Electromagnetic Waves

By the end of Chapter 6 you knew two facts. Oersted showed that an electric current produces a magnetic field. Faraday showed that a changing magnetic field produces an electric field (an induced emf). Maxwell asked the obvious next question: does a changing electric field produce a magnetic field? He found that Ampere's circuital law, as it stood, gave contradictory answers for a capacitor that is being charged. To remove the contradiction he added a new term, the displacement current, and with it the law became consistent and symmetric with Faraday's law.

Once that term was in place, the four equations of electricity and magnetism (now called Maxwell's equations) had a remarkable consequence. A changing electric field produces a changing magnetic field, which produces a changing electric field, and so on. The disturbance sustains itself and travels through empty space as a wave. Maxwell calculated its speed from two laboratory constants, c = 1/√(μ₀ε₀), and got 3 × 10⁸ m s⁻¹, the measured speed of light. His conclusion in 1864 was that light itself is an electromagnetic wave.

Hertz confirmed the prediction in 1887 by producing and detecting radio-frequency electromagnetic waves in the laboratory. Jagadis Chandra Bose worked at much shorter wavelengths (about 5 mm to 25 mm), and Marconi used the waves for long-distance communication, which is the beginning of the communication age. The chapter therefore has three parts: the displacement current, the nature of the wave, and the spectrum of such waves from radio to gamma rays.

Ampere's circuital law says ∮B·dl = μ₀ i, where i is the current crossing any surface bounded by the closed loop. Now take a parallel-plate capacitor being charged by a current i(t) in the connecting wire. Choose a point P just outside the wire and a circular loop of radius r through P, centred on the wire.

Surface 1: the flat disc bounded by the loop. The wire pierces it, so ∮B·dl = μ₀ i and, by symmetry, B(2πr) = μ₀ i, giving B = μ₀ i/2πr.

Surface 2: a pot-shaped surface with the same loop as its rim, but bulging out so that its bottom lies between the plates. No wire passes through it, so no conduction current crosses it, and the law gives B = 0 at the same point P. Both cannot be right. The law is missing a term that is non-zero between the plates.

What is present between the plates is a changing electric field. For plate area A and charge Q, the field is E = Q/(ε₀A), uniform and perpendicular to the plates. The electric flux through the bottom of surface 2 is Φ_E = EA = Q/ε₀. Differentiating, dΦ_E/dt = (1/ε₀) dQ/dt, so

ε₀ dΦ_E/dt = dQ/dt = i.

Maxwell's insight: the quantity i_d = ε₀ dΦ_E/dt behaves exactly like a current. He named it the displacement current and added it to the conduction current i_c. The generalised (Ampere–Maxwell) law is

∮B·dl = μ₀(i_c + i_d) = μ₀ i_c + μ₀ε₀ dΦ_E/dt.

Check it on the two surfaces. Through surface 1 there is conduction current i and no changing flux; through surface 2 there is no conduction current but a displacement current equal to i. Both give B = μ₀ i/2πr at P. The total current i_c + i_d is the same through every surface bounded by the loop, and it is continuous around the whole circuit: conduction current in the wire, displacement current in the gap.

The physical statement examiners want in words: a changing electric field produces a magnetic field, just as a changing magnetic field produces an electric field. This is the symmetry that makes electromagnetic waves possible.

Four consequences follow, and each has been an exam question.

1. Continuity of current. Kirchhoff's junction rule holds at each plate of the capacitor provided you count the displacement current: conduction current arrives at the plate, an equal displacement current leaves through the gap. This is why an ammeter in series with a capacitor and a dc battery shows a momentary deflection while the capacitor charges, and reads zero once it is fully charged (E between the plates is then constant, so i_d = 0 and i_c = 0).

2. A capacitor conducts ac. With an ac source the charge on the plates keeps changing, so the electric field and hence the displacement current never settle to zero. The circuit carries a continuous current although no charge crosses the gap.

