SECTION 8.1 – 8.2
Displacement Current
The Need for Displacement Current
Maxwell noticed an inconsistency in Ampere's circuital law when applied to a charging capacitor. Applying Ampere's law to two different surfaces with the same perimeter gave contradictory results for the magnetic field at a point P:
- Surface 1 (flat, through the wire): B(2πr) = μ₀i(t) → B ≠ 0
- Surface 2 (pot-shaped, between plates): No current passes → B = 0
- This contradiction arises because the electric field between capacitor plates changes with time — there must be something missing in Ampere's law.
Maxwell's Solution — Displacement Current
Maxwell argued that a changing electric field must also produce a magnetic field. He introduced the concept of displacement current:
- Displacement current: i_d = ε₀ (dΦ_E/dt)
- Between capacitor plates: i_c = 0, i_d = i (only displacement current)
- Outside capacitor plates: i_c = i, i_d = 0 (only conduction current)
- Total current: i = i_c + i_d = i_c + ε₀(dΦ_E/dt)
- Displacement current has the same physical effect as conduction current — it produces a magnetic field.
Displacement Current & Ampere-Maxwell Law
Displacement current: i_d = ε₀ dΦ_E/dt
Total current: i = i_c + i_d
Generalised Ampere-Maxwell Law:
∮ B·dl = μ₀i_c + μ₀ε₀ (dΦ_E/dt)
Symmetry: Changing B → E (Faraday's law)
Changing E → B (Ampere-Maxwell law)
Total current: i = i_c + i_d
Generalised Ampere-Maxwell Law:
∮ B·dl = μ₀i_c + μ₀ε₀ (dΦ_E/dt)
Symmetry: Changing B → E (Faraday's law)
Changing E → B (Ampere-Maxwell law)
Fig 8.1 — Parallel plate capacitor showing pot-shaped and tiffin-shaped surfaces; displacement current (NCERT Figure 8.1)
Fig 8.2 — Electric and magnetic fields E and B between capacitor plates at point M (NCERT Figure 8.2)
🎯 Practice MCQs — Displacement Current
Q1. A parallel plate capacitor (C = 100 pF, radius R = 6 cm) is connected to 230 V AC supply (ω = 300 rad/s). The displacement current amplitude between the plates is approximately:
X_C = 1/(ωC) = 1/(300×100×10⁻¹²) = 1/(3×10⁻⁸) = 3.33×10⁷ Ω. i_rms = V/X_C = 230/(3.33×10⁷) ≈ 6.9×10⁻⁶ A...
Actually: X_C = 1/ωC = 1/(300×10⁻¹⁰) = 10¹⁰/300 = 3.33×10⁷ Ω. i = 230/X_C ≈ 6.9 μA. i_d = i_c. Amplitude = √2×6.9 μA ≈ 6.9 mA (using correct calculation with C=100pF, ω=300).
Actually: X_C = 1/ωC = 1/(300×10⁻¹⁰) = 10¹⁰/300 = 3.33×10⁷ Ω. i = 230/X_C ≈ 6.9 μA. i_d = i_c. Amplitude = √2×6.9 μA ≈ 6.9 mA (using correct calculation with C=100pF, ω=300).
Q2. The displacement current between the plates of a capacitor is equal to:
By definition (Maxwell's modification): displacement current i_d = ε₀ × dΦ_E/dt. It equals the conduction current in the external circuit.
Q3. A circular plate capacitor with radius 12 cm is charged by a constant current of 0.15 A. The displacement current across the plates is:
The displacement current between the plates always equals the conduction current in the external circuit. So i_d = 0.15 A. Kirchhoff's junction rule is satisfied when displacement current is included.
SECTION 8.2 (continued)
Maxwell's Equations in Vacuum
The Four Maxwell's Equations
- 1. Gauss's Law for Electricity: ∮E·dA = Q/ε₀ — Electric field lines originate from charges.
- 2. Gauss's Law for Magnetism: ∮B·dA = 0 — No magnetic monopoles; B field lines are closed loops.
- 3. Faraday's Law: ∮E·dl = −dΦ_B/dt — Changing B gives rise to E.
- 4. Ampere-Maxwell Law: ∮B·dl = μ₀i_c + μ₀ε₀(dΦ_E/dt) — Changing E (or conduction current) gives rise to B.
Symmetry in Maxwell's Equations
- Faraday's law: changing B → generates E
- Ampere-Maxwell law: changing E → generates B
- This symmetry — time-varying E and B fields generate each other — is the basis for electromagnetic waves.
- The laws are still not perfectly symmetric: no magnetic monopoles exist (unlike electric charges).
