Light, Sound, and Wave Phenomena

🟢 Lite — Quick Review (1h–1d)

Rapid summary for last-minute revision.

Light — Reflection and Refraction

• Law of reflection: Angle of incidence (i) = Angle of reflection (r). Both measured from the normal (perpendicular to the surface).
• Refraction: Light bends when passing from one medium to another. This is why a pencil appears bent in water.
• Refractive index (n): n = c/v = sin i/sin r, where c = speed of light in vacuum (3 × 10⁸ m/s), v = speed in the medium.

Mirrors:

Mirror Type	Image Characteristics
Plane mirror	Virtual, upright, same size, laterally inverted
Concave mirror	Can be real (inverted) or virtual (magnified) depending on object position
Convex mirror	Always virtual, upright, diminished (smaller) image

Sign convention for mirrors (New Cartesian Sign Convention):

• Object is always on the left of the mirror.
• All distances measured from the pole (mirror’s centre point).
• Distances measured in the direction of light travel are positive; opposite is negative.
• Height above the principal axis is positive (upward).

Mirror formula: 1/f = 1/v + 1/u Where f = focal length, v = image distance, u = object distance. For concave mirror, f is negative. For convex mirror, f is positive.

Example (mirror): Object at 10 cm in front of a concave mirror with focal length 15 cm. 1/v = 1/f − 1/u = 1/(−15) − 1/10 = (−2/30) − (3/30) = −5/30 → v = −6 cm. Image is 6 cm in front of mirror (real, inverted).

Lenses:

Lens Type	Properties
Convex (converging)	Thick in middle, converges light, real images possible
Concave (diverging)	Thin in middle, diverges light, always virtual image

Lens formula: 1/f = 1/v − 1/u (sign convention: object distance u is always negative for lenses in standard orientation).

Power of lens: P = 1/f (in metres). Unit = Dioptre (D). Convex lens: positive power. Concave lens: negative power.

Example (lens): Convex lens with focal length 20 cm (= 0.2 m). Power P = 1/0.2 = +5 D.

⚡ Exam Tip: UPTET often asks: “An object is placed at 10 cm from a concave mirror of focal length 15 cm. Find the image distance.” Use 1/v = 1/f − 1/u. Substituting f = −15 cm, u = −10 cm: 1/v = −1/15 − (−1/10) = −1/15 + 1/10 = (−2 + 3)/30 = 1/30 → v = +30 cm. Wait — virtual image? Actually for concave mirror, if object is between F and pole (u > f in magnitude), image is virtual. Here u = 10 cm, f = 15 cm, object is beyond focal point → real inverted image. Recheck: v = 30 cm on the same side as object. Actually the standard formula gives v = 30 cm, real inverted.

⚡ Common Mistake: Confusing the focal length sign — concave mirror/spherical lens: f is negative. Convex mirror/spherical lens: f is positive.

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🟡 Standard — Regular Study (2d–2mo)

For students who want genuine understanding.

Human Eye — Structure and Defects

Structure of the Human Eye:

• Cornea: Transparent front layer, provides 2/3 of eye’s focusing power.
• Iris: Coloured part, controls the size of the pupil.
• Pupil: Opening in the iris, controls amount of light entering.
• Crystalline lens: Flexible lens, changes shape for accommodation (focusing at different distances).
• Retina: Light-sensitive layer at the back; contains rods (brightness) and cones (colour).
• Ciliary muscles: Change the shape of the lens for accommodation.

How the eye works: Light enters through cornea → lens focuses it on retina → retina sends signals to brain via optic nerve → brain interprets as image.

Power of accommodation: The ability of the eye to change the focal length of the lens. Near point = 25 cm (normal eye). Far point = infinity.

Common Eye Defects:

Defect	Cause	Correction
Myopia (Nearsightedness)	Eyeball too long or lens too convex	Concave (diverging) lens
Hypermetropia (Farsightedness)	Eyeball too short or lens too flat	Convex (converging) lens
Presbyopia	Age-related loss of accommodation	Bifocal or progressive lenses
Cataract	Clouding of the lens	Surgery (artificial lens)
Astigmatism	Irregular cornea curvature	Cylindrical lens

Myopia correction example: A myopic eye has far point of 2 m. To see distant objects clearly, a concave lens is used such that parallel rays appear to come from the far point. Power P = −1/f. For far point 2 m → f = −2 m → P = −0.5 D.

Astigmatism: Cylindrical lenses correct this by providing different focal lengths in different meridians.

