Why do electromagnetic waves use complex numbers?

[phabos]

why electromagnetic waves are represented by complex numbers?

 

[Lavabug]

My answer is a bit general but I think its pretty relevant:

Waves and harmonic oscillators are represented by sinusoidal functions. Using Euler's theorem you can rewrite them as the (real part) of an imaginary exponential, where the exponent is i*(arg), where the argument is the same one you would use for an oscillator(wt +phase) or a wave (kx - or +wt + phase).

Its a bit more convenient to work with imaginary exponentials since they're more compact, taking their time derivatives to get velocities for example.

Something worth trying to illustrate that example: show that the total energy (T+V) of a harmonic oscillator is proportional to the square of the amplitude. You can do this either way, but I think its more compact if you use y(t) = Re{Ae^(iwt)} instead of Acoswt as your starting point.

 

[yungman]

EM wave usually are of sinusoidal nature. It is easier to represent harmonic wave ( sinusoidal) in cosine wave:

$$\vec E =E_0 cos\;(\omega t -\vec k\cdot \vec R)\;=\; \Re e [E_0 e^{j\omega t}e^{-j\vec k \cdot \vec R}]$$And then use phasor form where $\tilde E = E_0 e^{-j\vec k \cdot \vec R} \;\hbox { and }\;\vec E = \Re e [\tilde E \;e^{j\omega t}]$The solution of homogeneous harmonic wave equation is something like:

$$\nabla ^2 E +\delta^2 \vec E = 0 \;\hbox { is } E^+ e^{-\delta \vec k \cdot \vec R} +E^- e^{\delta \vec k \cdot \vec R} \;\hbox { where } \delta = \alpha + j\beta$$It is not as common in Physics than in RF and microwave Electronics. In RF, we deal with transmission lines where we can assume the direction of propagation in z direction which really simplify the calculation tremendously. We avoid all the differential equations, PDE, integration and differentiation. In fact I learn in reverse order. I have been using phasor calculation to design filters, matching networks for years before I really start learning EM!