Maxwell's Equations (Summary)
1. ∮E·dA = Q/ε₀ (Gauss's Law — Electricity)
2. ∮B·dA = 0 (Gauss's Law — Magnetism)
3. ∮E·dl = −dΦ_B/dt (Faraday's Law)
4. ∮B·dl = μ₀i_c + μ₀ε₀ dΦ_E/dt (Ampere-Maxwell)
2. ∮B·dA = 0 (Gauss's Law — Magnetism)
3. ∮E·dl = −dΦ_B/dt (Faraday's Law)
4. ∮B·dl = μ₀i_c + μ₀ε₀ dΦ_E/dt (Ampere-Maxwell)
🎯 Practice MCQs — Maxwell's Equations
Q4. Which of Maxwell's equations tells us that magnetic monopoles do not exist?
Gauss's Law for Magnetism: ∮B·dA = 0. This states the net magnetic flux through any closed surface is zero — meaning there are no isolated magnetic poles (monopoles).
Q5. Maxwell's greatest contribution was to modify Ampere's law by adding the displacement current term. This modification predicts the existence of:
The symmetry between changing E producing B and changing B producing E (Faraday's law) means they sustain each other — predicting self-propagating electromagnetic waves in free space.
Q6. A plane electromagnetic wave travels in vacuum along z-direction with frequency 30 MHz. The directions of E and B fields are:
In an EM wave propagating along z, E and B are perpendicular to each other AND to the direction of propagation (transverse wave). E could be along x, B along y (or vice versa).
SECTION 8.3
Electromagnetic Waves — Properties
Sources of Electromagnetic Waves
- Accelerated charges radiate electromagnetic waves (key result of Maxwell's theory).
- Stationary charges → electrostatic fields only (no EM waves).
- Steady currents → static magnetic fields only (no EM waves).
- Oscillating charge at frequency ν → EM wave of same frequency ν.
- Hertz (1887): First to produce and detect EM waves in laboratory (radio waves, ~meter wavelength).
- J.C. Bose (Kolkata): Produced shorter wavelength EM waves (25 mm to 5 mm).
- Marconi: Transmitted EM waves over many km — beginning of wireless communication.
Nature and Properties of EM Waves
- E and B oscillate sinusoidally in space and time.
- E ⊥ B ⊥ direction of propagation (transverse wave).
- For wave propagating along z: E_x = E₀ sin(kz − ωt); B_y = B₀ sin(kz − ωt)
- E and B are in phase — reach maxima and minima simultaneously.
- Speed in vacuum: c = 1/√(μ₀ε₀) = 3 × 10⁸ m/s
- Relation: E₀/B₀ = c or B₀ = E₀/c
- No material medium required — EM waves are self-sustaining oscillations.
- Speed in medium: v = 1/√(με)
- Frequency relation: νλ = c
EM Wave Properties — Key Formulae
E_x = E₀ sin(kz − ωt) | B_y = B₀ sin(kz − ωt)
Wave vector: k = 2π/λ
Speed in vacuum: c = ω/k = 1/√(μ₀ε₀) = 3 × 10⁸ m/s
Speed in medium: v = 1/√(με)
Relation of amplitudes: B₀ = E₀/c
Frequency–wavelength: νλ = c
Example: If E = 6.3 V/m → B = E/c = 6.3/(3×10⁸) = 2.1×10⁻⁸ T
Wave vector: k = 2π/λ
Speed in vacuum: c = ω/k = 1/√(μ₀ε₀) = 3 × 10⁸ m/s
Speed in medium: v = 1/√(με)
Relation of amplitudes: B₀ = E₀/c
Frequency–wavelength: νλ = c
Example: If E = 6.3 V/m → B = E/c = 6.3/(3×10⁸) = 2.1×10⁻⁸ T
Fig 8.3 — Linearly polarised EM wave: E along x, B along y, propagation along z (NCERT Figure 8.3)
🎯 Practice MCQs — EM Wave Properties
Q7. The magnetic field amplitude of a harmonic EM wave in vacuum is B₀ = 510 nT. The electric field amplitude E₀ is:
E₀ = B₀ × c = 510×10⁻⁹ × 3×10⁸ = 510 × 0.3 = 153 V/m
Q8. In a plane EM wave, E₀ = 120 N/C and ν = 50 MHz. The angular frequency ω and wave number k are:
ω = 2πν = 2π × 50×10⁶ ≈ 3.14×10⁸ rad/s
λ = c/ν = 3×10⁸/(50×10⁶) = 6 m
k = 2π/λ = 2π/6 ≈ 1.05 rad/m
λ = c/ν = 3×10⁸/(50×10⁶) = 6 m
k = 2π/λ = 2π/6 ≈ 1.05 rad/m
Q9. An EM wave has electric field E oscillating at 2×10¹⁰ Hz with amplitude 48 V/m. The wavelength and magnetic field amplitude are:
λ = c/ν = 3×10⁸/(2×10¹⁰) = 0.015 m = 15 mm
B₀ = E₀/c = 48/(3×10⁸) = 1.6×10⁻⁷ T
B₀ = E₀/c = 48/(3×10⁸) = 1.6×10⁻⁷ T
SECTION 8.4
The Electromagnetic Spectrum
Overview of the EM Spectrum
All EM waves travel at the same speed in vacuum (c = 3×10⁸ m/s) but differ in wavelength and frequency. Classification is based on how waves are produced and detected. There are no sharp boundaries between regions.
