Electromagnetic Waves: NEET notes

1. The charging capacitor puzzle

NCERT §8.1, §8.2

• A current produces a magnetic field (Chapter 4), and a magnetic field that changes with time produces an electric field (Chapter 6). Maxwell argued that the converse also holds: an electric field that changes with time produces a magnetic field.
• He was led there by a flaw in Ampere's circuital law, ∮B·dl = μ₀i, which appears when the law is applied just outside a capacitor that is being charged by a time-dependent current i(t).
• Take a circular loop of radius r around the wire leading to the capacitor, perpendicular to it and centred on it. By symmetry B runs along the loop with the same size at every point, so the left side is B(2πr) and the law gives B(2πr) = μ₀i(t).
• Ampere's law lets us use any surface whose edge is the loop. A flat disc on the loop is pierced by the wire, so the current i crosses it.
• Now choose a pot-shaped surface with the same rim, or one shaped like an open tiffin box with a flat bottom, whose bottom lies in the gap between the plates. No charge crosses such a surface anywhere, so the right side becomes zero while the left side is unchanged.
• One way B at the point P is non-zero, the other way it is zero. That contradiction means the law was missing a term, one that gives the same B at P whichever surface is used.
• The clue is what does cross the surface in the gap: the electric field. With plate area A and charge Q, the field between the plates is E = (Q/A)/ε₀, perpendicular to the plates, uniform over the area A and zero outside it.

2. Displacement current

NCERT §8.2

• The electric flux through the flat bottom S in the gap is ΦE = EA = Q/ε₀, using Gauss's law.
• As the charge changes, dΦE/dt = (1/ε₀) dQ/dt = i/ε₀, so ε₀ dΦE/dt = i. This is the missing term.
• The current carried by moving charges in conductors is the conduction current ic. The new term, id = ε₀ dΦE/dt, is the displacement current (Maxwell's displacement current), caused by a changing electric field.
• The total current through a surface is i = ic + id. Outside the plates ic = i and id = 0; in the gap ic = 0 and id = i. So the total is the same for every surface on the loop and B at P comes out the same.
• The generalised law is the Ampere-Maxwell law: ∮B·dl = μ₀ic + μ₀ε₀ dΦE/dt.
• In every respect the displacement current acts like a conduction current; above all, it produces a magnetic field just as a real current would. The field measured at a point M between the plates equals the field just outside at P.
• A steady field in a conducting wire does not change, so there id = 0. In the charging capacitor ic and id sit in different regions; in most real media both are present together, because no medium conducts or insulates perfectly.
• There can be large regions of space with no conduction current at all but a displacement current from a changing E. A magnetic field exists there even though no current source is nearby.

3. Maxwell's equations

NCERT §8.1, §8.2

• Faraday's law can be read as a statement about fields. An emf between two points is the work done per unit charge, so an induced emf means an electric field: a magnetic field changing in time produces an electric field.
• The displacement current gives the matching statement: an electric field changing in time produces a magnetic field. Time-varying electric and magnetic fields generate each other.
• This makes the laws of electricity and magnetism more symmetric, though not perfectly: no magnetic monopoles, sources of B analogous to electric charges, are known.
• The four Maxwell equations in vacuum are Gauss's law ∮E·dA = Q/ε₀, Gauss's law for magnetism ∮B·dA = 0, Faraday's law ∮E·dl = −dΦB/dt, and the Ampere-Maxwell law ∮B·dl = μ₀ic + μ₀ε₀ dΦE/dt.
• Together with the Lorentz force formula, these equations express all the basic laws of electromagnetism.
• Their most important prediction is the electromagnetic wave: coupled electric and magnetic fields varying in time and travelling through space.
• The predicted speed came out very close to 3 × 10⁸ m/s, the speed of light found from optical measurements. So light is an electromagnetic wave, and Maxwell's work unified electricity, magnetism and light.
• Hertz demonstrated electromagnetic waves experimentally, and Marconi and others turned them into the technology of communication.

