Electromagnetic waves

• Power is the flux of the Poynting vector :

  \(P=\Pi s = \frac{1}{2}\varepsilon_0 c E_0^2s\).

  We can deduce \(E_0\).

  The magnetic field is then given by :

  \(B_0=E_0/c\)

• The conservation of energy gives :

  \(Pdt=n^*(s cdt)\;h\nu\)

  Where \(h\) is the Planck constant and \(\nu\) the frequency, which is \(\nu=c/\lambda\).

  Finally :

  \(n^*=\frac{P}{sch\nu}\)

The conductivity \(\gamma\) of a perfect conductor is considered infinite.

The electric and magnetic fields are zero inside a perfect conductor.

The skin thickness is assumed to be null.

The d'Alembert's equation is, for example for the propagation of an electric field \(\vec E\) in a vacuum :

\(\Delta \vec E-\frac{1}{c^2} \frac{\partial^2 \vec E}{\partial t^2}=\vec 0\)

With :

\(c^2=\frac {1}{\varepsilon_0 \mu_0}\)

• The standing wave (with \(k=\omega /c\)) :

  \(s(x,t) = Ccos \left( {kx-\varphi } \right)cos \left( {\omega t-\psi} \right)\)

  The nodes are the points where the amplitude \(cos(kx_N)=0\).

  The anti-nodes are the points where the amplitude \(cos(kx_V)=\pm 1\).

• The distance between two successive anti-nodes (two nodes) equals to \(\lambda/2\), with \(k=2\pi/\lambda\)

• The distance between a successive anti-node and a node is \(\lambda/4\).

The relation of structure for a plane wave :

\(\vec B = \frac {\vec k}{\omega}\wedge \vec E\)

If the wave propagates in the direction (Oz) at the speed \(c\) :

\(\vec B = \frac {\vec u_z}{c}\wedge \vec E\)

The velocity of the propagation of a MPPW in a medium of index \(n\) is :

\(v=\frac{c}{n}\)