the wavelength in the medium ($\lambda_m$).

Core Refractive Index Formulas:

$\mu = \frac{c}{v} \quad \Big| \quad \lambda_m = \frac{\lambda_0}{\mu}$

$_1\mu_2 = \frac{\mu_2}{\mu_1} = \frac{v_1}{v_2}$

2. Factors Influencing $\mu$

• Nature of Medium: Optically denser media hold smaller speeds of light, resulting in a higher refractive index (e.g., $\mu_{\text{glass}} \approx 1.5$, $\mu_{\text{diamond}} = 2.42$).
• Color/Wavelength of Light: Refractive index decreases with an increase in wavelength ($\mu \propto \frac{1}{\lambda}$). Thus, **$\mu$ is maximum for violet light** and **minimum for red light**.
• Temperature of Medium: As temperature increases, the speed of light in that medium generally increases due to a drop in density. Therefore, **$\mu$ decreases with a rise in temperature**.

3. Principle of Reversibility

• Core Statement: If the path of a ray of light is reversed after undergoing any number of refractions or reflections, it retraces its entire path exactly backward.
• Reciprocal Relation: The relative refractive index of medium 2 with respect to 1 is the exact mathematical reciprocal of the index of medium 1 with respect to 2.
• Multiple Interfaces: For three consecutive media (e.g., air, water, glass), the cyclic product satisfies: $_a\mu_w \times _w\mu_g \times _g\mu_a = 1$.

Reversibility & Multiple Media Link:

$_1\mu_2 = \frac{1}{_2\mu_1}$

$_w\mu_g = \frac{_a\mu_g}{_a\mu_w}$

❌ Common Error:

Writing $_w\mu_g$ as $\frac{\mu_w}{\mu_g}$ during numerical updates.
Fix: The symbol $_1\mu_2$ always places the absolute index of the second (destination) medium in the numerator. Therefore, **$_w\mu_g = \frac{\mu_g}{\mu_w}$**.

Forward Path: Ray enters ──🡪 [Medium 1 (Air)] ──🡪 [Medium 2 (Glass)]
                              $_a\mu_g = \frac{\sin i}{\sin r}$

Reversed Path: Ray reflects 🡨── [Medium 1 (Air)] 🡨── [Normal Mirror ┸]
                              $_g\mu_a = \frac{\sin r}{\sin i}$

🎯 Verification: $_a\mu_g \times _g\mu_a = 1$

Important Exam Diagram: Principle of Reversibility Setup

⚡ Fast Revision: Refraction at Plane Surfaces  - Parallel Glass Slab & Lateral Displacement

1. Emergence Characteristics

• Parallel Paths: When a ray of light passes through a parallel-sided glass slab, the final emergent ray is always **parallel to the original incident ray**.
• Angle Equality: Because the two refracting faces are parallel, the angle of incidence ($i$) at the first face equals the angle of emergence ($e$) at the second face ($\angle i = \angle e$).
• Net Deviation: The total angular deviation ($\delta$) suffered by the light ray across the entire parallel glass slab is strictly **zero** ($\delta = 0^\circ$).

2. Lateral Displacement ($X$)

• Definition: The perpendicular distance between the path of the emergent ray and the original un-deviated path of the incident ray produced forward.
• Direct Factors: Lateral displacement increases directly with an **increase in slab thickness ($t$)**, an **increase in the angle of incidence ($i$)**, and an **increase in the refractive index ($\mu$)** of the glass.
• Wavelength Link: It is inversely proportional to the wavelength of light ($\propto \frac{1}{\lambda}$). Hence, **lateral displacement is maximum for violet light** and minimum for red light.

