Physics · Ch 5 — Electromagnetic Waves

Production and Properties of Electromagnetic Waves

5.2.1

Production and Properties of Electromagnetic Waves

Maxwell's prediction of electromagnetic waves was experimentally confirmed by Heinrich Hertz in 1888 using an apparatus with two small spherical metal electrodes connected to an induction coil with a very large number of turns, built to generate a very high electromotive force. As the coil charges the electrodes to a high potential, the air between them ionises and a spark (an electrical discharge) jumps across the gap; this discharge is detected at a separate ring-shaped receiver electrode placed some distance away, confirming that energy is being transmitted through the intervening empty space as a wave -- what Hertz called, and what we now call, an electromagnetic wave. Crucially, when Hertz rotated the receiver by 90 $^\circ$ , no spark was observed at all, which confirmed that the wave is transverse (a longitudinal wave would have shown no such directional sensitivity); Hertz also measured the speed of these waves and found it equal to the known speed of light, $3\times10^8$ m/s, exactly as Maxwell's theory predicted. From this and further analysis, the complete list of properties of electromagnetic waves can be stated: (1) they are produced by any accelerated (i.e. non-uniformly-moving) electric charge; (2) they require no medium for propagation, making them non-mechanical waves, unlike sound or water waves; (3) they are transverse in nature -- the oscillating electric field vector $\vec E$ , the oscillating magnetic field vector $\vec B$ , and the propagation direction are all mutually perpendicular to one another; (4) in free space (vacuum) they all travel at the same speed $c=1/\sqrt{\mu_0\epsilon_0}\approx3\times10^8$ m/s, where $\epsilon_0$ is the permittivity and $\mu_0$ the permeability of free space; (5) inside a medium of permittivity $\epsilon$ and permeability $\mu$ , they travel slower, at $v=1/\sqrt{\mu\epsilon}$ , and this speed is related to the medium's refractive index by $n=c/v=\sqrt{\mu_r\epsilon_r}$ , where $\epsilon_r$ (also called the dielectric constant) and $\mu_r$ are the medium's relative permittivity and relative permeability; (6) they are not deflected by either an electric or a magnetic field, since they carry no net electric charge themselves; (7) like all waves, they can show interference, diffraction and polarisation; (8) they carry not just energy but also linear momentum and (perhaps surprisingly) angular momentum, exactly like material particles. Explicitly, if an electromagnetic wave propagates along the z-direction with the electric field oscillating along x and the magnetic field oscillating along y, the two fields can be written $E_x=E_0\sin(kz-\omega t)$ and $B_y=B_0\sin(kz-\omega t)$ , where $E_0$ and $B_0$ are the amplitudes, $k$ is the wave number, $\omega$ is the angular frequency (equal for both fields, …

Figure 5.7(a) Heinrich Rudolf Hertz (b) Schematic diagram of Hertz apparatus

What this figure shows. Panel (a) is a portrait photograph of the German physicist Heinrich Rudolf Hertz, who in 1888 became the first person to experimentally generate and detect electromagnetic waves. Panel (b) is a schematic circuit diagram of his transmitter-receiver apparatus: an induction coil with a very large number of turns builds up a very high emf across two small spherical metal electrodes (marked + and -) connected to larger spheres, which together form the transmitter; a separate ring-shaped receiver electrode, not a fully closed loop, is placed some distance away. When the transmitter's coil charges its electrodes to a high enough potential, the air between them ionises and a spark jumps across the gap, and this discharge is shown radiating outward as waves that induce a matching spark at the distant receiver -- the schematic that let Hertz …

Figure 5.8Electromagnetic wave -- transverse wave

What this figure shows. A three-dimensional Cartesian sketch with x, y and z axes drawn from a common origin. An oscillating electric field vector $\vec E$ is shown along the y-axis as a sinusoidal curve in the x-y plane, and an oscillating magnetic field vector $\vec B$ is shown along the z-axis as a sinusoidal curve in the x-z plane, with both curves sharing the same x-axis and oscillating in phase with each other. A separate arrow labelled "Direction of propagation" points along the positive x-axis, illustrating the defining geometric feature of an electromagnetic wave: the electric field, the magnetic field, and the direction the wave travels in are all mutual …

Misc Example 5.2Refractive index from relative permeability and permittivity

Worked out. A medium has relative magnetic permeability $\mu_r=2.5$ and relative electrical permittivity (dielectric constant) $\epsilon_r=2.25$ , and the task is to find its refractive index. Since the speed of an electromagnetic wave in a medium is $v=1/\sqrt{\mu\epsilon}=c/\sqrt{\mu_r\epsilon_r}$ and the refractive index is defined as $n=c/v$ , it follows directly that $n=\sqrt{\mu_r\epsilon_r}$ . Substituting the given values, $n=\sqrt{2.25\times2.5}=\sqrt{5.625}\approx2.37$ . This example demonstrates the direct link between a medium's electromagnetic properties (how strongly it responds to electric and magnetic fields) and its familiar optical property, the refractive index -- both are really just two ways of describing how much slowe …

Misc Example 5.3Speed of an electromagnetic wave from its field amplitudes

Worked out. The amplitude of the electric field of an electromagnetic wave travelling through some medium is given as $E_0=3\times10^4$ N C $^{-1}$ and the amplitude of the magnetic field is $B_0=2\times10^{-4}$ T, and the task is to find the wave's speed in that medium. Because in any medium the ratio of the instantaneous (and therefore also the amplitude) electric and magnetic field values equals the wave's speed, $v=E_0/B_0$ , substituting the given numbers gives $v=(3\times10^4)/(2\times10^{-4})=1.5\times10^8$ m/s. Since this value is less than $c=3\times10^8$ m/s, it correctly signals that the wave is travelling through a material medium rather than free space, consistent with the general rule $v<c$ inside any medium with permi …

Misc ~note-optical-tweezersMomentum and angular momentum of electromagnetic waves -- optical tweezers

Worked out. A short enrichment box connecting the wave properties just listed to a 2018 Nobel Prize in Physics, awarded for the invention of optical tweezers and the production of high-intensity light pulses. An optical tweezer is simply a tightly focused laser beam whose linear momentum is used to trap and move microscopic particles or molecules -- for instance, separating bacteria and viruses from healthy tissue, or isolating cancerous cells from normal cells, entirely by the mechanical push of light's momentum. The box also notes that a comet's tail points away from the Sun because sunlight's linear momentum physically pushes the comet's loosened dust and gas away from the Sun as it approaches, and it describes an angular-momentum analogue: in a setup of oppositely charged coaxial cylindrical shells with a solenoid between them, switching off the solenoid's current makes the two shells spin in opposite directions, because the collapsing electromagnetic field carries away angular momentum that is transferred to the shells -- direct laboratory evidence that electromagnetic wave …

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