Light and Geometrical Optics

🟢 Lite — Quick Review (1h–1d)

Rapid summary for last-minute revision before your exam.

Light and Geometrical Optics — Key Facts for JAMB

Reflection Laws: Angle of incidence $i$ = angle of reflection $r$, both measured from the normal. The incident ray, reflected ray, and normal all lie in the same plane. For a plane mirror: image is laterally inverted, same distance behind mirror as object is in front ($d_i = d_o$), virtual and upright.

Spherical Mirrors: Mirror formula: $\frac{1}{f} = \frac{1}{u} + \frac{1}{v}$ where $f$ = focal length, $u$ = object distance (negative by convention), $v$ = image distance.

• Concave mirror: $f < 0$ (real focus), $R = 2f$
• Convex mirror: $f > 0$ (virtual focus), always produces a diminished, virtual, upright image
• Magnification: $m = -\frac{v}{u} = \frac{h_i}{h_o}$; $|m| > 1$ means magnified, $|m| < 1$ means diminished

Refraction — Snell’s Law: $n_1 \sin\theta_1 = n_2 \sin\theta_2$ where $n$ = refractive index. For light going from rarer to denser medium: bends towards normal ($\theta_2 < \theta_1$). Absolute refractive index $n = c/v$ where $c = 3 \times 10^8$ m/s.

Critical Angle and Total Internal Reflection (TIR): When light goes from denser to rarer: $\sin\theta_c = n_2/n_1$. If $\theta_i > \theta_c$, total internal reflection occurs.钻石 $n = 2.42$, so $\theta_c = \sin^{-1}(1/2.42) = 24.4°$. That’s why diamonds sparkle — low critical angle leads to many internal reflections.

⚡ Exam tip: TIR only occurs when light travels from a denser to a rarer medium AND the angle of incidence exceeds the critical angle. A common mistake is trying to apply TIR when going from rarer to denser.

---

🟡 Standard — Regular Study (2d–2mo)

Standard content for students with a few days to months.

Light and Geometrical Optics — JAMB UTME Study Guide

Lens Formula: $\frac{1}{f} = \frac{1}{u} + \frac{1}{v}$. For thin lenses:

• Convex (converging) lens: $f > 0$; can produce real inverted images (when object beyond $f$) or virtual upright magnified images (object within $f$)
• Concave (diverging) lens: $f < 0$; always produces a virtual, diminished, upright image regardless of object position

Power of Lens: $P = 1/f$ (in metres), unit = dioptre (D). For a convex lens of focal length 0.25 m, $P = +4$ D. Lens combinations: $P_{total} = P_1 + P_2 + …$ (in contact). For separated lenses: $\frac{1}{F} = \frac{1}{f_1} + \frac{1}{f_2} - \frac{d}{f_1 f_2}$ where $d$ = separation.

Refractive Index Details: Refractive index $n = \frac{c}{v} = \frac{\sin\theta_{air}}{\sin\theta_{medium}}$. For water $n = 1.33$; for glass $n ≈ 1.5$ (varies with type). Wavelength decreases in a medium: $\lambda_{medium} = \lambda_{air}/n$. Frequency $f$ stays constant when light enters a different medium.

Dispersion: White light splits into constituent colours because $n$ depends on wavelength (Cauchy’s equation: $n = A + B/\lambda^2$). Violet light bends more than red light. In a prism: red ($1.50n$) deviates least, violet ($1.54n$) deviates most. Angular dispersion = $(n_v - n_r) \times A$ where $A$ = prism angle.

Optical Phenomena:

• Apparent depth: $n = \frac{real\ depth}{apparent\ depth}$. A fish 1 m below water appears at $1/1.33 = 0.75$ m.
• Shift of object in a refracting medium: Shift $S = t(1 - 1/n)$ where $t$ = actual thickness.
• Atmospheric refraction: Stars appear slightly higher than actual position due to Earth’s atmosphere (lower $n$ at higher altitude). This causes twinkling.

---

🔴 Extended — Deep Study (3mo+)

Comprehensive coverage for students on a longer study timeline.

Light and Geometrical Optics — Comprehensive Physics Notes

Derivation of Mirror and Lens Formulae:

For a concave mirror with object at $O$ (distance $u$), focus at $F$ (distance $f$), image at $I$ (distance $v$):

From geometry of similar triangles: $\frac{h_i}{h_o} = \frac{v - f}{f} = \frac{f}{f - u}$. So $m = -\frac{h_i}{h_o} = -\frac{v - f}{f}$ (negative because image is inverted). Also $m = -\frac{v}{u}$. Setting equal: $\frac{v}{u} = \frac{v - f}{f}$. Cross-multiplying: $vf = uv - uf$. Dividing by $uvf$: $\frac{1}{u} = \frac{1}{v} + \frac{1}{f}$ (convention: $u$ is negative, $f$ is negative for concave mirror, $v$ is positive for real image). Rearranged: $\frac{1}{f} = \frac{1}{u} + \frac{1}{v}$.

Lens-maker’s Formula: $\frac{1}{f} = (n - 1)\left(\frac{1}{R_1} - \frac{1}{R_2}\right)$ where $R_1, R_2$ are radii of curvature of the two surfaces (sign convention: light travels from left to right; centre of curvature on the incoming side gives negative $R$). For a thin lens in air with equal curvature: $f = R/(2(n-1))$.

