Chapter 8: Electromagnetic Waves – Study Notes

• 8.1 Introduction

• Maxwell showed that a time-varying electric field can generate a magnetic field, just as a moving charge/current can generate a magnetic field.

• Ampere’s circuital law needed a correction to be logically consistent when charging a capacitor; this led Maxwell to introduce the displacement current.

• Displacement current is not a real current of moving charges but a term proportional to the time rate of change of electric flux, ε0 dΦ_E/dt, which completes Ampere’s law.

• Maxwell’s equations couple electric and magnetic fields and their sources (charge and current densities). Together with the Lorentz force, they express the fundamental laws of electromagnetism.

• A key prediction of Maxwell’s equations is the existence of electromagnetic waves: coupled, time-varying electric and magnetic fields that propagate through space.

• The speed of these waves, as predicted by the equations, matches the speed of light measured optically, c ≈ 3 × 10^8 m/s, suggesting light is an electromagnetic wave.

• Hertz experimentally demonstrated electromagnetic waves in 1885; later developments by Bose and Marconi led to modern communication technologies.

• This chapter discusses the need for displacement current, the qualitative description of EM waves, the broad EM spectrum, and related concepts.

$\oint<em>C \mathbf{B}\cdot d\boldsymbol{\ell} = \mu</em>0 I<em>{enc} + \mu</em>0 \epsilon<em>0 \frac{d\Phi</em>E}{dt}$

8.2 Displacement Current

• Recap: A steady current in a conductor produces a magnetic field. A changing electric field also produces a magnetic field (the displacement current).

• Consider charging a parallel-plate capacitor C in a circuit with a time-varying current i(t).

• Ampere’s law around a loop outside the capacitor would give a nonzero B if only the conduction current in the wire were considered, but a surface bounded by the loop that passes between the plates would yield zero current through it, leading to a contradiction.

• The resolution is that a changing electric flux between the plates produces a magnetic effect equivalent to a current: the displacement current.

• Electric flux through a surface S between the plates (area A, charge Q on plates):
  $\Phi<em>E = \int</em>S \mathbf{E}\cdot d\mathbf{A} = \frac{Q}{\epsilon_0}.$

• If Q changes with time (i.e., charging the capacitor), then
  $\frac{d}{dt} \Phi<em>E = \frac{1}{\epsilon</em>0} \frac{dQ}{dt} = \frac{i}{\epsilon_0}.$

• Therefore, the displacement current is
  $I<em>d = \epsilon</em>0 \frac{d\Phi_E}{dt} = i.$

• The total current through any surface bounded by the loop is the sum of the conduction current and the displacement current:
  $i = i<em>c + I</em>d = i<em>c + \epsilon</em>0 \frac{d\Phi_E}{dt}.$

• Generalised Ampere–Maxwell law (same form as Ampere’s law but with total current):
  $\oint<em>C \mathbf{B}\cdot d\boldsymbol{\ell} = \mu</em>0 \left(i<em>c + \epsilon</em>0 \frac{d\Phi<em>E}{dt}\right) = \mu</em>0 i_{tot}.$

• Outside the capacitor plates: conduction current exists (ic = i), displacement current term zero (dΦE/dt = 0).

• Between the plates: conduction current is zero (ic = 0), but dΦE/dt ≠ 0, so displacement current I_d = i, and the magnetic field is the same as just outside.

• The displacement current is a real source of magnetic fields, so the Ampere–Maxwell law is symmetric with respect to changing electric and magnetic fields.

• Physical picture: the displacement current makes the magnetic field continuous and well-defined for all surfaces bounded by the same loop, eliminating the Kapitza-like contradiction.

• Consequences of the displacement current:

  • Time-varying electric fields can exist in regions with no physical conduction current.

  • Electromagnetic waves can propagate through vacuum, since changing E and B fields sustain each other in time and space.

  • The symmetry with Faraday’s law (changing magnetic field induces an electric field) is enhanced.

8.3 Electromagnetic Waves

8.3.1 Sources of electromagnetic waves

• Stationary charges emit only electrostatic fields; charges in uniform motion (steady currents) produce static magnetic fields but not radiation.

• Accelerated charges radiate electromagnetic waves. Rough reasoning: an oscillating charge (a simple accelerating charge) creates an oscillating E field, which in turn creates an oscillating B field, and so on, producing a propagating wave in space.

• The frequency of the EM wave equals the frequency of the charge’s oscillation.

• Energy carried by the wave comes from the energy supplied by the source; the oscillating charge loses energy to the radiated field.

• In practice, producing visible light waves directly by oscillating current is difficult because visible frequencies are ~6 × 10^14 Hz, far higher than typical electronic circuit frequencies. This is why early demonstrations used lower-frequency radio waves (Hertz) to test Maxwell’s theory.