| Type | Frequency | Wavelength | Production | Key Uses |
|---|---|---|---|---|
| Radio waves | 500 kHz–1000 MHz | >0.1 m | Accelerated electrons in aerials | Radio, TV, mobile phones (UHF). AM: 530kHz–1710kHz; FM: 88–108 MHz |
| Microwaves | GHz range | 0.1 m to 1 mm | Klystrons, magnetrons, Gunn diodes | Radar, aircraft navigation, speed guns, microwave ovens (resonant freq of H₂O) |
| Infrared (IR) | ~10¹² Hz | 1 mm to 700 nm | Hot bodies and molecules | Heat therapy, greenhouse effect, TV remotes, satellite imaging, IR photography |
| Visible Light | 4×10¹⁴ – 7×10¹⁴ Hz | 700–400 nm | Electrons in atoms (energy level transitions) | Vision; red (700nm) to violet (400nm) |
| Ultraviolet (UV) | ~10¹⁵ Hz | 400 nm to 1 nm | Very hot bodies, special lamps, sun | LASIK surgery, water purification, kills germs. Absorbed by ozone layer. Causes tanning. |
| X-rays | ~10¹⁸ Hz | 1 nm to 10⁻³ nm | High-energy electrons bombarding metal target | Medical imaging, cancer treatment (radiotherapy) |
| Gamma rays | >10¹⁸ Hz | <10⁻³ nm | Nuclear reactions, radioactive decay | Cancer treatment (destroy cancer cells) |
Fig 8.4 — The electromagnetic spectrum: frequency vs wavelength with all bands labeled (NCERT Figure 8.4)
Special Notes for AAI ATC Exam
- All EM waves travel at c = 3×10⁸ m/s in vacuum regardless of frequency.
- Infrared = heat waves: absorbed by water molecules in food (microwave oven frequency matches water resonance).
- UV absorbed by ozone layer (~40–50 km altitude); CFCs deplete ozone — international concern.
- UV absorbed by ordinary glass (cannot get sunburn through glass window).
- X-rays: wavelength 1nm to 10⁻³nm; generated by bombarding metal with high energy electrons.
- Gamma rays: shortest wavelength, produced by nuclear reactions/radioactive decay.
- Radio waves: AM band 530 kHz–1710 kHz; FM band 88–108 MHz; TV waves 54–890 MHz.
- Visible spectrum: VIBGYOR — Violet (400nm) to Red (700nm).
🎯 Practice MCQs — EM Spectrum
Q10. A radio can tune from 7.5 MHz to 12 MHz. The corresponding wavelength band is:
λ = c/ν. For ν₁ = 12 MHz: λ₁ = 3×10⁸/(12×10⁶) = 25 m. For ν₂ = 7.5 MHz: λ₂ = 3×10⁸/(7.5×10⁶) = 40 m. Wavelength band: 25 m to 40 m.
Q11. Which of the following EM waves is used in microwave ovens and why?
Microwave ovens use microwaves whose frequency is tuned to match the resonant frequency of water molecules. This efficiently transfers energy to water molecules, increasing their kinetic energy and heating the food.
Q12. The physical quantity that is the same for X-rays (λ = 10⁻¹⁰ m), red light (λ = 6800 Å) and radio waves (λ = 500 m) is:
All EM waves — regardless of wavelength or frequency — travel at the same speed c = 3×10⁸ m/s in vacuum. This is a fundamental property of all electromagnetic radiation.
📋 Chapter Summary
⚡ Displacement Current
i_d = ε₀ dΦ_E/dt. Between capacitor plates (no conduction current). Produces B just like conduction current.
📐 Ampere-Maxwell Law
∮B·dl = μ₀i_c + μ₀ε₀(dΦ_E/dt). Total current = conduction + displacement current.
🌊 EM Wave Speed
c = 1/√(μ₀ε₀) = 3×10⁸ m/s. In medium: v = 1/√(με). All EM waves same speed in vacuum.
📏 Wave Properties
E ⊥ B ⊥ propagation. E₀/B₀ = c. In phase. νλ = c. Self-sustaining oscillations.
📻 Radio & Micro
Radio: 500kHz–1GHz. FM: 88–108 MHz. Micro: GHz range. Radar, speed guns, microwave ovens.
☀️ IR & Visible
IR: 1mm–700nm, heat waves, greenhouse effect, TV remotes. Visible: 700–400nm. VIBGYOR.
☢️ UV, X-ray, γ-ray
UV: 400nm–1nm, ozone layer. X-rays: 1nm–10⁻³nm, medical imaging. γ-rays: nuclear reactions, cancer therapy.
👨🔬 Key Scientists
Maxwell (theory), Hertz (1887, experimental verification), J.C. Bose (short λ), Marconi (wireless communication).
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