Electromagnetic Waves
Maxwell's displacement current, accelerating charges, transverse E & B wave speed, energy & momentum, and complete EM spectrum uses.
Topic 8.2: Ampere-Maxwell Consistency & Displacement Current
Charge a capacitor from a battery: current zooms down the copper wire, hits the empty air gap between the plates, and zooms out the far wire! Draw a flat loop around the wire and Ampere's Law finds magnetic field
B. Now bulge that same rim into a deep tiffin-box dipping BETWEEN the empty plates where zero electrons cross — suddenly Ampere's old law predicts B = 0 at the exact same rim! James Clerk Maxwell solved this puzzle: a changing electric field in the gap acts just like current!DISPLACEMENT CURRENT
id = ε₀ dΦEdt (created by changing electric flux, zero electron flow!)AMPERE-MAXWELL LAW
∮ B · dl = μ₀ (ic + id) = μ₀ [ ic + ε₀ dΦEdt ]CAPACITOR CONTINUITYWire:
ic = I, id = 0 • Air gap: ic = 0, id = I • Total ic + id is continuous everywhere!🧠 Memory Trick: Faraday proved changing B → E; Maxwell completed the mirror twin: changing E → B via Displacement Current
id = ε₀ dΦE/dt!💡 Simple Kid-Check: Inside a charging capacitor's empty gap, no electrons jump across the air — yet the electric field
E = Q / (ε₀A) grows stronger every millisecond. That growing electric field creates circular magnetic field loops outside the plates identical to real wire current!- 1The Tiffin-Box Inconsistency: Two surfaces sharing the same circular loop — a flat disc cutting the wire (
ienc = I) and a pot/tiffin-box passing inside the capacitor gap (ienc = 0) — gave contradicting values for∮ B · dlin old Ampere's Law! - 2Proving
id = icin a Capacitor: Electric flux between plates isΦE = E A = (Q / ε₀ A) A = Q / ε₀.
Differentiating:id = ε₀ dΦEdt = ε₀ (1ε₀ dQdt) = dQdt = ic! - 3Capacitor Voltage Shortcut: Since
Q = C V, displacement current inside a capacitor is simplyid = C (dVdt)!
🧒 Super-Easy Example: A
4 μF parallel-plate capacitor is charged so the potential difference across its plates rises at 5 × 10⁵ V/s. What is the displacement current between the plates? id = C (dV/dt) = (4 × 10⁻⁶) × (5 × 10⁵) = 2.0 A (exact match to wire conduction current!).
Watch three classic DHSE traps: (1) Inside the capacitor gap while charging,
id = ic numerically, yet ZERO conduction electrons cross the gap! (2) Once the capacitor is fully charged by steady DC (V constant), dΦE/dt = 0 ⇒ BOTH ic = 0 and id = 0! (3) Even though called "displacement current", its SI unit is still Ampere (A)!Topic 8.3.1: Accelerating Charges as Sources of EM Waves
Park an electric charge calmly on your desk — it creates only a static electric field. Slide it smoothly in a straight line at steady speed — it adds a steady magnetic field, still zero light! Only when you shake or accelerate that charge back and forth (like electrons bouncing inside a radio tower antenna!) do ripples of coupled
E and B break loose and fly away as EM waves!CHARGE STATES → FIELDSRest (
v=0): E only • Constant velocity (a=0): E + B • Accelerating (a≠0): Radiates EM Waves!OSCILLATING FREQUENCYCharge oscillating at frequency
ν emits EM wave of the EXACT same frequency ν!SELF-SUSTAINING LOOPChanging
E creates changing B → creates changing E → needs NO medium!🧠 Memory Trick: Stationary = E; Steady Cruise = E + B; Shake/Accelerate = EM Wave! And Hertz (1887) proved Maxwell right in the lab using a spark-gap oscillator!
💡 Simple Kid-Check: Dip your finger still in a pond → no water ripples. Move a toy boat steadily → water moves smoothly around it. Shake a float up and down → circular ripples keep racing outward even after they leave your hand! Accelerating charges shake the electromagnetic field the exact same way!