• Historical note: Hertz demonstrated EM waves in the radio region; Bose and Marconi contributed to shorter wavelengths and long-distance radio communication, respectively.

8.3.2 Nature of electromagnetic waves

• In an EM wave in vacuum, electric and magnetic fields are perpendicular to each other and to the direction of propagation.

• For a plane wave propagating in the z-direction, with Ex along x and By along y, we have:
  $E<em>x = E</em>0 \sin(kz - \omega t), \qquad B<em>y = B</em>0 \sin(kz - \omega t).$

• The wave relations:

  • Wavenumber and angular frequency: $k = \frac{2\pi}{\lambda}, \qquad \omega = 2\pi \nu.$

  • Phase velocity: $v = \frac{\omega}{k} = c,$ in vacuum, with $c = \frac{1}{\sqrt{\mu<em>0 \epsilon</em>0}} \approx 3 \times 10^8 \text{ m s}^{-1}.$

  • Relation between E and B amplitudes: $B<em>0 = \frac{E</em>0}{c}.$

  • Frequency-wavelength product: $\nu \lambda = c.$

• In a material medium, the speed is reduced to
  $v = \frac{1}{\sqrt{\mu \epsilon}}.$

• Electric and magnetic field magnitudes in EM waves satisfy E and B are in phase and related by the same propagation constant.

• Light is an EM wave; EM waves in vacuum propagate at speed c independent of wavelength; in media, the speed depends on the medium’s properties (permittivity and permeability).

8.3.3 Additional remarks

• EM waves are self-sustaining oscillations of E and B fields in free space; no material medium is required for propagation in vacuum.

• The energy carried by EM waves can be quantified by energy densities: uE = (1/2) ε0 E^2 and uB = (1/2) (B^2 / μ0) with B = μ0 H.

• The constancy of c in vacuum is a fundamental constant and underpins metrology (defining length standards via light).

8.4 Electromagnetic Spectrum

• Maxwell’s theory predicts EM waves across an enormous range of wavelengths; no sharp boundaries between regions; classifications are based on production/detection and practical uses.

8.4.1 Radio waves

• Produced by accelerated charges in conductors (circuits); used in radio and TV.

• Frequency range roughly from 500 kHz to about 1000 MHz.

• Sub-bands: AM 530 kHz–1710 kHz; short-wave up to ~54 MHz; TV 54–890 MHz; FM 88–108 MHz; cellular communications in the UHF band.

• Transmission and reception in these bands are discussed in Chapter 15.

8.4.2 Microwaves

• Frequency in the GHz range; produced by vacuum tubes such as klystrons, magnetrons, and Gunn diodes.

• Applications include radar (aircraft navigation), speed guns for sports and traffic, microwave ovens (heating via resonant absorption by water molecules).

8.4.3 Infrared waves

• Produced by hot bodies and molecular vibrations; closely related to heat.

• Role in greenhouse effect: Earth’s surface absorbs visible light and radiates infrared; greenhouse gases trap IR and warm the planet.

• Detectors: IR detectors used in Earth observation satellites; IR thermography and remote sensing; IR LEDs and remote controls.

8.4.4 Visible rays

• The spectrum visible to the human eye: roughly from 700 nm to 400 nm (about 4 × 10^14 Hz to 7 × 10^14 Hz).

• Provides information about the world around us; other animals have different sensitivities.

8.4.5 Ultraviolet rays

• Wavelengths from about 400 nm down to 6 × 10^-10 m (0.6 nm).

• Produced by very hot bodies and UV lamps; the Sun is a major source.

• UV is largely absorbed by the ozone layer; UV can cause tanning and skin damage, and it is absorbed by ordinary glass (so glass windows block part of UV).

• Applications include LASIK surgery and germicidal UV lamps.

• Ozone layer protects against high-energy UV; depletion by CFCs is an environmental concern.

8.4.6 X-rays

• Wavelengths from about 10^-8 m (10 nm) down to 10^-13 m (0.1 Å).

• Generated by X-ray tubes or nuclear processes; used in medical imaging and cancer treatment.

• Care must be taken to avoid overexposure due to ionising effects.

8.4.7 Gamma rays

• Wavelengths from about 10^-10 m down to <10^-14 m (much higher frequencies).

• Produced in nuclear reactions and radioactive decay; used in medicine (cancer therapy) and industry.

Table 8.1 (Types of EM waves)

• Radio: Wavelength > 0.1 m; Production: Rapid acceleration and deceleration of electrons in aerials; Detection: Receiver’s aerials.

• Microwave: 0.1 m to 1 mm; Production: Klystron valve or magnetron valve; Detection: Point-contact diodes.