3. Displacement current produces a magnetic field exactly as conduction current does. For circular plates of radius R, at a point between the plates a distance r from the axis (r ≤ R), the displacement current enclosed by a circle of radius r is i_d(r²/R²) because the field is uniform. So B(2πr) = μ₀ i_d r²/R², giving B = μ₀ i_d r/(2πR²). Outside the plates (r ≥ R) the full current is enclosed and B = μ₀ i_d/2πr.

NCERT Example 8.1 uses this. Plates of radius 1 m, C = 1 nF, charged through R = 1 MΩ from a 2 V battery; find B halfway to the rim at t = 10⁻³ s. Time constant τ = RC = 10⁻³ s, so i = (V/R)e^−t/τ = 2 × 10⁻⁶ × e⁻¹ = 0.74 μA. Then B = μ₀ i r/(2πR²) = 2 × 10⁻⁷ × 0.74 × 10⁻⁶ × 0.5/1 = 0.74 × 10⁻¹³ T.

4. Maxwell's equations. With the new term, the laws of electricity and magnetism in free space are: Gauss's law for electricity ∮E·dA = Q/ε₀; Gauss's law for magnetism ∮B·dA = 0 (no magnetic monopoles); Faraday's law ∮E·dl = −dΦ_B/dt; Ampere–Maxwell law ∮B·dl = μ₀ i_c + μ₀ε₀ dΦ_E/dt. Together with the Lorentz force F = q(E + v × B), they describe every electromagnetic phenomenon in this course. The two circulation laws are now symmetric: a time-varying B gives E, a time-varying E gives B.

Maxwell's equations predict that a time-varying electric field generates a time-varying magnetic field and vice versa. Neither field can then stay confined; the pair propagates outward as an electromagnetic wave. The wave needs no material medium, which is why light reaches us from the Sun across empty space.

Which charges radiate? A stationary charge produces only an electrostatic field. A charge in uniform motion is a steady current and produces a magnetic field that does not change with time. Neither radiates. Only an accelerated charge produces a changing electric field, which produces a changing magnetic field, which in turn produces a changing electric field, and so on. So accelerated charges are the sources of electromagnetic waves.

The important case is a charge oscillating with frequency ν. The electric field it produces oscillates at ν, so does the magnetic field, and the wave that travels out has the same frequency ν. Its wavelength in vacuum is λ = c/ν. The energy carried by the wave is supplied by the source, that is, by whatever does work to keep the charge accelerating. This is why a radio transmitter needs power.

Frequency sets what you can build. Hertz (1887) used a spark-gap oscillator to produce waves of a few metres wavelength and detected them with a loop with a small gap in which sparks appeared. Even modern electronic circuits reach only about 10¹¹ Hz, the radio and microwave range. Visible light, at about 10¹⁴ Hz (yellow light is close to 6 × 10¹⁴ Hz), is far beyond any circuit; it comes from electrons within atoms and molecules. X-rays come from inner-shell electron transitions and gamma rays from the nucleus, so the source of a wave is a good guide to its band.

Take a plane wave travelling along the z-axis. The electric field is along x and the magnetic field along y:

E_x = E₀ sin(kz − ωt), B_y = B₀ sin(kz − ωt),

where k = 2π/λ is the propagation constant (wave number) and ω = 2πν is the angular frequency. The wave moves along +z with speed ω/k. Three facts about this picture carry marks.

Directions. E, B and the direction of propagation are mutually perpendicular. The wave travels along E × B. So if E is along x and B along y, propagation is along +z; if E is along y and B along z, propagation is along +x. Because the fields oscillate perpendicular to the direction of travel, electromagnetic waves are transverse. This is why light can be polarised, which Chapter 10 uses.

Phase. E and B oscillate in phase: they reach their maxima together and pass through zero together. The wave is not like a see-saw where one field is large when the other is small.

Speed and amplitude relation. Maxwell's equations give the speed in vacuum as c = 1/√(μ₀ε₀) = 3 × 10⁸ m s⁻¹, the same for every frequency. They also fix the ratio of the amplitudes: B₀ = E₀/c, i.e. E₀/B₀ = c. Given one field you can write the other immediately: same phase, same k and ω, amplitude scaled by c, direction fixed by E × B.

