Electromagnetic waves represent one of the most profound unifications in the history of physics. James Clerk Maxwell, in the 1860s, noticed a critical inconsistency in Ampere's circuital law when applied to a circuit containing a charging capacitor. The original Ampere's law (∮B·dl = μ_{0}I_c) correctly predicted the magnetic field due to conduction current in a wire, but when applied to a charging capacitor it gave contradictory results depending on which surface bounded by the Amperian loop was considered. One surface cutting the wire enclosed a current; another surface passing through the capacitor gap enclosed no current — yet both surfaces were bounded by the same loop. This was a fundamental inconsistency that demanded resolution.

Maxwell's genius lay in identifying that although no charges crossed the capacitor gap, the electric field between the plates was changing as the capacitor charged. He introduced the concept of displacement current: Id = ε_{0}(dΦ_E/dt), where ε_{0} is the permittivity of free space and Φ_E is the electric flux between the plates. This term has the units of amperes and behaves exactly like a real current in producing a magnetic field — even though no actual charges move. The displacement current in the gap exactly equals the conduction current in the connecting wire at every instant, restoring current continuity throughout the circuit. The modified Ampere-Maxwell law became: ∮B·dl = μ_{0}(I_c + ε_{0} dΦ_E/dt), completing Maxwell's four equations of electromagnetism.

These four equations — Gauss's law for electricity (charges produce diverging E fields), Gauss's law for magnetism (no magnetic monopoles; B field lines are always closed loops), Faraday's law (changing B field produces E field), and the Ampere-Maxwell law (electric currents and changing E fields produce B fields) — embody a beautiful symmetry. Faraday's law says a changing B creates E; the Ampere-Maxwell law says a changing E creates B. This mutual coupling means that a disturbance in one field self-perpetuates by creating the other, which creates the first again, and so on — propagating outward as an electromagnetic wave. Accelerating charges produce such disturbances: the oscillating E and B fields radiate outward as a transverse wave.

Properties of EM waves in vacuum: the E and B fields are mutually perpendicular to each other and to the direction of propagation (transverse nature); they oscillate in phase — crests, zeros, and troughs coincide at any given point in time; the ratio of their amplitudes is $E_{0}$/$B_{0}$ = c = $3 \times 10^{8}$ m/s; they travel at the same speed c = 1/√(μ_{0}ε_{0}) for all frequencies. This last point deserves emphasis: unlike sound waves where speed depends on the medium's properties alone, EM waves in vacuum always travel at exactly c regardless of frequency, energy, or polarisation. In a material medium, however, different wavelengths travel at different speeds (dispersion), giving rise to phenomena like the rainbow.

EM waves carry energy (intensity I = (1/2)ε_{0}c$E_{0}^{2}$, units W/$m^{2}$) and momentum (p = U/c for absorption, p = 2U/c for total reflection), exerting radiation pressure on surfaces they strike.

The electromagnetic spectrum orders all EM waves by frequency (increasing) or wavelength (decreasing): Radio waves (10^{3}–10^{9} Hz; source: oscillating circuits; use: communication and broadcasting), Microwaves (10^{9}–10^{12} Hz; source: magnetron/klystron; use: RADAR, microwave ovens, satellite communication), Infrared radiation (10^{12}–$4 \times 10^{14}$ Hz; source: hot bodies; use: night vision devices, physiotherapy, greenhouse effect), Visible light ($4 \times 10^{14}$–$7.5 \times 10^{14}$ Hz; 400–700 nm; VIBGYOR from violet to red; use: vision, photosynthesis), Ultraviolet radiation ($7.5 \times 10^{14}$–10^{17} Hz; source: Sun, mercury vapour lamps; use: sterilisation, LASIK surgery, vitamin D synthesis), X-rays (10^{16}–10^{21} Hz; source: Coolidge tube, electron deceleration; use: medical imaging, CT scans, crystal diffraction), and Gamma rays (10^{18}–10^{24} Hz; source: radioactive nuclear decay; use: cancer radiotherapy, sterilisation of medical equipment).

A critical distinction that NEET tests annually: X-rays and gamma rays have overlapping frequency ranges (10^{18}–10^{21} Hz is common to both). They are distinguished solely by their source: X-rays originate from the deceleration of electrons or from atomic inner-shell electron transitions (bremsstrahlung); gamma rays originate from nuclear transitions during radioactive decay. You cannot classify a photon at 10^{19} Hz as X-ray or gamma ray without knowing its origin.

The mnemonic "Rich Men In Visible Uniforms eXude Glamour" reliably orders the spectrum from lowest to highest frequency: Radio → Microwave → Infrared → Visible → Ultraviolet → X-rays → Gamma. The VIBGYOR mnemonic orders visible light from highest frequency (Violet, 400 nm) to lowest frequency (Red, 700 nm). For application matching in NEET: microwaves = RADAR and ovens; UV = sterilisation and LASIK; IR = night vision and physiotherapy; X-rays = medical imaging and crystallography; Gamma = cancer treatment. These pairings are tested nearly every year.