Chapter 8 · Introduction

Chapter 8 — Electromagnetic Waves

A changing electric field makes a magnetic field, just as a changing magnetic field makes an electric one. Together they can travel through empty space on their own — as radio waves, light and X-rays — at $3\times10^8$ m/s.

When you send a message from your phone, the words leave the handset as a pattern of electric and magnetic fields oscillating about two thousand million times a second. They cross the street to a tower at the speed of light without any wire, without any air being needed, and are picked up by an antenna a few centimetres long. The warmth of sunlight on your face, the image on an X-ray film, the signal from a satellite $36\,000$ km above the equator and the faint hiss picked up by the dishes of the Giant Metrewave Radio Telescope near Pune are all the same kind of thing: electromagnetic waves, differing only in their wavelength.

The idea that such waves exist came from a piece of pure reasoning. In the 1860s James Clerk Maxwell noticed that Ampère's circuital law gives contradictory answers for a capacitor that is being charged. He repaired it by adding a new term, the displacement current: a changing electric field produces a magnetic field. With that term, the laws of electricity and magnetism — now called Maxwell's equations — predict waves of electric and magnetic field travelling at the speed $1/\sqrt{\mu_0\varepsilon_0}$. Putting in the measured values of $\mu_0$ and $\varepsilon_0$ gives $3\times10^8$ m/s, the speed of light. Light was an electromagnetic wave. In 1887 Heinrich Hertz made such waves with sparks and detected them across his laboratory; in 1895 Jagadish Chandra Bose, in Kolkata, produced waves between a few millimetres and a few centimetres long and studied their reflection and refraction; and in 1901 Guglielmo Marconi sent radio signals across the Atlantic. This chapter follows that path, from the capacitor puzzle to the whole electromagnetic spectrum.

Ampère's law fails for a charging capacitor→Displacement current $\varepsilon_0\,d\Phi_E/dt$→Maxwell's equations→Accelerated charges radiate→$c=1/\sqrt{\mu_0\varepsilon_0}$, $E_0=cB_0$→The spectrum, radio to gamma
🎯 By the end of this chapter you can…
  • explain why Ampère's circuital law is incomplete for a charging capacitor, define the displacement current $i_d=\varepsilon_0\,d\Phi_E/dt$ and show that it equals the conduction current in the leads;
  • use the Ampère–Maxwell law to find the magnetic field between the plates of a charging capacitor;
  • state Maxwell's four equations in words and say what each one describes;
  • explain how accelerated charges produce electromagnetic waves, and describe the wave: transverse, $\mathbf E\perp\mathbf B$, in phase, travelling along $\mathbf E\times\mathbf B$ at $c=1/\sqrt{\mu_0\varepsilon_0}$ with $E_0=cB_0$;
  • calculate wavelength, frequency, field amplitudes, energy density and intensity of an electromagnetic wave;
  • name the bands of the electromagnetic spectrum in order, with rough wavelength ranges, how each is produced and detected, and its main uses;
  • solve board, JEE Main, JEE Advanced and NEET problems on all of the above.
Teaser — A Wave Leaving an Antenna

A charge oscillates up and down in the antenna at the left. Far away, the electric field at distance $x$ copies what the charge was doing a time $x/c$ earlier, because the news travels outwards at the speed of light. Each red arrow is that field (the fall-off of the amplitude with distance is not drawn). The result is a wave, and in one period it moves on by exactly one wavelength $\lambda=c/\nu$. The motion is shown enormously slowed down. Pick a source to change the scale.

frequency $\nu$—
wavelength $\lambda=c/\nu$—
period $T=1/\nu$—
slowed down by—

Sections in this chapter

8.1

Displacement Current and Maxwell's Equations

The capacitor puzzle in Ampère's law, the displacement current, the Ampère–Maxwell law, the field between capacitor plates and Maxwell's four equations.

$i_d=\varepsilon_0\dfrac{d\Phi_E}{dt}$
Open section →8.2

Electromagnetic Waves and the Electromagnetic Spectrum

Sources and nature of electromagnetic waves, their speed, field amplitudes, energy and intensity, and the spectrum from radio waves to gamma rays.

$c=\dfrac{1}{\sqrt{\mu_0\varepsilon_0}}$, $\ E_0=cB_0$
Open section →
✅ Before you start
This chapter ties together almost everything so far. You need electric flux and Gauss's law (Section 1.3), the parallel-plate capacitor and $C=\varepsilon_0A/d$ (Section 2.3), the energy density $\tfrac12\varepsilon_0E^2$ of an electric field (Section 2.4), the constant $\mu_0$ (Section 4.2), Ampère's circuital law and the field of a long straight wire (Section 4.3) and Gauss's law for magnetism (Section 5.1). Faraday's law of induction (Chapter 6) appears as one of Maxwell's equations. From Class 11 Physics you need the description of a travelling wave, $y=a\sin(kx-\omega t)$ with $k=2\pi/\lambda$, $\omega=2\pi\nu$ and $v=\nu\lambda$ (Class 11 Physics, Chapter 14 — Waves). From mathematics you need derivatives; the only integrals are $\oint$ along circles and flux through flat surfaces.
← 7.3 Power in AC Circuits and the Transformer8.1 Displacement Current and Maxwell's Equations →