4. Sources of electromagnetic waves

NCERT §8.3.1

• Charges at rest produce only electrostatic fields. Charges moving uniformly (steady currents) produce magnetic fields that do not change with time. Neither is a source of electromagnetic waves.
• Accelerated charges radiate electromagnetic waves. This follows from Maxwell's theory; the proof is beyond the syllabus.
• An oscillating charge is an accelerating charge. It sets up an oscillating electric field, which sets up an oscillating magnetic field, which in turn sets up an oscillating electric field, and so on: the two fields keep regenerating each other as the wave moves out.
• The frequency of the wave equals the frequency with which the charge oscillates. The energy the wave carries comes from the source, the accelerated charge. An oscillating electric dipole is a basic source.
• Testing Maxwell with an ac circuit at the frequency of light is impossible: yellow light has a frequency of about 6 × 10¹⁴ Hz, while even modern electronic circuits barely reach about 10¹¹ Hz. So the first demonstration used low-frequency radio waves.
• Hertz produced and detected electromagnetic waves in the laboratory in 1887, with wavelengths of the order of a few metres, confirming Maxwell's prediction.
• Seven years after Hertz, Jagdish Chandra Bose, working in Calcutta (now Kolkata), produced and observed much shorter waves, 25 mm to 5 mm, also within the laboratory.
• Around the same time Guglielmo Marconi in Italy sent electromagnetic waves over many kilometres, the start of communication by electromagnetic waves.
• The wavelength radiated often matches the size of the radiating system: an aerial radiates best at a wavelength about its own length, and nuclei give gamma rays of 10⁻¹⁴ to 10⁻¹⁵ m. Visible light from atoms is the exception, much longer than an atom.

5. Nature of electromagnetic waves

NCERT §8.3.2

• The E and B fields of an electromagnetic wave are at right angles to each other, and both are at right angles to the direction of travel: the wave is transverse. The capacitor hints at this: E between the plates is perpendicular to them, and the B it produces runs in circles parallel to them.
• For a plane wave travelling along z with E along x and B along y: Ex = E₀ sin(kz − ωt) and By = B₀ sin(kz − ωt). Both vary sinusoidally with position and time and are in phase.
• Here k = 2π/λ is the size of the wave vector (propagation vector) k, whose direction is the direction of travel, and ω is the angular frequency. The wave moves with speed ω/k.
• Maxwell's equations give ω = ck with c = 1/√(μ₀ε₀). Written with ν = ω/2π and λ = 2π/k this is νλ = c.
• The amplitudes are related by B₀ = E₀/c. Because c is so large, the magnetic field in tesla is a tiny number next to the electric field in V/m.
• The wave travels in the direction of E × B. Given two of the three directions, this fixes the third.
• Example 8.1: a 25 MHz wave travels along x, and at some point E = 6.3 ĵ V/m. Then B = E/c = 6.3/(3 × 10⁸) = 2.1 × 10⁻⁸ T, and since ĵ × k̂ = î, B = 2.1 × 10⁻⁸ k̂ T.
• Example 8.2: By = (2 × 10⁻⁷ T) sin(0.5 × 10³x + 1.5 × 10¹¹t). Then λ = 2π/(0.5 × 10³) m = 1.26 cm and ν = (1.5 × 10¹¹)/2π = 23.9 GHz. E₀ = B₀c = 60 V/m, and E lies along z: Ez = 60 sin(0.5 × 10³x + 1.5 × 10¹¹t) V/m.

6. Speed in vacuum and in media

NCERT §8.3.2

• Electromagnetic waves are self-sustaining oscillations of E and B in free space. Unlike sound or waves on a string, they need no material medium: nothing material vibrates.
• In vacuum the speed is c = 1/√(μ₀ε₀). With μ₀ = 4π × 10⁻⁷ T m A⁻¹ and ε₀ = 8.854 × 10⁻¹² C² N⁻¹ m⁻², this is about 3 × 10⁸ m/s.
• In a material medium of permittivity ε and permeability μ these replace ε₀ and μ₀, and the speed becomes v = 1/√(με). The speed of light depends on the electric and magnetic properties of the medium.
• The refractive index of one medium with respect to another equals the ratio of the speeds of light in them (taken up in the next chapter).
• Measured with waves of many different wavelengths, the vacuum speed does not depend on wavelength; the results agree to a few m/s out of 3 × 10⁸ m/s. The value is known so precisely that it is used to define the standard of length.
• All electromagnetic waves, from radio waves to gamma rays, travel at the same speed c in vacuum.
• Electromagnetic waves carry energy from one place to another. Radio and TV signals carry energy, and light carries the sun's energy to the earth, which makes life on earth possible.