- 1Why Acceleration Is Essential: Maxwell predicted that a harmonically oscillating electric dipole (electrons surging up & down a metal rod at frequency
ν) generates time-varyingEandBfields that regenerate each other and detach into space! - 2Frequency Lock: If an electron oscillates at
νcycles per second, the emitted EM wave MUST carry frequencyνand wavelengthλ = c / ν(only speedcand wavelength change inside glass/water, NEVER frequency!). - 3Hertz Experiment (1887) & J.C. Bose: Heinrich Hertz produced ~5 m radio waves with a spark gap and caught tiny sparks on a distant detector loop; Jagadish Chandra Bose generated millimetre-scale microwaves in Kolkata!
🧒 NCERT Exercise 8.6 Check: A charged particle oscillates about its mean position at frequency
• Frequency =
ν = 10⁹ Hz (1 GHz). What is the frequency and wavelength of the emitted EM wave in vacuum?• Frequency =
10⁹ Hz (exact match!) • Wavelength λ = c / ν = (3 × 10⁸) / 10⁹ = 0.3 m (Microwave!).
Top 1-mark conceptual trap: Does a charge moving at UNIFORM VELOCITY (zero acceleration,
v = constant) radiate electromagnetic waves? NO! It creates both E and B fields carried along with it, but radiates ZERO EM waves! Only accelerated charges (including charges circling at constant speed, because centripetal acceleration a ≠ 0!) emit EM radiation!Topic 8.3.2: Transverse Nature of E, B & Propagation c = 1/√(μ₀ε₀)
Picture two friends shaking skipping ropes at right angles: one waves up-and-down (
E) while the other waves side-to-side (B) in perfect lockstep, and the ripple shoots forward along the third perpendicular axis (E × B)! When Maxwell combined electric ε₀ and magnetic μ₀ constants, out popped 3 × 10⁸ m/s — proving light itself is an EM wave!TRANSVERSE & DIRECTION
E ⊥ B ⊥ v • Travels along E × B • IN PHASE (peak together!)SPEED & E₀/B₀ RATIO
c = 1√(μ₀ε₀) = 3 × 10⁸ m/s • E₀B₀ = c • Medium: v = 1√(με) = cn!WAVE EQUATIONS
Ex = E₀ sin(kz − ωt), By = B₀ sin(kz − ωt) with k = 2πλ, ω = 2πν, c = ωk!🧠 Memory Trick:
E and B are 90° apart in SPACE (axes), yet 0° apart in TIME (strictly in phase)! Check propagation axis with E × B!💡 Simple Kid-Check: Remember
î × ĵ = k̂! If electric vector E wiggles up along +y (ĵ) and magnetic vector B wiggles sideways along +z (k̂), then ĵ × k̂ = +î — so the wave blasts forward along +x!- 1Vacuum Speed from Constants: Plugging
μ₀ = 4π × 10⁻⁷ T·m/Aandε₀ = 8.854 × 10⁻¹² C²/(N·m²)into Maxwell's wave speed givesc = 1√(μ₀ε₀) = 3.0 × 10⁸ m/s— matching the measured speed of light! Inside a medium,v = c / √(μr εr). - 2Same Speed for ALL Wavelengths in Vacuum: Gamma rays, X-rays, visible light, and radio waves ALL travel at the exact same speed
c = 3 × 10⁸ m/sin vacuum! - 3Reading Wave Equations: Given
By = 2×10⁻⁷ sin(0.5×10³ x + 1.5×10¹¹ t): notice the+sign means wave moves along−x! Alsok = 500 m⁻¹ ⇒ λ = 2π/500 = 1.26 cm,ω = 1.5×10¹¹ ⇒ ν = ω/(2π) = 23.9 GHz, andE₀ = c B₀ = 60 V/m!
🧒 NCERT Example 8.1: A plane EM wave travels in free space along
• Magnitude:
• Direction: Need
+x. At an instant, E = 6.3 ĵ V/m. Find B.• Magnitude:
B = E / c = 6.3 / (3 × 10⁸) = 2.1 × 10⁻⁸ T.• Direction: Need
E × B along +î. Since (+ĵ) × (+k̂) = +î, B = 2.1 × 10⁻⁸ k̂ T!