The Lateral Displacement Formula:

$X = \frac{t \cdot \sin(i - r)}{\cos r}$

(Where $t$ is slab thickness, $i$ is angle of incidence, and $r$ is angle of refraction)

Variable Altered	Change Imposed	Effect on Lateral Displacement ($X$)
Thickness of Slab ($t$)	Increases 🡩	Increases 🡩
Angle of Incidence ($i$)	Increases 🡩	Increases 🡩
Wavelength of Light ($\lambda$)	Increases (Red Light) 🡩	Decreases 🡫

❌ Common Error:

Stating that light passing through a glass slab gets permanently shifted in direction.
Fix: The direction does not change (the emergent ray stays perfectly parallel to the original trajectory). It only undergoes a parallel sideways translation called **lateral shift**.

Incident Ray ➘           │ Normal 1
────────────\───────────┿──────────── Top Face of Slab
             \  r      │
              \         │ [Thickness t]
               \        │
────────────────\───────┿──────────── Bottom Face of Slab
                 \  e   │ Normal 2
                  ▼ Emergent Ray ──▶ [ X ] 🡨── Original Path produced forward
                                 (Lateral Shift)

Important Exam Diagram: Ray Path and Lateral Shift in Glass Slab

⚡ Fast Revision: Refraction at Plane Surfaces - Real and Apparent Depth

1. Real vs Apparent Depth Mechanics

• The Visual Shift: When an object placed in an optically denser medium is viewed obliquely from a rarer medium, it appears to be raised up toward the surface.
• Refractive Index Linking: The ratio of the true structural depth (**Real Depth**) to the optically observed depth (**Apparent Depth**) equals the refractive index of the denser medium.
• Apparent Shift ($S$): The net vertical distance by which the object appears to be brought closer to the boundary interface.

Core Depth & Shift Formulas:

$\mu = \frac{\text{Real Depth }(R)}{\text{Apparent Depth }(A)}$

$\text{Apparent Shift }(S) = \text{Real Depth} \times \left(1 - \frac{1}{\mu}\right)$

2. Dependencies of Shift ($S$)

• Real Depth: Apparent shift is directly proportional to the true depth of the medium ($S \propto R$). Deeper tanks show larger shifts.
• Refractive Index ($\mu$): Shift increases with a higher refractive index value of the liquid. A tank filled with glass shifts light more than water.
• Wavelength Dependency: Shift decreases with an increase in light wavelength ($\propto \frac{1}{\lambda}$). Hence, **shift is maximum for violet light** and minimum for red light.

❌ Common Error:

Using the shift value ($S$) directly as the denominator in the refractive index equation.
Fix: $\mu$ is equal to Real Depth divided by **Apparent Depth ($A$)**, NOT Shift. Remember that $\text{Apparent Depth} = \text{Real Depth} - \text{Shift}$.

Medium Type	Refractive Index ($\mu$)	Apparent Depth ($A$)	Apparent Shift ($S$)
Water Tank	$4/3 \approx 1.33$	$9\text{ cm}$	$3\text{ cm}$
Glass Block	$3/2 = 1.50$	$8\text{ cm}$	$4\text{ cm}$

👁️ Observer in Air (Rarer)
  \
───\───────────────────────── Boundary Interface
    \ 🡩 Apparent Depth (A)
     ▲ Apparent Image (I) 🡩 🌟 Shift (S)
      \
       ● Real Object (O)
    ◀─────── Real Depth (R) ───────▶

Important Exam Diagram: Real vs Apparent Depth Ray Mechanics

⚡ Fast Revision: Refraction at Plane Surfaces - Critical Angle & Total Internal Reflection

1. Critical Angle Conditions

• Definition: The angle of incidence in the optically denser medium for which the corresponding angle of refraction in the rarer medium is exactly **$90^\circ$**.
• The Grazing Ray: At this specific angle, the refracted light ray does not escape; it grazes along the boundary surface separating the two media.
• Index Relationship: The sine of the critical angle is inversely proportional to the refractive index of the denser medium ($\sin C = \frac{1}{\mu}$).