Combination of Thin Lenses: Two thin lenses separated by distance $d$: effective focal length $F$ satisfies: $\frac{1}{F} = \frac{1}{f_1} + \frac{1}{f_2} - \frac{d}{f_1 f_2}$. If lenses are in contact ($d = 0$): $\frac{1}{F} = \frac{1}{f_1} + \frac{1}{f_2}$. Power: $P = P_1 + P_2$.

Total Internal Reflection — Full Derivation: When light goes from medium 1 ($n_1$) to medium 2 ($n_2 < n_1$): Snell’s law $n_1 \sin\theta_1 = n_2 \sin\theta_2$. As $\theta_1$ increases, $\theta_2$ increases faster. Maximum $\theta_2 = 90°$ when $\sin\theta_1 = n_2/n_1$. This $\theta_1 = \theta_c$ is the critical angle. If $\theta_1 > \theta_c$, no refraction is possible — all light is reflected.

Applications of TIR:

• Optical fibres: Core ($n ≈ 1.50$) surrounded by cladding ($n ≈ 1.48$). $\theta_c ≈ 79°$. Light enters within the acceptance cone: $\sin\theta_a = \sqrt{n_1^2 - n_2^2}$.
• Prism binoculars: 45°-90°-45° prisms used to invert images twice, giving an upright final image.
• Diamond sparkle: Low critical angle (24.4°) means many facets cause TIR, creating brilliant sparkle.
• Rainbow: Internal reflection in water droplets — red (longer $\lambda$) at top, violet at bottom.

Aberrations (for spherical mirrors and thin lenses):

• Spherical aberration: Parallel rays near the edge focus at a different point than paraxial rays. For mirrors: paraboloid eliminates this. For lenses: combining positive and negative lenses reduces it.
• Chromatic aberration: Different wavelengths focus at different points because $n$ depends on $\lambda$. For lenses: chromatic aberration $\propto P/(V-1)$ where $V$ = Abbe number (dispersion). Corrected by achromatic doublet (convex + concave lens of different glasses).

Double Refraction (Birefringence): In calcite ($CaCO_3$) and quartz, unpolarised light splits into two rays: ordinary ( obeys Snell’s law, $n = 1.658$) and extraordinary (doesn’t obey Snell’s law, $n$ varies 1.658–1.486). The optic axis is the direction along which both rays travel at the same speed.

Human Eye Defects:

• Myopia (nearsighted): Eyeball too long or cornea too curved. Corrected by concave lens. Far point is closer than infinity. If far point = $D$, corrective lens $P = -1/D$ metres.
• Hypermetropia (farsighted): Eyeball too short or lens too flat. Near point > 25 cm. Corrected by convex lens.
• Presbyopia: Loss of accommodation with age. Corrected by bifocal lens.
• Astigmatism: Cornea not spherical. Corrected by cylindrical lens.

Angular Magnification: For a simple magnifier (convex lens): $M = D/f + 1$ where $D$ = least distance of distinct vision = 25 cm (normal eye). For relaxed eye: $M = D/f$. Microscope: $M = (L/f_o)(D/f_e)$ where $L$ = tube length, $f_o$ = objective focal length, $f_e$ = eyepiece focal length. Telescope: $M = -f_o/f_e$ (refracting) or $M = -f_o/f_e$ (reflecting).

JAMB Pattern Analysis: JAMB questions frequently ask: (1) Image characteristics from mirror/lens formula, (2) Critical angle and TIR conditions, (3) Refractive index from apparent depth, (4) Power of lens combinations. Classic JAMB question: “An object is placed 20 cm from a concave mirror of focal length 10 cm. Find the image distance.” Answer: $1/f = 1/u + 1/v$; $1/10 = 1/(-20) + 1/v$; $0.1 = -0.05 + 1/v$; $1/v = 0.15$; $v = 6.67$ cm (real, inverted).

---

---

📊 JAMB Exam Essentials

Detail	Value
Questions	180 MCQs (UTME)
Subjects	4 subjects (language + 3 for course)
Time	2 hours
Marking	+1 per correct answer
Score	400 max (used for university admission)
Registration	January – February each year

🎯 High-Yield Topics for JAMB

• Use of English (Grammar + Comprehension) — 60 marks
• Biology for Science students — 40 marks
• Chemistry (Organic + Physical) — 40 marks
• Physics (Mechanics + Optics) — 35 marks
• Mathematics (Algebra + Geometry) — 40 marks

📝 Previous Year Question Patterns

• Q: “The process of photosynthesis requires…” [2024 Biology]
• Q: “The electronic configuration of Fe is…” [2024 Chemistry]
• Q: “Find the value of x if 2x + 5 = 15…” [2024 Mathematics]

💡 Pro Tips

• Use of English carries the most weight — master grammar rules and comprehension strategies
• JAMB syllabus is your Bible — questions come directly from it. Download and use it.
• Past questions are highly predictive — repeat patterns appear every year
• For Science students, Biology and Chemistry are high-scoring if you study NCERT-level content

🔗 Official Resources

• JAMB Official
• JAMB Syllabus

---

Content adapted based on your selected roadmap duration. Switch tiers using the pill selector above.