33-1 Electromagnetic Waves#
Prompts
What is an electromagnetic wave? How does it differ from the mechanical waves we studied in Chapters 16 and 17?
Describe the electromagnetic spectrum. Where does visible light sit? How do AM radio, FM radio, infrared, ultraviolet, and x rays compare in wavelength and frequency?
How does an LC oscillator and antenna generate an electromagnetic wave? Why does the antenna act like an oscillating electric dipole?
For an EM wave traveling in vacuum: what is the speed \(c\)? How are \(E\) and \(B\) oriented relative to each other and to the direction of travel? Are they in phase?
Derive or explain \(c = 1/\sqrt{\mu_0\varepsilon_0}\) and \(E/B = c\). Why do EM waves not require a medium?
Lecture Notes#
Overview#
An electromagnetic (EM) wave consists of oscillating electric and magnetic fields that propagate together through space.
EM waves span a vast spectrum (Maxwell’s rainbow): radio, microwaves, infrared, visible light, ultraviolet, x rays, gamma rays—all travel at the same speed \(c\) in vacuum.
Unlike mechanical waves (string, sound), EM waves require no medium; they propagate through vacuum.
The electric field \(\vec{E}\) and magnetic field \(\vec{B}\) are perpendicular to each other and to the direction of travel; they vary sinusoidally and in phase.
The electromagnetic spectrum#
The spectrum has no gaps—all EM waves share the same nature. Labels (radio, visible, x rays, etc.) denote wavelength ranges where certain sources and detectors are common.
Region |
Wavelength (approx.) |
Frequency (approx.) |
Examples |
|---|---|---|---|
Radio (AM) |
\(10^2\)–\(10^3\) m |
\(10^5\)–\(10^6\) Hz |
Broadcasting |
FM / TV |
\(1\)–\(10\) m |
\(10^7\)–\(10^8\) Hz |
FM radio, television |
Infrared |
\(10^{-5}\)–\(10^{-3}\) m |
\(10^{11}\)–\(10^{14}\) Hz |
Heat radiation |
Visible |
400–700 nm |
\(\sim 10^{15}\) Hz |
Light we see |
Ultraviolet |
\(10^{-8}\)–\(10^{-7}\) m |
\(10^{15}\)–\(10^{17}\) Hz |
Sunburn, fluorescence |
X rays |
\(10^{-11}\)–\(10^{-8}\) m |
\(10^{16}\)–\(10^{19}\) Hz |
Medical imaging |
Gamma rays |
\(< 10^{-11}\) m |
\(> 10^{19}\) Hz |
Nuclear decay |
Visible light: center ~555 nm (yellow-green); human eye sensitive roughly 430–690 nm.
All EM waves in vacuum travel at \(c \approx 3.0 \times 10^8\) m/s.
Generation: LC oscillator and antenna#
For wavelengths \(\lambda \gtrsim 1\) m, EM waves are generated by macroscopic sources:
LC oscillator: Establishes angular frequency \(\omega = 1/\sqrt{LC}\). Charges and currents vary sinusoidally.
Antenna: Two conducting rods coupled to the oscillator. Charge oscillates along the rods at frequency \(\omega\)—the antenna acts as an electric dipole whose dipole moment varies sinusoidally.
Radiation: The changing \(\vec{E}\) and \(\vec{B}\) from the dipole do not change everywhere at once; the changes propagate outward at speed \(c\). Together they form a traveling EM wave.
Example: Dipole radiation
An oscillating electric dipole—two opposite charges moving back and forth—produces electromagnetic radiation. Here’s the physical mechanism:
Field generation: Separating the charges creates electric field lines between them.
Detachment: As the charges move toward each other and neutralize, those field lines can no longer stay attached to charges. They pinch off and form closed loops.
Self-sustaining waves: The changing electric field induces a perpendicular magnetic field (Faraday’s law), and the changing magnetic field induces a perpendicular electric field (Ampère–Maxwell law). These fields sustain each other and propagate outward as electromagnetic radiation.
An oscillating electric dipole produces electromagnetic radiation.
The traveling EM wave: qualitative features#
At a distant point (plane wave approximation), the wave has these key properties:
Transverse: \(\vec{E}\) and \(\vec{B}\) are perpendicular to the direction of travel.
Mutually perpendicular: \(\vec{E} \perp \vec{B}\).
Direction: \(\vec{E} \times \vec{B}\) gives the propagation direction.
Sinusoidal and in phase: Both fields vary with the same frequency and phase.
Poll: E and B directions from propagation
An EM wave is traveling in the \(+y\) direction. At a certain time and position, the electric field points in the \(-z\) direction. What is the direction of the magnetic field at that same time and position?
(A) \(+x\) direction
(B) \(-x\) direction
(C) Neither of these
Sinusoidal form and wave speed#
For a wave traveling in the \(+x\) direction, with \(\vec{E}\) along \(y\) and \(\vec{B}\) along \(z\):
Wave speed (vacuum):
Amplitude ratio: \(E_m/B_m = c\).
Instantaneous magnitude ratio: \(E/B = c\) at any point and time.
Important
The two fields induce each other: a changing \(\vec{B}\) induces \(\vec{E}\) (Faraday), and a changing \(\vec{E}\) induces \(\vec{B}\) (Ampère–Maxwell). They cannot exist independently; together they form a self-sustaining wave.
Poll: Which B-field describes a wave in \(+z\)?
Which of the following could describe the \(\vec{B}\)-field of an EM wave traveling in the \(+z\) direction?
(A) \(\vec{B} = B_0 \cos(kz - \omega t)\,\hat{y}\)
(B) \(\vec{B} = B_0 \cos(kz - \omega t)\,\hat{z}\)
(C) \(\vec{B} = B_0 \cos(ky - \omega t)\,\hat{x}\)
(D) \(\vec{B} = B_0 \cos(kz + \omega t)\,\hat{y}\)
(E) More than one of these
No medium required#
Unlike mechanical waves (string, sound, seismic), EM waves require no medium. They propagate through vacuum—e.g., light from the Sun to Earth.
The speed of light in vacuum is defined as \(c = 299\,792\,458\) m/s (exact).
In special relativity, \(c\) is the same in all inertial frames.
Summary#
EM wave: oscillating \(\vec{E}\) and \(\vec{B}\), perpendicular to each other and to propagation; sinusoidal, in phase.
Spectrum: radio → visible → x rays; all travel at \(c\) in vacuum.
Generation: LC oscillator + antenna (oscillating dipole) → changing fields propagate at \(c\).
Speed: \(c = 1/\sqrt{\mu_0 \varepsilon_0}\); \(E/B = c\).
No medium: EM waves propagate through vacuum.