Watch three classic traps: (1) Many students guess
E and B are 90° out of phase in time! FALSE: E and B peak together and hit zero together — phase difference is 0° (they are perpendicular only in spatial direction!). (2) Direction is ALWAYS E × B (NOT B × E!). (3) Watch (kx + ωt) vs (kx − ωt): − travels along +x, + travels along −x!Topic 8.3.2: Energy Density, Momentum & Wave Intensity
Stand under warm Kerala sunshine and you instantly feel energy arriving from 150 million kilometres away — yet sunlight ALSO gives every surface a gentle physical push (Radiation Pressure), strong enough in deep space to blow a comet's dust tail straight away from the Sun and push solar sails!
50% – 50% ENERGY SPLITAvg electric
uE = ½ ε₀ Erms² EQUALS avg magnetic uB = Brms²2μ₀!TOTAL ENERGY DENSITY
uavg = ½ ε₀ E₀² = ε₀ Erms² = B₀²2μ₀ = Brms²μ₀ (J/m³)INTENSITY & MOMENTUM
I = uavg · c = ½ ε₀ c E₀² (W/m²) • Momentum: p = Uc (absorb) or 2Uc (reflect)!🧠 Memory Trick: Electric and Magnetic fields share wave energy equally 50–50! And a mirror gets DOUBLE push (
2U/c) compared to a black absorber (U/c)!💡 Simple Kid-Check: Even though
E₀ = c B₀ makes electric numbers look 3 × 10⁸ times bigger than magnetic Tesla numbers, plug them into energy formulas and both carry the EXACT same 50% share of wave energy!- 1Proving Equal Energy Share: Instantaneous electric energy density is
uE = ½ ε₀ E²and magnetic isuB = B²/(2μ₀). UsingB = E/candc² = 1/(μ₀ε₀):uB = E²2μ₀ c² = ½ ε₀ E² = uE! Total average densityuavg = 2 uE,avg = ½ ε₀ E₀² = ε₀ Erms²! - 2Wave Intensity (
I): Energy crossing unit perpendicular area per second:I = EnergyArea · Time = uavg · c = ε₀ c Erms²(W/m²). - 3Momentum & Radiation Pressure: Total energy
Uabsorbed completely transfers momentump = U / c(pressurePrad = I / c). On a 100% reflecting mirror, momentum change doubles top = 2U / c(Prad = 2I / c)!
🧒 NCERT-Style Numerical: An EM beam of intensity
• Energy delivered:
• Momentum transferred:
I = 90 W/m² hits a 2.0 m² black solar absorber for 10 s:• Energy delivered:
U = I · A · t = 90 × 2.0 × 10 = 1800 J.• Momentum transferred:
p = U / c = 1800 / (3 × 10⁸) = 6.0 × 10⁻⁶ kg·m/s (if mirror → 1.2 × 10⁻⁵ kg·m/s!).
Watch two sneaky numerical traps: (1) Using peak
E₀ vs RMS Erms: total average energy density is ½ ε₀ E₀² (with ½!) OR ε₀ Erms² (NO ½, because Erms² = E₀²/2 doubled for E + B!). (2) Always check whether the surface absorbs (U/c) or perfectly reflects (2U/c)!Topic 8.4.1 – 8.4.4: Radio, Microwaves, Infrared & Visible Spectrum Uses
Look around a typical Kerala living room: FM music drifts from the kitchen speaker (Radio waves), an oven warms appam curry (Microwaves), your TV remote blinks silently (Infrared), and sunshine lights the courtyard (Visible light)! Every single one is the SAME Maxwell EM wave — differing ONLY in wavelength and frequency!
RADIO (> 0.1 m)Accelerated wire electrons; AM (530–1710 kHz), TV (54–890 MHz), FM (88–108 MHz), UHF mobile!
MICROWAVES (0.1 m – 1 mm)Klystron / Magnetron; Aircraft radar & ovens (resonate with water molecules!).
IR & VISIBLEIR (1 mm – 700 nm): Heat waves, remotes, Greenhouse effect • Visible (700–400 nm): Human eye!
🧠 Memory Trick: Order from Longest λ to Shortest λ: "Raging Martians Invaded Venus Using X-ray Guns" → Radio, Micro, IR, Visible, UV, X-ray, Gamma!
