What this chapter is about
This chapter explains how changing electric and magnetic fields create waves that travel through space without needing any medium. You will learn how James Clerk Maxwell predicted these waves by adding a crucial term — the displacement current — to the equations governing electricity and magnetism. This unification showed that light itself is an electromagnetic wave.
A Class 12 student meets this topic now because it connects everything learned about electrostatics, magnetism and electromagnetic induction into one coherent picture. Understanding electromagnetic waves also prepares you for optics, communication technology and modern physics. After studying this chapter, you should be able to describe how EM waves are produced and propagated, state their properties, and identify different regions of the electromagnetic spectrum along with their sources and uses.
The chapter also develops the idea that electromagnetic waves carry energy and momentum, which explains phenomena like radiation pressure. You will see that all EM waves — radio waves, microwaves, infrared, visible light, ultraviolet, X-rays and gamma rays — differ only in frequency and wavelength, yet share identical fundamental nature.
Key ideas
- A time-varying electric field produces a magnetic field, and a time-varying magnetic field produces an electric field; together they sustain each other and propagate as an electromagnetic wave.
- Maxwell introduced displacement current, Id = ε₀ × (dΦE/dt), to account for the magnetic field produced by a changing electric flux even when no conduction current flows (for example, between capacitor plates).
- Electromagnetic waves are transverse: the electric field vector E, magnetic field vector B, and the direction of propagation are mutually perpendicular.
- In free space, EM waves travel at speed c = 1/√(μ₀ε₀) ≈ 3.00 × 10⁸ m/s; at every instant, E₀/B₀ = c.
- EM waves carry energy; the average energy density is u = (1/2)ε₀E² + (1/2)(B²/μ₀), and intensity is the energy crossing unit area per unit time.
- The electromagnetic spectrum spans from radio waves (longest wavelength, lowest frequency) through microwaves, infrared, visible, ultraviolet, X-rays to gamma rays (shortest wavelength, highest frequency).
- EM waves also carry momentum; when absorbed by a surface, they exert radiation pressure p = I/c (for total absorption).
Formulas and facts to remember
- Displacement current: Id = ε₀ × (dΦE/dt), where ΦE is electric flux. It has the same unit (ampere) as conduction current.
- Speed of EM waves in vacuum: c = 1/√(μ₀ε₀) ≈ 3.00 × 10⁸ m/s.
- Relation between field amplitudes: E₀ = c × B₀, or at any instant E = c × B.
- Wave equation form: E = E₀ sin(kx − ωt), B = B₀ sin(kx − ωt), with k = 2π/λ and ω = 2πν.
- Energy density: Total average energy density u = ε₀E² (since electric and magnetic contributions are equal in a travelling wave).
- Intensity: I = (1/2) × c × ε₀ × E₀² (average power per unit area).
- Radiation pressure: p = I/c (complete absorption) or p = 2I/c (perfect reflection).
- Spectrum order by increasing frequency: Radio < Microwave < Infrared < Visible < Ultraviolet < X-ray < Gamma ray.
Worked examples
Example 1 — Displacement current in a charging capacitor
A parallel-plate capacitor with circular plates of radius 6.0 cm is being charged. The electric field between the plates increases at 2.0 × 10¹² V/(m·s). Find the displacement current.
Solution
Area of plate, A = π × (0.06)² = 1.13 × 10⁻² m².
Electric flux ΦE = E × A (uniform field, area perpendicular to E).
Rate of change of flux, dΦE/dt = A × (dE/dt) = 1.13 × 10⁻² × 2.0 × 10¹² = 2.26 × 10¹⁰ V·m/s.
Displacement current, Id = ε₀ × (dΦE/dt) = 8.85 × 10⁻¹² × 2.26 × 10¹⁰ ≈ 0.20 A.
So the displacement current is 0.20 A — the same as the conduction current in the wires feeding the capacitor.
Example 2 — Speed of light from constants
Using μ₀ = 4π × 10⁻⁷ T·m/A and ε₀ = 8.85 × 10⁻¹² C²/(N·m²), verify the speed of light.
Solution
c = 1/√(μ₀ε₀).
μ₀ε₀ = 4π × 10⁻⁷ × 8.85 × 10⁻¹² = 1.11 × 10⁻¹⁷ (in SI units).
√(μ₀ε₀) = 3.33 × 10⁻⁹ s/m.
c = 1/(3.33 × 10⁻⁹) ≈ 3.00 × 10⁸ m/s.
This matches the measured speed of light, confirming Maxwell's prediction.
Example 3 — Electric and magnetic field amplitudes
A laser beam has an intensity of 1.2 × 10³ W/m². Find E₀ and B₀.
Solution
Intensity, I = (1/2) × c × ε₀ × E₀².
Rearranging, E₀² = 2I / (c × ε₀) = (2 × 1.2 × 10³) / (3 × 10⁸ × 8.85 × 10⁻¹²).
E₀² = 2.4 × 10³ / 2.66 × 10⁻³ = 9.02 × 10⁵.
E₀ = 9.5 × 10² V/m (approximately 950 V/m).
B₀ = E₀ / c = 950 / (3 × 10⁸) ≈ 3.2 × 10⁻⁶ T = 3.2 μT.
Common mistakes
- Thinking displacement current is a flow of charges → it is not; it is ε₀ × (dΦE/dt), arising from a changing electric field.
- Assuming EM waves need a material medium to travel → they propagate perfectly through vacuum.
- Confusing the direction of E or B with the direction of wave propagation → E, B and the propagation direction are all mutually perpendicular.
- Using E₀ = B₀ instead of E₀ = c × B₀ → the magnitudes are related by the speed of light, not equality.
- Mixing up frequency and wavelength trends in the spectrum → higher frequency means shorter wavelength.
Quick revision
- Displacement current Id = ε₀ × (dΦE/dt) completes Ampere's law for time-varying fields.
- EM waves are transverse; E ⊥ B ⊥ direction of propagation.
- Speed in vacuum: c = 1/√(μ₀ε₀) ≈ 3 × 10⁸ m/s; field ratio E = cB.
- Spectrum order (low to high frequency): Radio, Microwave, Infrared, Visible, UV, X-ray, Gamma.
- EM waves carry energy (intensity I) and exert radiation pressure I/c on an absorbing surface.