Class 12 Physics · Chapter 8 NotesElectromagnetic Waves

While applying Ampere's circuital law to a parallel plate capacitor being charged by a time-dependent current, Maxwell noticed a contradiction. For a circular loop outside the capacitor, the conduction current i(t) passes through the flat surface bounded by the loop, giving B(2πr) = μ₀i(t). But if the same loop is taken as the rim of a pot-shaped surface passing between the plates, no conduction current crosses it, so the same law gives zero magnetic field. Since the magnetic field at a point cannot depend on the choice of surface, Ampere's law had to be incomplete. Maxwell resolved this by noting that the electric flux between the plates changes with time, and added a term ε₀(dΦ_E/dt) to the law. This extra term is the displacement current.

• Displacement currenti_d = ε₀ (dΦ_E/dt)
• Total currenti = i_c + i_d
• Outside the capacitori_c = i, i_d = 0
• Inside the capacitori_c = 0, i_d = i

Maxwell expressed all the basic laws of electromagnetism as a set of four equations involving electric and magnetic fields and their sources, the charge and current densities. Together with the Lorentz force formula, these equations mathematically describe electricity and magnetism. The generalised form of Ampere's circuital law, called the Ampere-Maxwell law, states that the line integral of B around a closed loop equals μ₀ times the sum of the conduction current and ε₀ times the rate of change of electric flux through the surface bounded by the loop. Faraday's law says a time-varying magnetic field produces an electric field; the displacement current says a time-varying electric field produces a magnetic field. This symmetry is the key to electromagnetic waves.

• Gauss's law for electricity∮E·dA = Q/ε₀
• Gauss's law for magnetism∮B·dA = 0
• Faraday's law∮E·dl = –dΦ_B/dt
• Ampere-Maxwell law∮B·dl = μ₀i_c + μ₀ε₀ (dΦ_E/dt)

Stationary charges produce only electrostatic fields, and charges in uniform motion produce magnetic fields that do not vary with time, so neither can be a source of electromagnetic waves. Maxwell's theory shows that accelerated charges radiate electromagnetic waves. A charge oscillating with some frequency is an example of an accelerating charge. It produces an oscillating electric field, which produces an oscillating magnetic field, which in turn produces an oscillating electric field, and so on. The fields regenerate each other as the wave travels through space. The frequency of the wave equals the frequency of oscillation of the charge, and the energy of the wave comes from the energy of the accelerated charge.

• Accelerated charges radiate electromagnetic waves
• An oscillating charge produces waves of the same frequency
• An electric dipole is a basic source of electromagnetic waves

In an electromagnetic wave, the electric field E and magnetic field B are perpendicular to each other and to the direction of propagation, so the wave is transverse. For a plane wave travelling along the z-direction, E can be taken along the x-axis and B along the y-axis, each varying sinusoidally with z and t. Maxwell's equations give ω = ck, or νλ = c, and relate the field amplitudes as E₀/B₀ = c. The speed of electromagnetic waves in vacuum is c = 1/√(μ₀ε₀), which equals the speed of light obtained from optical measurements. In a material medium the speed becomes v = 1/√(με), so it depends on the electric and magnetic properties of the medium. No material medium is needed for their propagation.

• E and B are mutually perpendicular and perpendicular to the direction of propagation
• E_x = E₀ sin(kz – ωt), B_y = B₀ sin(kz – ωt)
• νλ = c and E₀/B₀ = c
• c = 1/√(μ₀ε₀) ≈ 3 × 10⁸ m/s in vacuum
• In a medium, v = 1/√(με)

Electromagnetic waves cover an infinite range of wavelengths, and different regions are known by different names. There is no sharp division between one kind and the next; the classification is based roughly on how the waves are produced and detected. In order of increasing wavelength, the spectrum runs from gamma rays, X-rays, ultraviolet rays, visible rays, infrared rays, microwaves and radio waves, stretching from about 10⁻¹² m to 10⁶ m. All these waves travel through vacuum with the same speed c, so they differ mainly in wavelength or frequency and in the way they interact with matter.

• Gamma rays< 10⁻³ nm
• X-rays1 nm to 10⁻³ nm
• Ultraviolet400 nm to 1 nm
• Visible700 nm to 400 nm
• Infrared1 mm to 700 nm
• Microwaves0.1 m to 1 mm
• Radio waves> 0.1 m

Radio waves are produced by the accelerated motion of charges in conducting wires and are used in radio and television communication. Their frequency range is about 500 kHz to 1000 MHz, with the AM band from 530 kHz to 1710 kHz, short wave bands up to 54 MHz, TV waves from 54 MHz to 890 MHz, and the FM band from 88 MHz to 108 MHz. Microwaves have frequencies in the gigahertz range and are produced by special vacuum tubes such as klystrons, magnetrons and Gunn diodes. They are used in radar systems for aircraft navigation and in microwave ovens, where the frequency is matched to the resonant frequency of water molecules. Infrared waves are produced by hot bodies and molecules and are often called heat waves because water molecules and gases such as CO₂ and NH₃ absorb them readily, increasing thermal motion. They are used in physical therapy, remote switches and Earth satellites, and play a role in the greenhouse effect.

• Radio wavesproduced by accelerated charges in aerials; detected by receiver aerials
• Microwavesproduced by klystron or magnetron valves; detected by point contact diodes
• Infraredproduced by vibration of atoms and molecules; detected by thermopiles and bolometers

Visible light is the part of the spectrum detected by the human eye, running from about 4 × 10¹⁴ Hz to 7 × 10¹⁴ Hz, or a wavelength range of about 700 nm to 400 nm. It is emitted or reflected by objects around us and gives us information about the world. Ultraviolet radiation covers wavelengths from about 400 nm down to 0.6 nm and is produced by special lamps and very hot bodies; most solar UV is absorbed in the ozone layer. It causes tanning and can harm humans, so welders use special glass goggles. X-rays cover about 10 nm down to 10⁻⁴ nm and are commonly generated by bombarding a metal target with high energy electrons; they are used in medical diagnosis and cancer treatment. Gamma rays have the shortest wavelengths, from about 10⁻¹⁰ m to less than 10⁻¹⁴ m, are produced in nuclear reactions and by radioactive nuclei, and are used in medicine to destroy cancer cells.

• Visibleemitted when electrons move to lower energy levels; detected by the eye and photocells
• Ultravioletproduced by inner shell electrons; detected by photocells and photographic film
• X-raysproduced in X-ray tubes; detected by photographic film, Geiger tubes and ionisation chambers
• Gamma raysproduced by radioactive decay of the nucleus