💡 Simple Kid-Check: Why does a microwave oven heat up sambar fast while the dry ceramic bowl stays cool? Because microwaves are tuned to the natural resonant frequency of water molecules — making water spin and heat up straight inside the food!
| Band | Wavelength λ | Source / Production | Detector & Key Uses |
|---|---|---|---|
| Radio Waves | > 0.1 m | Oscillating electrons in LC/aerials | Receiver aerials; AM/FM radio, TV, UHF mobile phones |
| Microwaves | 0.1 m – 1 mm | Klystron, Magnetron, Gunn diode | Point-contact diodes; Radar, satellites, microwave ovens |
| Infrared (IR) | 1 mm – 700 nm | Hot bodies, vibrating atoms/molecules | Thermopiles, bolometer; Remotes, physiotherapy, greenhouse warmth |
| Visible Light | 700 – 400 nm | Valence atomic electron transitions | Human eye, photocells, camera film; Vision & photosynthesis |
🧒 NCERT Exercise 8.5 Style: An FM radio station broadcasts at
ν = 100 MHz (10⁸ Hz). Find its wavelength:λ = c / ν = (3 × 10⁸) / (100 × 10⁶) = 3.0 m (Right inside the Radio band!).
Top 3 NCERT Board questions on lower-frequency bands: (1) Name the valve used to produce Microwaves: Klystron or Magnetron! (2) Which EM wave keeps Earth warm via the Greenhouse Effect? Infrared (IR) (visible enters, Earth re-emits longer-wavelength IR trapped by CO₂/water vapour!). (3) Why are microwaves chosen for radar & satellite communication? Short wavelength gives narrow beams and pierces Earth's ionosphere!
Topic 8.4.5 – 8.4.7: Ultraviolet, X-Rays & Gamma Rays Applications & Safety
Beyond violet, electromagnetic waves pack so much punch per photon (
E = hν) that they become double-edged tools: Ultraviolet (UV) kills bacteria in water purifiers, X-rays peer through skin to photograph broken bones, and nuclear Gamma rays blast cancer tumours — yet over-exposure can harm living cells unless properly shielded!ULTRAVIOLET (400 – 1 nm)Sun & special lamps • Blocked by Ozone (O₃) & glass • LASIK eye surgery, water purifiers, welding goggles!
X-RAYS (1 nm – 10⁻³ nm)High-energy electrons hitting metal target • Bone imaging, cancer treatment, crystal study!
GAMMA RAYS (< 10⁻³ nm)Radioactive nuclear decay • Highest frequency/penetration • Cancer radiotherapy & nuclear medicine!
🧠 Memory Trick: Watch the SOURCE ladder: UV = outer/inner atomic shells; X-ray = fast electrons hitting metal; Gamma = straight from inside the ATOMIC NUCLEUS!
💡 Simple Kid-Check: Why do welders wear dark glass masks while working? Electric welding arcs blast blinding Ultraviolet (UV) rays that burn eyes — ordinary glass absorbs UV so the welder stays safe! Above Earth, the stratospheric Ozone layer works as our giant UV sunglasses!
| Band | Wavelength λ | Production | Uses & Safety |
|---|---|---|---|
| Ultraviolet (UV) | 400 nm – 1 nm | Inner-shell atomic transitions, sun, UV lamps | Water purifiers (kills germs), LASIK eye surgery; Ozone blocks solar UV-B/C; glass absorbs UV |
| X-Rays | 1 nm – 10⁻³ nm | Fast electrons bombarding heavy metal target | Medical fracture imaging, crystal structure, security scans; avoid over-exposure (cell/DNA damage) |
| Gamma Rays | < 10⁻³ nm | Nuclear reactions & radioactive nuclear decay | Destroy cancer cells (radiotherapy), sterilisation; needs thick lead/concrete shielding |
🧒 NCERT Exercise 8.3 Golden Check: What physical quantity stays the EXACT SAME in vacuum for X-rays (
Answer: SPEED IN VACUUM (
10⁻¹⁰ m), red light (6800 Å), and radio waves (500 m)?Answer: SPEED IN VACUUM (
c = 3 × 10⁸ m/s)! Only frequency ν and wavelength λ differ!
Watch three high-yield DHSE exam traps: (1) X-rays vs Gamma rays: Even though their wavelength ranges overlap slightly, X-rays come from electrons hitting metal targets / inner atomic shells, whereas Gamma rays come strictly from radioactive atomic NUCLEI! (2) Welders' goggles protect specifically against UV rays (NOT IR or X-rays!). (3) Stratospheric Ozone (O₃) shields life from UV radiation!