Critical Angle Formulas:

$\sin C = \frac{1}{_r\mu_d} = \frac{\mu_{\text{rarer}}}{\mu_{\text{denser}}}$

For Glass ($\mu = 1.5$), $C \approx 42^\circ$ | For Water ($\mu = 1.33$), $C \approx 49^\circ$

2. Total Internal Reflection (TIR)

• Definition: When a ray of light traveling in a denser medium strikes the interface of a rarer medium at an angle of incidence greater than the critical angle ($i > C$), the ray is completely reflected back into the denser medium.
• Two Mandatory Conditions: 1. Light must travel from a **denser medium to a rarer medium**.
  2. The angle of incidence must be **greater than the critical angle** ($i > C$).
• Energy Efficiency: Unlike a silvered glass mirror which absorbs some light, TIR reflects **100% of the incident light energy**, producing exceptionally bright reflections.

❌ Common Error:

Assuming TIR can occur when light travels from air into glass or water.
Fix: TIR is strictly impossible when moving from a rarer to a denser medium. The light ray **must** be inside the denser medium trying to cross into a rarer environment.

Condition	Behavior of Light Ray	Resulting Angles
$i < C$	Normal refraction occurs into the rarer medium.	$r < 90^\circ$ (Bends away from normal)
$i = C$	Critical State (Grazing Emergence).	$r = 90^\circ$
$i > C$	Total Internal Reflection (TIR).	Follows laws of reflection ($i = r'$)

                    [ RARER MEDIUM (AIR) ]
════════════════════┿══════════════════┿══════════════════ Interface
                   │ 🡪 r=90°             \
                  ▲│                      \ Reflected back
                  /│                       ▼
                 / │ i = C                / │ i > C
        Incident ➘ │             Incident ➘ │
                    [ DENSER MEDIUM (GLASS) ]

Important Exam Diagram: Transition from Critical Angle to TIR

⚡ Fast Revision: Refraction at Plane Surfaces - Total Internal Reflection in Prisms

1. Total Internal Reflection in Right-Angled Prisms

• Deviating Light by 90°: When a ray strikes one perpendicular face normally, it hits the hypotenuse face at $45^\circ$. Since $45^\circ > 42^\circ$ (critical angle of glass), it suffers TIR and emerges through the other perpendicular face (used in Periscopes).
• Deviating Light by 180°: When light enters normally through the hypotenuse face, it strikes both perpendicular internal faces at $45^\circ$, undergoing TIR twice to emerge parallel but inverted (used in Binoculars).
• Erecting an Image (0° Deviation): Light travels parallel to the hypotenuse base, undergoes TIR twice inside, and emerges without angular deviation but with the top-to-bottom orientation inverted (Erecting Prism).

2. Other High-Frequency Prism Geometries

• Equilateral Prism TIR: A ray entering normally through one face strikes the adjacent face internally at $60^\circ$. Because $60^\circ > 42^\circ$, it undergoes TIR rather than refracting out.
• 30°-60°-90° Prism: Light entering normally through the face opposite to $60^\circ$ strikes the hypotenuse at $60^\circ$, causing TIR and routing the beam uniquely for specific optical layouts.
• Critical Condition: For TIR to happen in any glass prism, the ray inside must always hit the boundary wall at an angle strictly **greater than 42°**.

Prism Action	Angle of Incidence at Face	Total Deviation Angle	Practical Instrument
Periscope Type	$45^\circ$ at Hypotenuse	$90^\circ$	Submarine Periscope
Binocular Type	$45^\circ$ at both legs	$180^\circ$	Prism Binoculars
Erecting Type	Refracts first, then $45^\circ$ TIR	$0^\circ$ (Parallel)	Slide Projectors

❌ Common Error:

Drawing a light ray bending at the first surface even when it enters the prism completely at a right angle ($90^\circ$).
Fix: Any ray that is **normal to the surface ($\angle i = 0^\circ$)** passes straight through without any deviation. Do not bend it until it hits the next internal wall.

                  / \
                 / \
                / 45° \ 🡪 (Internal TIR because 45° > 42°)
Incident Ray ──🡪/ \
(Normal Entry) /__________\
                    │
                    ▼ Emergent Ray (Deviated by 90°)

Important Exam Diagram: 90° Deviation in Total Reflecting Prism