7. The electromagnetic spectrum

NCERT §8.4

• When Maxwell made his prediction, visible light was the only familiar electromagnetic wave; ultraviolet and infrared were barely established. X-rays and gamma rays were discovered by the end of the nineteenth century.
• Arranging electromagnetic waves by frequency gives the electromagnetic spectrum. In order of decreasing wavelength: radio waves, microwaves, infrared, visible light, ultraviolet, X-rays, gamma rays.
• There is no sharp boundary between one kind and the next, and neighbouring regions overlap. The names follow roughly how each kind is produced or detected.
• The spectrum runs, in principle, over an unlimited range; the named regions span about 10⁻¹² m (gamma rays) to 10⁶ m (long radio waves).
• Since all of them have the same speed in vacuum, the only basic difference between the types is their wavelength or frequency. That is why they interact with matter so differently.
• They act on matter through their electric and magnetic fields, which set the charges in all matter oscillating. How strongly a wave is absorbed or scattered depends on its wavelength and on the atoms and molecules of the medium.
• Wavelength ranges (Table 8.1): radio above 0.1 m; microwave 0.1 m to 1 mm; infrared 1 mm to 700 nm; light 700 nm to 400 nm; ultraviolet 400 nm to 1 nm; X-rays 1 nm to 10⁻³ nm; gamma rays below 10⁻³ nm.

8. Radio waves and microwaves

NCERT §8.4.1, §8.4.2

• Radio waves are produced by accelerated charges in conducting wires: rapid acceleration and deceleration of electrons in aerials. They are detected by receivers' aerials and used in radio and television communication.
• Their usual range is roughly 500 kHz to 1000 MHz. AM (amplitude modulated) radio uses 530-1710 kHz; short-wave bands go up to 54 MHz; television uses 54-890 MHz; FM (frequency modulated) radio uses 88-108 MHz.
• Cellular phones carry voice on radio waves in the ultrahigh frequency (UHF) band.
• Microwaves are short-wavelength radio waves with frequencies in the gigahertz range, produced by special vacuum tubes (klystrons, magnetrons) and by Gunn diodes, and detected by point-contact diodes.
• Their short wavelength suits them to radar for aircraft navigation. Radar is also the basis of the speed guns that time fast balls, tennis serves and vehicles.
• In a microwave oven the frequency is chosen to match the resonant frequency of water molecules, so energy passes efficiently from the waves to the kinetic energy of the molecules and any food containing water gets hot.
• With λ = c/ν: the FM band runs from 3 × 10⁸/88 × 10⁶ = 3.41 m down to 3 × 10⁸/108 × 10⁶ = 2.78 m, and 530 kHz on the AM band has λ = 566 m.

9. Infrared and visible light

NCERT §8.4.3, §8.4.4

• Infrared waves are produced by hot bodies and by molecules, through the vibration of atoms and molecules. The band lies next to the red (low-frequency, long-wavelength) end of the visible spectrum, from about 1 mm to 700 nm.
• They are sometimes called heat waves. Water molecules, present in most materials, absorb infrared readily (so do CO₂, NH₃ and many others); the absorbed energy increases thermal motion, so the material and its surroundings warm up.
• Infrared, being lower in frequency than light, sets whole atoms and molecules vibrating, not only electrons. This raises the internal energy and hence the temperature.
• Uses: infrared lamps in physical therapy; infrared detectors on earth satellites, for military purposes and to watch the growth of crops; and infrared-emitting LEDs in the remote controls of TV sets and music systems. Detectors include thermopiles, bolometers and infrared photographic film.
• Greenhouse effect: visible sunlight passes fairly easily through the air and is absorbed by the ground, which re-radiates the energy as longer-wavelength infrared. Greenhouse gases such as carbon dioxide and water vapour trap this infrared and keep the earth warm.
• Visible light is the band the human eye detects: about 4 × 10¹⁴ Hz to about 7 × 10¹⁴ Hz, or wavelengths of about 700 nm to 400 nm. It is emitted when electrons in atoms move from a higher energy level to a lower one, and detected by the eye, photocells and photographic film.
• Other animals see other ranges: snakes can detect infrared, and the visible range of many insects reaches well into the ultraviolet.
• Our eyes are most sensitive near the centre of the sun's wavelength distribution, having evolved to see best the wavelengths the sun gives out most strongly.

