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Class XII · Chapter 08

Electromagnetic Waves

The missing piece Maxwell needed to complete his equations, and why light, radio, and gamma rays are all fundamentally the same phenomenon at different frequencies.

● Easy ⏱ 18 min read 🎯 24 practice questions 📊 Not yet revised
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1Displacement Current

Ampere's Circuital Law (Chapter 4) works perfectly for steady currents, but breaks down in one specific situation: a charging capacitor. Current flows in the wires up to the plates, but no actual charge crosses the gap between them — so which current do you use in Ampere's Law for a Gaussian surface that bulges between the plates?

Maxwell resolved this by proposing that a changing electric field is, in every meaningful sense, equivalent to a current — even where no charge is physically moving.

Displacement Current
Id = ε₀ (dΦE / dt)
Inside a charging capacitor, the displacement current between the plates exactly equals the conduction current in the connecting wires — keeping current "continuous" through the whole circuit, even across the gap.

2Maxwell's Equations and Electromagnetic Waves

With displacement current added, Ampere's Law becomes the Ampere–Maxwell Law:

Ampere–Maxwell Law
∮ B · dl = μ₀ Ic + μ₀ε₀ (dΦE / dt)
This completes a symmetric picture: Faraday's Law (Chapter 6) says a changing B creates an E; this equation says a changing E creates a B. Together with Gauss's Laws for E and B, these four equations — Maxwell's Equations — fully describe classical electromagnetism, and predict that the two fields can sustain each other, propagating through empty space as a self-perpetuating wave.
Speed of an Electromagnetic Wave in Vacuum
c = 1 / √(μ₀ε₀)
Plugging in μ₀ and ε₀ gives c ≈ 3 × 10⁸ m/s — exactly the known speed of light. This was one of the most striking results in physics: Maxwell's equations, built purely from electricity and magnetism, predicted the speed of light without ever mentioning optics, revealing that light itself is an electromagnetic wave.

3Sources of Electromagnetic Waves

Any accelerating electric charge radiates an electromagnetic wave. A charge oscillating back and forth (as in an antenna) produces a continuous EM wave at the frequency of oscillation — this is exactly how radio and TV transmitters work: an oscillating current in the antenna generates a matching electromagnetic wave that propagates outward.

4Nature of Electromagnetic Waves

  • EM waves are transverse: both E and B oscillate perpendicular to the direction the wave travels.
  • E and B are also perpendicular to each other, and they oscillate in phase — reaching maximum and zero at the same points and times.
  • They require no medium — unlike sound, EM waves travel through a perfect vacuum, which is exactly how sunlight reaches us across empty space.
Relation Between Field Amplitudes
E₀ = c B₀
The electric and magnetic amplitudes aren't independent — their ratio is fixed by the speed of light.

EM waves carry energy and momentum, distributed between the electric and magnetic fields:

Energy Density
uE = ½ε₀E², uB = B² / 2μ₀
On average, over a full oscillation, the energy carried by the electric field equals the energy carried by the magnetic field — a clean, symmetric split.

5The Electromagnetic Spectrum

Visible light is just one narrow slice of a vast continuum of electromagnetic waves, all travelling at exactly the same speed c, distinguished only by frequency (and correspondingly, wavelength).

  • Radio waves (lowest frequency): produced by accelerating charges in antennas; used for broadcasting and communication.
  • Microwaves: produced by special vacuum tubes; used in radar, satellite communication, and microwave ovens (which excite water molecules).
  • Infrared: emitted by warm objects; used in remote controls, thermal imaging, and responsible for much of the greenhouse effect.
  • Visible light: the narrow band our eyes detect, roughly 400–700 nm.
  • Ultraviolet: from the Sun and specialised lamps; causes sunburn, ionizes atoms, absorbed by the ozone layer, used for sterilization.
  • X-rays: produced by decelerating fast electrons; penetrate soft tissue, used in medical imaging and crystallography.
  • Gamma rays (highest frequency): from nuclear decay and cosmic sources; highly penetrating, used in cancer treatment and studied in nuclear/astrophysics.
Key Insight Every one of these is physically the same kind of wave — oscillating E and B fields moving at c. What changes across the spectrum is purely frequency and wavelength, which is what determines how each type interacts with matter.

Formula Summary

Displacement Current
I_d = ε₀ dΦ_E/dt
Ampere–Maxwell Law
∮B·dl = μ₀I_c + μ₀ε₀ dΦ_E/dt
Speed of Light
c = 1/√(μ₀ε₀)
Amplitude Ratio
E₀ = cB₀
Electric Energy Density
u_E = ½ε₀E²
Magnetic Energy Density
u_B = B²/2μ₀
Wave Speed Relation
c = fλ

Solved Examples

Example 1 · Displacement Current

A parallel plate capacitor is being charged, and a conduction current of 2 A flows through the connecting wires. What is the displacement current between the plates?

Solution: For the Ampere–Maxwell Law to be consistent regardless of which surface is chosen, displacement current must exactly equal conduction current: Id = Ic = 2 A.

Example 2 · Wavelength from Frequency

Find the wavelength of an electromagnetic wave with frequency 5×10¹⁴ Hz.

Solution: λ = c/f = (3×10⁸) / (5×10¹⁴) = 6×10⁻⁷ m = 600 nm — squarely in the visible light range (orange light).

Quick Check

1. In an electromagnetic wave, the electric field E and magnetic field B are:
Parallel to each other
Parallel to the direction of propagation
✓ Perpendicular to each other and to propagation (correct)
Out of phase by 90°
2. Displacement current arises due to:
Moving free charges only
✓ A changing electric field (correct)
A changing magnetic field
Resistive heating
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Alternating Current

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Ray Optics and Optical Instruments

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