Sound — Propagation and Characteristics

• Sound is a longitudinal wave (particles of the medium vibrate parallel to the direction of wave propagation).
• Cannot travel through vacuum — requires a medium (solid, liquid, or gas).
• Speed of sound in air ≈ 340 m/s at 22°C. Increases with temperature (~0.6 m/s per °C rise) and is faster in liquids and solids.

Speed of sound in different media:

• Air: ~340 m/s
• Water: ~1500 m/s
• Steel: ~5000 m/s

Wave characteristics:

• Amplitude (A): Maximum displacement from rest position. Determines loudness (energy).
• Frequency (f): Number of vibrations per second. Measured in Hertz (Hz). Determines pitch.
• Wavelength (λ): Distance between two consecutive crests (or troughs). In metres.
• Speed (v): v = fλ (fundamental wave equation).

Example: A sound wave with frequency 500 Hz and wavelength 0.68 m. Speed v = 500 × 0.68 = 340 m/s.

Echo: Sound reflected from a surface and received by the listener. Minimum distance for echo in air: d = v × t/2. At 22°C: for distinct echo, minimum distance is ~17.2 m (since v = 340 m/s, time between original and echo must be ≥ 0.1 s → distance ≥ 17 m).

Resonance: When the frequency of an external force matches the natural frequency of a body, the body vibrates with maximum amplitude. Examples:

• Singer breaking a glass — frequency matches glass’s natural frequency.
• Bridge swaying — soldiers marching in step avoided by design.
• Tuning forks.

Infrasound and Ultrasound:

• Infrasound: f < 20 Hz. Produced by elephants, whales, earthquakes.
• Ultrasound: f > 20,000 Hz. Used in medical imaging (echocardiography, foetal scans), industrial flaw detection, sonar.

⚡ UPTET Common Mistakes:

1. Confusing frequency with amplitude — frequency determines pitch, amplitude determines loudness. A high-pitched but soft sound has high frequency, low amplitude.
2. Confusing concave and convex lenses — concave diverges light (corrects myopia), convex converges light (corrects hypermetropia).
3. Mixing up mirror and lens formulas — 1/v + 1/u for mirrors vs 1/v − 1/u for lenses. Check the sign convention carefully.
4. Thinking sound can travel in vacuum — it cannot, unlike light.

Total Internal Reflection (TIR)

When light travels from a denser to a rarer medium, at angles beyond the critical angle, the light reflects entirely back into the denser medium. This is the working principle of:

• Optical fibres: Used in telecommunications to transmit data over long distances with minimal loss.
• DiamondSparkle: The brilliance of a cut diamond is due to TIR — light enters, undergoes multiple internal reflections, and exits as brilliant flashes.
• Mirage: Refraction of light through layers of hot air near the ground causes TIR, creating the illusion of water.

Critical angle for water (n ≈ 1.33): sin C = 1/n → C ≈ 49°. For glass (n ≈ 1.5): C ≈ 41°.

Practical Applications of Light:

• Periscope: Uses two plane mirrors at 45° to see over obstacles (submarines, trenches).
• Kaleidoscope: Multiple reflections create symmetrical patterns.
• Endoscope: Medical instrument using optical fibres to view inside the body.

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🔴 Extended — Deep Study (3mo+)

Comprehensive theory for students with extended preparation time.

Mirror Formula Derivation and Magnification

For a spherical mirror, the mirror formula is derived using geometry and sign conventions:

1/f = 1/v + 1/u

Linear magnification (m): Ratio of image height to object height. m = h′/h = −v/u

|m| > 1: Image larger than object. |m| < 1: Image smaller than object.

Sign of m:

• m positive → image is upright (virtual).
• m negative → image is inverted (real).

Example: Object height 2 cm, image height −4 cm. m = −4/2 = −2. Image is inverted, 2× magnification.

Lens Formula Derivation:

1/f = 1/v − 1/u

Magnification: m = h′/h = v/u (sign differs from mirrors convention).

Power of Lens and Combination:

For lenses in contact: P_total = P₁ + P₂ + P₃ + …

Example: Two lenses of power +3 D and +2 D placed in contact. Total power = +5 D. Focal length f = 1/5 = 0.2 m = 20 cm.

Refraction Through a Glass Slab

A rectangular glass slab shifts the emergent ray parallel to the incident ray. The lateral shift d depends on:

• Thickness of slab (t)
• Angle of incidence (i)
• Refractive index of the slab (n)

d = t × sin(i − r) / cos(r)

Important: White light splits into seven colours through a glass prism — dispersion. Violet deviates the most (highest n), red deviates the least (lowest n). This is because the refractive index is slightly different for different wavelengths — n(λ_blue) > n(λ_red).