• Infra-red: 1 mm to 700 nm; Production: Vibration of atoms and molecules; Detection: Bolometer, infrared photographic film.

• Visible light: 700 nm to 400 nm; Production: Electrons in atoms emit light when moving between energy levels; Detection: The eye; Photocells; Photographic film.

• Ultraviolet: 400 nm to 1 nm; Production: Inner shell electrons in atoms moving between energy levels; Detection: Photocells; Photographic film.

• X-rays: 1 nm to 10^-3 nm; Production: X-ray tubes or inner-shell electrons; Detection: Photographic film; Geiger tubes; Ionisation chambers.

• Gamma rays: <10^-3 nm; Production: Radioactive decay of the nucleus; Detection: Photographic film; -do-.

8.5 SUMMARY

• The displacement current term ε0 dΦ_E/dt fixes inconsistencies in Ampere’s law and acts as a source of magnetic field, making electricity and magnetism symmetrical in time-varying scenarios.

• Accelerating charges radiate EM waves; the emitted wave frequency matches the charge’s oscillation frequency; energy is drawn from the source.

• EM waves can be produced and detected across a broad spectrum, from radio to gamma rays; Hertz first demonstrated EM waves in the lab; EM waves propagate in vacuum with speed c, independent of wavelength.

• In vacuum, Ex and By components in a plane wave propagate in such a way that E, B, and the direction of propagation are mutually perpendicular, with E and B in phase.

• In a medium, the speed is v = 1/√(µε).

• The energy carried by EM waves is shared between electric and magnetic fields according to the relations E and B satisfy in vacuum, and the spectrum extends over many orders of magnitude with practical uses tied to production/detection methods.

8.6 POINTS TO PONDER

• The basic difference between EM waves of different types lies in their wavelengths or frequencies, yet they all travel in vacuum at the same speed; their interaction with matter varies.

• Accelerated charges radiate EM waves; the wavelength often correlates with the characteristic size of the radiating system (e.g., gamma from nuclei, X-rays from heavy atoms, radio waves from circuits, visible light from atomic transitions).

• Infrared waves (lower frequencies than visible) involve vibrational modes of whole molecules, manifesting as heat.

• The peak sensitivity of human vision roughly matches the solar spectrum distribution because humans evolved to be sensitive to the most intense wavelengths from the Sun.

Exercises (8.1 to 8.10)

8.1

• Figure 8.5 shows a capacitor made of two circular plates each of radius 12 cm, separated by 5.0 cm. The capacitor is charged by an external source. The charging current is constant and equal to 0.15 A.

  • (a) Calculate the capacitance and the rate of change of the potential difference between the plates.

  • (b) Obtain the displacement current across the plates.

  • (c) Is Kirchhoff’s first rule (junction rule) valid at each plate of the capacitor? Explain.

8.2

• A parallel-plate capacitor with circular plates radius R = 6.0 cm has C = 100 pF. It is connected to a 230 V AC supply with angular frequency ω = 300 rad s^-1.

  • (a) What is the RMS value of the conduction current?

  • (b) Is the conduction current equal to the displacement current?

  • (c) Determine the amplitude of B at a point 3.0 cm from the axis between the plates.

8.3

• What physical quantity is the same for X-rays of wavelength 10^-10 m, red light of wavelength 6800 Å (6.8 × 10^-7 m), and radio waves of wavelength 500 m?

8.4

• A plane EM wave travels in vacuum along the z-direction. What can you say about the directions of its electric and magnetic field vectors? If the frequency is 30 MHz, what is its wavelength?

8.5

• A radio can tune in to any station in the 7.5 MHz to 12 MHz band. What is the corresponding wavelength band?

8.6

• A charged particle oscillates about its mean position with a frequency of 10^9 Hz. What is the frequency of the emitted electromagnetic wave?

8.7

• The amplitude of the magnetic field part of a harmonic EM wave in vacuum is B0 = 510 nT. What is the amplitude of the electric field part of the wave?

8.8

• Suppose E0 = 120 N C^-1 and frequency ν = 50.0 MHz.

  • (a) Determine B0, ω, k, and v.

  • (b) Write expressions for E and B.

8.9

• Using E = hν (photon energy), obtain the photon energy in eV for different parts of the EM spectrum. Explain how these photon energies relate to the likely sources of the radiation.

8.10

• In a plane EM wave, E oscillates sinusoidally at frequency ν = 2.0 × 10^10 Hz and amplitude E0 = 48 V m^-1.

  • (a) What is the wavelength of the wave?

  • (b) What is the amplitude of the oscillating magnetic field?

  • (c) Show that the time-averaged energy density of the E field equals that of the B field. (Use c = 3 × 10^8 m s^-1.)