In a material medium of permittivity ε and permeability μ the speed becomes v = 1/√(με), which is less than c. The frequency does not change on entering a medium (it is set by the source), so the wavelength decreases to λ/n, where n = c/v is the refractive index.

What is waving in vacuum? Nothing material: the Michelson–Morley experiment found no aether, and electromagnetic waves travel through empty space with no medium at all.

An electromagnetic wave carries energy because the fields themselves store energy. In a region with electric field E the electric energy density is u_E = ½ε₀E², and in a region with magnetic field B the magnetic energy density is u_B = B²/2μ₀. In an electromagnetic wave these two are equal at every instant, not only on average.

To see this, use the amplitude relation for the wave, E = cB, together with c² = 1/(μ₀ε₀):

u_B = B²/2μ₀ = (E/c)²/2μ₀ = E²/(2μ₀c²) = E²μ₀ε₀/(2μ₀) = ½ε₀E² = u_E.

So the energy carried by the wave is shared equally between its electric and magnetic parts at every point and at every instant, and therefore on average as well. This is exactly the result NCERT Exercise 8.10(c) asks you to prove for a given numerical E₀ and ν, so practise substituting numbers into this same chain: find B₀ = E₀/c first, then show ½ε₀E₀² equals B₀²/2μ₀.

All electromagnetic waves are the same phenomenon; they differ only in frequency (or wavelength, since νλ = c in vacuum). The whole range, from wavelengths of kilometres down to less than 10⁻¹⁴ m, is the electromagnetic spectrum. Historical names label its parts, and there is no sharp boundary between one part and the next; neighbouring bands overlap. What distinguishes them in practice is how they are produced and how they are detected.

Memorise the bands in order of decreasing wavelength (increasing frequency and photon energy) with NCERT's approximate wavelength ranges:

• Radio waves: greater than 0.1 m
• Microwaves: 0.1 m to 1 mm
• Infrared: 1 mm to 700 nm
• Visible light: 700 nm to 400 nm (frequency 4 × 10¹⁴ Hz to 7 × 10¹⁴ Hz)
• Ultraviolet: 400 nm to 1 nm
• X-rays: 1 nm to 10⁻³ nm
• Gamma rays: less than 10⁻³ nm

Two quantities let you place any wave. Wavelength and frequency are linked by λ = c/ν, so a frequency of 3 × 10¹³ Hz means λ = 10⁻⁵ m = 10 μm, which is infrared. Photon energy is E = hν = hc/λ; the handy form is E (in eV) = 1240/λ (in nm). A 1 mm microwave photon has about 1.2 × 10⁻³ eV, a 620 nm red photon 2 eV, a 1 nm X-ray photon 1.24 keV and a 10⁻³ nm gamma photon 1.24 MeV. These energy scales match the sources: circuit oscillations for radio, molecular rotations and vibrations for microwave and infrared, outer-electron transitions in atoms for visible and ultraviolet, inner-shell transitions for X-rays, and nuclear transitions for gamma rays. Whatever the band, the speed in vacuum is the same c.

Radio waves (wavelength above 0.1 m) are produced by the accelerated motion of charges in conducting wires, that is, by oscillating currents in an antenna fed by an LC oscillator or a transmitter circuit. They are used in radio and television communication. NCERT gives the bands: AM radio 530 kHz to 1710 kHz, TV 54 MHz to 890 MHz, FM radio 88 MHz to 108 MHz, and cellular phones in the UHF (ultra high frequency) band. Detection is by an antenna and a tuned receiver circuit. The frequency of the emitted wave equals the frequency of the oscillating current, so a 100 MHz FM station radiates at λ = c/ν = 3 m.