Atmospheric Refraction:

• Stars appear slightly higher than their actual position because light from stars bends through Earth’s atmosphere (denser lower layer → rarer upper layer → refraction towards the normal).
• Twinkling of stars: Fluctuations in atmospheric density cause varying refractive effects.
• Advance sunrise and delayed sunset: Because of atmospheric refraction, the Sun is visible even when it is slightly below the horizon.

Sound Waves — Deep Dive

Speed of sound formula (for gases): v = √(γP/ρ), where γ = adiabatic index, P = pressure, ρ = density. For air at 22°C, v ≈ 340 m/s.

Temperature dependence: v ∝ √T (absolute temperature). At 0°C, v ≈ 332 m/s. At 30°C, v ≈ 349 m/s.

Sound intensity level (β): Measured in decibels (dB). β = 10 log(I/I₀), where I₀ = 10⁻¹² W/m² (threshold of hearing).

Examples: Whisper ~30 dB, normal conversation ~60 dB, road traffic ~80 dB, jet engine ~140 dB. Sounds above 85 dB cause hearing damage with prolonged exposure.

Echo-ranging (Sonar):

• SONAR = Sound Navigation and Ranging.
• Used to measure ocean depth and detect underwater objects.
• A sound pulse is sent; the time for echo to return is measured. Depth = v × t/2 (v in water ≈ 1500 m/s).
• Bats and dolphins use echolocation for navigation and hunting.

Ultrasound in Medicine:

• Echocardiography: Ultrasound imaging of the heart.
• Pregnancy scans: Monitoring foetal development without radiation.
• Lithotripsy: Sound waves break kidney stones non-invasively.
• Cleaning: Ultrasound cleaning of delicate instruments.
• Frequency used: 1–10 MHz (above human hearing range).

Doppler Effect:

When a source of sound and an observer are moving relative to each other, the observed frequency differs from the actual frequency.

f′ = f × (v ± v₀)/(v ∓ vₛ)

Where v = speed of sound in air (~340 m/s), v₀ = observer’s speed, vₛ = source’s speed.

Example: An ambulance moving at 20 m/s (vₛ = 20) with siren at 500 Hz approaches a stationary observer (v₀ = 0). As it approaches: f′ = 500 × (340)/(340 − 20) = 500 × 340/320 ≈ 531 Hz (higher pitch). After passing, it recedes: f′ = 500 × (340)/(340 + 20) = 500 × 340/360 ≈ 472 Hz (lower pitch). This is why a siren’s pitch seems to suddenly drop as the vehicle passes you.

Applications: Police radar (speed detection), weather forecasting (tracking storms), blood flow measurement in medicine.

⚡ Previous Year UPTET Focus: Questions on the human eye (defects and corrections), speed of sound, and mirror/lens formulas are very common. A typical question: “A person cannot see distant objects clearly but can read a book held close. Which defect? Answer: Myopia. Corrected with concave lens.” Another: “Calculate the speed of sound in air if frequency is 512 Hz and wavelength is 0.66 m. Answer: 338 m/s.”

Numerical Examples for Practice:

1. Mirror: Object 30 cm from concave mirror, focal length 15 cm. Find image distance. Using 1/v = 1/f − 1/u: 1/v = 1/(−15) − 1/(−30) = −1/15 + 1/30 = (−2 + 1)/30 = −1/30 → v = −30 cm. Image is real, inverted, at 30 cm.

2. Lens: Convex lens focal length 10 cm. Object placed 25 cm away. Find image position. 1/v = 1/f − 1/u = 1/10 − (−1/25) = 1/10 + 1/25 = (5 + 2)/50 = 7/50 → v = 7.14 cm. Real, inverted.

3. Sound echo: A person claps near a cliff and hears echo after 2 seconds. How far is the cliff? (Speed of sound = 340 m/s). Distance = 340 × 2 / 2 = 340 m.

4. Doppler: Train whistle at 400 Hz. Train moves at 30 m/s toward stationary observer. Speed of sound = 340 m/s. f′ = 400 × (340)/(340 − 30) = 400 × 340/310 ≈ 439 Hz.

⚡ Common Errors to Flag:

1. In the lens formula, u is always negative — don’t forget the negative sign when substituting.
2. Mixing up magnification m = −v/u (mirror) with m = v/u (lens) — the sign conventions differ.
3. Forgetting that real images are formed on the same side as the object for mirrors, and on the opposite side for lenses.
4. Confusing the speed of light (3 × 10⁸ m/s) with speed of sound (~340 m/s) — vastly different orders of magnitude.
5. In ray diagrams, drawing the focal point incorrectly — for concave mirrors, focus is in front; for convex mirrors, focus is behind the mirror.