Electromagnetic Fields and Waves: Notes, Solved Problems and Interactive Lab

If the time dependence of an electromagnetic wave is sinusoidal, each Cartesian component of $E$ or $H$ at the point $r$ can be written in the form

u (r; t) = A (r) cos [ω t - ϕ (r)], (5.6.1)

where $u$ stands for any one of the six components $E_{x}, \dots, H_{z}$ and

$A (r) \geq 0$ is the amplitude, in $V/m$ for $E$ and $A/m$ for $H$ ,
$ω$ is the angular frequency or pulsation, in $rad/s$ ,
$ω t - ϕ (r)$ is the phase, and $ϕ (r)$ its part that depends on position,
$T = 2 π / ω$ is the period, in $s$ , and $f = 1/ T = ω /2 π$ the frequency, in $Hz$ .

Such a wave is called monochromatic, a word borrowed from optics, where one frequency is one colour. At a fixed point $r_{0}$ the field is an ordinary sinusoid of time (Figure 5.6.1): it oscillates between $\pm A (r_{0})$ , repeats every $T$ , and its first maximum is delayed by $ϕ (r_{0}) / ω$ after $t = 0$ . All the dependence on position is in the two functions $A (r)$ and $ϕ (r)$ .

A monochromatic wave observed at a fixed point \vec r_0. The field is a sinusoid of amplitude A(\vec r_0) and period T=2\pi/\omega, and its first maximum comes \phi(\vec r_0)/\omega after t=0. Drawn for \phi(\vec r_0)=\pi/3. — **Figure 5.6.1.** A monochromatic wave observed at a fixed point $r_{0}$ . The field is a sinusoid of amplitude $A (r_{0})$ and period $T = 2 π / ω$ , and its first maximum comes $ϕ (r_{0}) / ω$ after $t = 0$ . Drawn for $ϕ (r_{0}) = π /3$ .

Why one frequency is enough

Any signal in time is a sum of sinusoids, a Fourier series if it is periodic and a Fourier integral if it is not. In a linear medium, one in which $ε$ , $μ$ and $σ$ do not depend on the field, Maxwell's equations are linear, so each sinusoid travels on its own, at its own $ω$ , without disturbing the others. Solve the problem for a single $ω$ , and the response to any signal follows by superposition. This is exactly how circuits are analysed with phasors and impedances, and it is why everything in this section is written for one frequency.

5.6.1 Monochromatic waves