“Light: Reflection and Refraction”

🟢 Lite — Quick Review (1h–1d)

Rapid summary for last-minute revision before your WAEC exam.

Laws of Reflection:

1. The incident ray, reflected ray, and normal all lie in the same plane
2. Angle of incidence ($i$) = Angle of reflection ($r$)

Plane Mirror Characteristics:

• Image is the same distance behind the mirror as the object is in front
• Image is laterally inverted (left-right reversed)
• Image is virtual (cannot be projected on a screen)
• Image size = Object size (magnification = 1)

Refraction - Snell’s Law: $$n_1 \sin i = n_2 \sin r$$

Where $n$ is the refractive index (ratio of speed of light in vacuum to speed in medium).

Critical Angle and Total Internal Reflection:

• Critical angle: $\sin c = \frac{n_2}{n_1}$ (when $n_1 > n_2$)
• Total internal reflection occurs when $i > c$ (angle of incidence in denser medium exceeds critical angle)

⚡ WAEC Tip: For total internal reflection, light MUST be travelling from a denser to a rarer medium. If going from glass (n=1.5) to air (n=1.0), critical angle $\sin c = 1/1.5 = 0.667$, so $c ≈ 42°$.

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🟡 Standard — Regular Study (2d–2mo)

For students who want genuine understanding.

Real Depth and Apparent Depth:

When light travels from a denser medium to a rarer medium: $$\text{Apparent depth} = \frac{\text{Real depth}}{n}$$

The refractive index $n = \frac{\text{Real depth}}{\text{Apparent depth}}$

Worked Example: A coin appears to be 6cm below the surface of water when viewed from above. If refractive index of water is 1.33, find the actual depth.

Solution: $n = \frac{\text{Real depth}}{\text{Apparent depth}} = \frac{\text{Real depth}}{6}$ $1.33 = \frac{\text{Real depth}}{6}$ Real depth = $1.33 \times 6 = 7.98,\text{cm} ≈ 8,\text{cm}$

Refraction Through a Glass Slab:

• Light enters parallel to normal: no deviation (but slows down)
• Light enters at angle: ray bends TOWARDS normal in denser medium
• Emergent ray is parallel to incident ray (but displaced)
• Lateral shift $x = t \frac{\sin(i-r)}{\cos r}$

Where $t$ = thickness of slab, $i$ = angle of incidence, $r$ = angle of refraction.

Spherical Mirrors:

Mirror Type	Focal Point	Image Characteristics
Concave	In front of mirror (real focus)	Can be real or virtual
Convex	Behind mirror (virtual focus)	Always virtual, smaller

Mirror Formula (Sign Convention - New Cartesian): $$\frac{1}{f} = \frac{1}{u} + \frac{1}{v}$$

Where:

• $f$ = focal length (positive for concave, negative for convex)
• $u$ = object distance (negative if object is in front of mirror)
• $v$ = image distance

Linear Magnification: $$m = \frac{\text{Image height}}{\text{Object height}} = \frac{v}{u}$$

⚡ Common Mistake: Using the wrong sign convention. In the “real-is-positive” convention (used in many textbooks), distances measured against the direction of incident light are positive. In the “new Cartesian” convention, distances are positive in the direction of incident light. Stick to ONE convention throughout.

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🔴 Extended — Deep Study (3mo+)

Comprehensive theory for serious exam preparation.

Lens Formula and Optical Instruments:

Thin Lens Formula: $$\frac{1}{f} = \frac{1}{u} + \frac{1}{v}$$

For converging lens (convex): $f > 0$ For diverging lens (concave): $f < 0$

Lens Types:

Type	Shape	Function	f
Biconvex	Bulging outward	Converges light	+
Plano-convex	One flat, one bulging	Converges light	+
Biconcave	Thinner at centre	Diverges light	-
Plano-concave	One flat, one hollow	Diverges light	-

Power of Lens: $$P = \frac{1}{f} \text{ (in metres)}$$

Unit: Dioptre (D) For a combination of lenses in contact: $$P_{\text{total}} = P_1 + P_2 + P_3$$

The Eye - Accommodation:

• Far point = infinity (normal eye)
• Near point = 25 cm (normal reading distance)
• Ciliary muscles change lens shape for accommodation
• Defects: Myopia (near-sighted, concave lens), Hypermetropia (far-sighted, convex lens), Presbyopia (age-related, bifocal lens)

Magnifying Glass: Angular magnification $M = \frac{\text{Angular size of image}}{\text{Angular size of object at near point}} = 1 + \frac{D}{f}$

Where $D$ = 25 cm (least distance of distinct vision)

Compound Microscope: $$M = \frac{L}{f_o} \times \frac{D}{f_e}$$

Where $L$ = tube length, $f_o$ = objective focal length, $f_e$ = eyepiece focal length.

Astronomical Telescope (Keplerian): $$M = \frac{f_o}{f_e} \text{ (for relaxed eye)}$$

For normal adjustment: $M = \frac{f_o}{f_e} + 1$$

Refractive Index - Wavelength Dependence:

Dispersion: Different wavelengths refract by different amounts. $$n_{\text{violet}} > n_{\text{red}}$$

This is why a prism separates white light into a spectrum. Rainbow forms through dispersion + total internal reflection in water droplets.

Critical Angle for Crown Glass: For crown glass ($n = 1.52$): $\sin c = 1/1.52 = 0.658$, so $c ≈ 41°$ For water ($n = 1.33$): $\sin c = 1/1.33 = 0.752$, so $c ≈ 49°$

⚡ WAEC Previous Year Pattern:

Year	Question	Concept
2023	Glass slab - lateral shift	Refraction through parallel surfaces
2022	Total internal reflection in prism	Critical angle
2021	Lens formula	Object/image distance

Optical Fibre Communication:

• Uses total internal reflection
• Core (n₁) surrounded by cladding (n₂) where n₁ > n₂
• Light travels through core without escaping
• Used in telecommunications and endoscopy

Derivation - Snell’s Law from Wave Theory: When light enters a different medium, its speed changes. Since frequency remains constant: $\lambda_1 f = \lambda_2 f$ Therefore: $\frac{\sin i}{\sin r} = \frac{\lambda_1}{\lambda_2} = \frac{v_1}{v_2} = \frac{n_2}{n_1}$ Rearranging: $n_1 \sin i = n_2 \sin r$

⚡ Exam Strategy: Draw ray diagrams carefully with a ruler. Label all angles clearly. For mirror problems, remember: object between F and mirror → virtual, upright, magnified. Object beyond C (concave) → real, inverted, diminished.

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