10. Ultraviolet, X-rays and gamma rays

NCERT §8.4.5, §8.4.6, §8.4.7

• Ultraviolet runs from about 4 × 10⁻⁷ m (400 nm) down to 6 × 10⁻¹⁰ m (0.6 nm). Special lamps and very hot bodies emit it, and the sun is a major source. In atoms it comes from inner-shell electrons dropping to a lower level, and it is detected by photocells and photographic film.
• Most of the sun's ultraviolet is absorbed by the ozone layer, at a height of about 40-50 km. Depletion of this layer by chlorofluorocarbons (CFCs) such as freon is a matter of international concern.
• Large doses of UV are harmful. Exposure makes the skin produce more melanin, which causes tanning. Ordinary glass absorbs UV, so no one tans or burns through a glass window, and welders wear glass goggles or face masks against the UV of the welding arc.
• Its short wavelength lets UV be focused into very narrow beams for precise work such as LASIK eye surgery. UV lamps kill germs in water purifiers.
• X-rays lie beyond the UV, from about 10⁻⁸ m (10 nm) down to 10⁻¹³ m (10⁻⁴ nm). A common way to make them is to strike a metal target with high-energy electrons (an X-ray tube). They are detected by photographic film, Geiger tubes and ionisation chambers.
• X-rays are used to diagnose illness and to treat some forms of cancer. They damage or destroy living tissue, so unnecessary or excessive exposure must be avoided.
• Gamma rays occupy the top of the frequency range, with wavelengths from about 10⁻¹⁰ m to below 10⁻¹⁴ m. They come from nuclear reactions and radioactive nuclei, and are used in medicine to destroy cancer cells.
• The regions overlap: UV is quoted down to 0.6 nm while X-rays are quoted from about 10 nm, and the X-ray range (down to 10⁻¹³ m) reaches well into the gamma range (from 10⁻¹⁰ m). The boundaries follow how the radiation is produced, not a sharp wavelength.

Must-know facts

1. Displacement current id = ε₀ dΦE/dt; it produces a magnetic field exactly as a conduction current does.
2. In a charging capacitor: ic = i in the wires, id = i in the gap; total current is continuous.
3. Ampere-Maxwell law: ∮B·dl = μ₀ic + μ₀ε₀ dΦE/dt.
4. Maxwell's four equations: Gauss (E), Gauss (B), Faraday, Ampere-Maxwell.
5. ∮B·dA = 0 because no magnetic monopoles are known.
6. Accelerated charges radiate; charges at rest or in uniform motion do not.
7. The frequency of the wave equals the frequency of the oscillating charge.
8. Hertz produced and detected em waves in 1887; J. C. Bose produced waves of 25 mm to 5 mm.
9. E, B and the direction of travel are mutually perpendicular; travel is along E × B.
10. Ex = E₀ sin(kz − ωt), By = B₀ sin(kz − ωt): E and B in phase.
11. c = 1/√(μ₀ε₀) ≈ 3 × 10⁸ m/s; ω = ck; νλ = c; B₀ = E₀/c.
12. In a medium v = 1/√(με).
13. em waves need no medium, and all of them travel at c in vacuum.
14. Spectrum by decreasing wavelength: radio, microwave, infrared, visible, UV, X-rays, gamma.
15. Visible: about 400-700 nm, 4 × 10¹⁴ to 7 × 10¹⁴ Hz.
16. FM 88-108 MHz, AM 530-1710 kHz, TV 54-890 MHz.
17. Microwaves: klystron, magnetron, Gunn diode; radar and ovens.
18. Infrared = heat waves; used in remotes and physical therapy; causes the greenhouse effect.
19. Ozone layer (40-50 km) absorbs most solar UV; glass absorbs UV.
20. X-rays: electrons striking a metal target; gamma rays: radioactive nuclei.

Common traps

Formulas

Key terms

Conduction current

Current carried by charges moving through a conductor.

Displacement current

ε₀ times the rate of change of electric flux; acts as a source of B like a real current.

Ampere-Maxwell law

Ampere's law with the displacement current added to the conduction current.

Maxwell's equations

The four laws of electromagnetism: Gauss's laws for E and B, Faraday's law and the Ampere-Maxwell law.

Electromagnetic wave

Coupled E and B fields oscillating at right angles to each other and to the direction of travel.

Wave vector (k)

Vector of size 2π/λ pointing along the direction of travel.

Transverse wave

A wave whose oscillations are perpendicular to the direction it travels.

Electromagnetic spectrum

The full range of em waves arranged by frequency or wavelength.

Microwaves

Short-wavelength radio waves in the GHz range, from 0.1 m to 1 mm.

Infrared (heat waves)

Waves from 1 mm to 700 nm that set molecules vibrating and so heat matter.

Greenhouse effect

Warming of the earth as CO₂ and water vapour trap the infrared the ground re-radiates.

Ozone layer

Layer about 40-50 km up that absorbs most of the sun's ultraviolet.

Klystron, magnetron

Vacuum tubes that generate microwaves.

Gunn diode

A solid-state source of microwaves.