Solved Previous Year Questions (Kerala DHSE & NCERT)
Q1 (Kerala HSE March • 2 Marks): Define displacement current. Write the modified Ampere-Maxwell circuital law.
Displacement Current (id): Apparent current arising from a time-varying electric field/flux:
Ampere-Maxwell Law:
id = ε₀ (dΦE/dt).Ampere-Maxwell Law:
∮ B · dl = μ₀ [ ic + ε₀ (dΦE/dt) ].Q2 (Kerala HSE Say • 2 Marks): A parallel-plate capacitor of capacitance 2 μF is being charged so that the voltage across plates changes at 10⁵ V/s. Find the displacement current between the plates.
ΦE = Q/ε₀ = C V / ε₀ ⇒ id = ε₀ (dΦE/dt) = C (dV/dt).id = (2 × 10⁻⁶ F) × (10⁵ V/s) = 0.2 A.Q3 (Kerala HSE March • 2 Marks): State any four properties of electromagnetic waves.
1. Produced by accelerating/oscillating charges.
2. Purely transverse:
3. Travel in vacuum at
4. Carry energy (50% electric, 50% magnetic) and momentum
2. Purely transverse:
E ⊥ B ⊥ propagation (E × B), oscillating in phase.3. Travel in vacuum at
c = 1/√(μ₀ε₀) = 3 × 10⁸ m/s (no medium required).4. Carry energy (50% electric, 50% magnetic) and momentum
p = U/c.Q4 (NCERT / HSE Model • 3 Marks): The magnetic field in a plane EM wave is B_y = 2 × 10⁻⁷ sin(0.5 × 10³ x + 1.5 × 10¹¹ t) T. Find (a) wavelength & frequency, (b) direction of propagation, and (c) electric field equation.
(a)
(b) Plus sign in
(c)
k = 0.5×10³ m⁻¹ ⇒ λ = 2π/k ≈ 1.26 cm; ω = 1.5×10¹¹ rad/s ⇒ ν = ω/(2π) ≈ 23.9 GHz.(b) Plus sign in
(kx + ωt) ⇒ wave moves along −x.(c)
E₀ = c B₀ = 3×10⁸ × 2×10⁻⁷ = 60 V/m. Since E × B must point −î and B is along +ĵ, E is along +k̂: E_z = 60 sin(0.5×10³ x + 1.5×10¹¹ t) V/m.Q5 (Kerala HSE March • 2 Marks): Name the EM waves used in (i) aircraft radar & microwave ovens, and (ii) TV remote controls & greenhouse warmth. State how microwaves heat food.
• (i) Microwaves (produced by Klystron/Magnetron valves; resonant frequency matches water molecules, making them oscillate and heat food!).
• (ii) Infrared (IR) waves.
• (ii) Infrared (IR) waves.
Q6 (Kerala HSE Say • 2 Marks): Which electromagnetic radiation are welders protected against by special glass goggles? Why is the stratospheric ozone layer vital for life on Earth?
Welders' goggles absorb harmful Ultraviolet (UV) radiation produced by electric arcs.
Similarly, Earth's stratospheric ozone (O₃) layer absorbs damaging solar UV radiation before it reaches living organisms.
Similarly, Earth's stratospheric ozone (O₃) layer absorbs damaging solar UV radiation before it reaches living organisms.
Q7 (Kerala HSE Model • 2 Marks): Give one source and one medical application each for X-rays and Gamma rays.
• X-rays: Generated by fast electrons bombarding a heavy metal target; used to image bone fractures.
• Gamma rays: Emitted during radioactive decay of atomic nuclei; used in radiotherapy to destroy cancer tumours.
• Gamma rays: Emitted during radioactive decay of atomic nuclei; used in radiotherapy to destroy cancer tumours.
Q8 (NCERT / DHSE Numerical • 2 Marks): Light with energy flux 18 W/cm² falls at normal incidence on a non-reflecting surface of area 20 cm² for 30 minutes. Find total momentum delivered.
Total energy
For complete absorption:
U = I · A · t = 18 × 20 × (30 × 60) = 6.48 × 10⁵ J.For complete absorption:
p = U / c = (6.48 × 10⁵) / (3 × 10⁸) = 2.16 × 10⁻³ kg·m/s.