Microwaves (wavelength 0.1 m to 1 mm, frequency in the gigahertz range) are short-wavelength radio waves. They are produced by special vacuum tubes, the klystron and the magnetron, and by Gunn diodes. Their three standard uses:

• Radar and aircraft navigation. Because the wavelength is short, microwaves can be sent as a narrow, directed beam with little spreading, and they reflect well from objects of aircraft size. The time delay of the echo gives the distance.
• Microwave ovens. Water molecules in food have a natural (resonant) frequency of rotation in the microwave range. The oven's frequency is chosen to match it, so water molecules absorb energy strongly, and the food heats from within through its water content while the dry container stays cooler. Ask: why match the resonant frequency? Because energy transfer is most efficient at resonance.
• Speed guns, satellite links and mobile communication.

Infrared waves (1 mm to 700 nm) lie just beyond the red end of the visible spectrum. They are produced by hot bodies and by vibrating molecules; every object above absolute zero emits them. They are called heat waves because water molecules present in most materials readily absorb infrared radiation (so do carbon dioxide, ammonia and other molecules); after absorption the molecules vibrate more energetically, their thermal motion increases, and the material warms up along with its surroundings.

Uses that NCERT lists and the board asks: infrared lamps in physical therapy (treating muscular strain); remote controls of TV sets and other household electronics, where a semiconductor light-emitting diode emits an infrared signal; infrared detectors on Earth satellites for military use and for observing the growth of crops; infrared photography and night-vision equipment.

Infrared also maintains the Earth's average temperature through the greenhouse effect. Sunlight (mostly visible) passes through the atmosphere and warms the ground. The warm ground re-radiates energy as infrared. Greenhouse gases in the atmosphere, chiefly water vapour and carbon dioxide, absorb this infrared and radiate part of it back to the surface, so the Earth stays warmer than it would otherwise be. If the Earth had no atmosphere, the infrared would escape directly to space and the average surface temperature would be lower. This 'higher or lower' question has appeared in a 3-mark set.

Visible light (700 nm to 400 nm, frequency 4 × 10¹⁴ Hz to 7 × 10¹⁴ Hz) is the part our eyes detect; the retina is a detector tuned to this band. It is emitted by atoms and molecules when their outer electrons drop from a higher to a lower energy level (the Sun, lamps, LEDs, lasers). Red lies at the long end near 700 nm and violet at the short end near 400 nm.

Ultraviolet (UV) radiation (NCERT: 4 × 10⁻⁷ m down to 6 × 10⁻¹⁰ m, i.e. 400 nm to about 0.6 nm) is produced by special lamps and by very hot bodies. The Sun is an important source, and so is a welding arc, which is why welders wear special goggles or face masks with glass windows that absorb UV. Most of the Sun's UV is absorbed by the ozone layer at the top of the stratosphere; that thin layer is crucial for human survival because UV in large quantities harms living tissue, causing skin cancer and cataracts. In moderate doses UV tans the skin and helps the body make vitamin D. Uses: LASIK eye surgery (UV lasers can be focused into very narrow beams), and UV lamps to kill germs in water purifiers.

X-rays (about 10⁻⁸ m, 10 nm, down to 10⁻¹³ m, 10⁻⁴ nm) are produced when a metal target is bombarded by high-energy electrons in an X-ray tube: the sudden deceleration of the electrons and inner-shell transitions in the target atoms produce the radiation. They penetrate soft tissue but are stopped by bone, so they are used as a diagnostic tool in medicine, and also to treat certain forms of cancer. Because they damage living tissue, exposure must be kept low. They are also used to study crystal structure (X-ray diffraction) and in security scanning.

Gamma rays (about 10⁻¹⁰ m to below 10⁻¹⁴ m) have the shortest wavelength and highest frequency, hence the largest penetrating power. They are produced in nuclear reactions and are emitted by radioactive nuclei during decay. In medicine they are used to destroy cancer cells (radiotherapy) and for imaging with radioactive tracers. They cannot be produced by any electronic circuit; the source is always nuclear.

A pattern for the last three bands: the source moves inward in the atom as the frequency rises, from outer electrons (UV) to inner-shell electrons (X-rays) to the nucleus (gamma).