Electromagnetic waves

The tangential component of the electric field should be continuous, therefore :

\({E_{0r}} = - {E_0}\)

The boundary condition for the electric field :

\({\vec E_{metal}(0^+,t)} - {\vec E_{vide}(0^-,t)} = - {\vec E_{vide}(0^-,t)} = \frac{\sigma }{{{\varepsilon _0}}}{\vec u_x}\)

Show that \(\sigma =0\).

The boundary condition for the magnetic field gives :

\({\vec B_{metal}(0^+,t)} - {B_{vide}(0^-,t)} = - {\vec B_{vide}(0^-,t)} = {\mu _0}{\vec j_s} \wedge {\vec u_x}\)

The incident magnetic field is :

\({\underline {\vec B} _i} = \frac{{{{\vec e}_x}}}{c} \wedge {\underline {\vec E} _i} = \frac{{{{\vec e}_x}}}{c} \wedge {E_0}{e^{j(\omega t - kx)}}{\vec e_y} = \frac{{{E_0}}}{c}{e^{j(\omega t - kx)}}{\vec e_z}\)

The reflected magnetic field is :

\({\underline {\vec B} _r} = - \frac{{{{\vec e}_x}}}{c} \wedge {\underline {\vec E} _r} = \frac{{{{\vec e}_x}}}{c} \wedge {E_0}{e^{j(\omega t + kx)}}{\vec e_y} = \frac{{{E_0}}}{c}{e^{j(\omega t + kx)}}{\vec e_z}\)

The resulting field in the vacuum for \(x=0^-\) is :

\({\vec B_{vide}}(0^-,t) = {({\underline {\vec B} _i} + {\underline {\vec B} _r})_{x=0^-}} = 2\frac{{{E_0}}}{c}{e^{j\omega t}}{\vec e_z}\)

We can deduce :

\({\vec B_{vide}(0^-,t)} = 2\frac{{{E_0}}}{c}{e^{j\omega t}}{\vec e_z} = - {\mu _0}{\vec j_s} \wedge {\vec u_x}\;\;\;\;\;\;\;\;\;\;so\;\;\;\;\;\;\;\;\;\;{\vec j_s} = 2\frac{{{E_0}}}{{{\mu _0}c}}{e^{j\omega t}}{\vec e_y} = 2{\varepsilon _0}c{E_0}{e^{j\omega t}}{\vec e_y}\)

Real notation :

\({\vec j_s} = 2{\varepsilon _0}c{E_0}\cos (t){\;\vec e_y}\)

The resultant electric field is :

\(\vec E = {E_0}{e^{j(\omega t - kx)}}{\vec e_y} - {E_0}{e^{j(\omega t + kx)}}{\vec e_y} = - 2j{E_0}\sin (kx){e^{j\omega t}}{\vec e_y}\)

Either in real notation :

\(\vec E = 2{E_0}\sin (kx)\sin (\omega t){\;\vec e_y}\)

Similarly, for the magnetic field :

\(\vec B = {\underline {\vec B} _i} + {\underline {\vec B} _r} = \frac{{{E_0}}}{c}{e^{j(\omega t - kx)}}{\vec e_z} + \frac{{{E_0}}}{c}{e^{j(\omega t + kx)}}{\vec e_z} = 2\frac{{{E_0}}}{c}\cos (kx){e^{j\omega t}}{\vec e_z}\)

Either in real notation :

\(\vec B = 2\frac{{{E_0}}}{c}\cos (kx)\cos (\omega t){\;\vec e_z}\)

This type of solutions, called standing plane wave is very different from a progressive plane wave. Spatial and temporal dependencies occur separately  : spatial dependence occurs in the amplitude of the time oscillation and no longer in phase, so that all points vibrate in phase or in opposition of phase.

At certain points (called nodes of vibrations), the field is always zero.

In other points (called vibration antinodes), the amplitude of vibration is maximum.

We can calculate Poynting vector and verify that its average value is zero : a standing wave does not transport energy.

The electromagnetic field exerted on a mirror surface \( dS\) a force \(d\vec F\) whose expression is, in a real notation : (note that \(\sigma =0\))

\(d\vec F = \frac{1}{2}{\vec j_s} \wedge \vec B\;dS\)

Indeed, the surface \(dS\) is subjected to the action of the magnetic field external to it ; so do not take into account the magnetic field created by the charged surface \(dS\) and traveled by volume current.

The factor \(1/2\) takes into account the remarks.

The average value of the force becomes :

\(\left\langle {d\vec F} \right\rangle = \frac{1}{2}{\mathop{\rm Re}\nolimits} \left( {\frac{1}{2}{{\vec j}_s} \wedge {{\vec B}^*}dS} \right) = \frac{1}{4}(2{\varepsilon _0}c{E_0}{e^{j\omega t}}{\vec e_y}) \wedge (2\frac{{{E_0}}}{c}{e^{ - j\omega t}}{\vec e_z})dS = {\varepsilon _0}E_0^2{\;\vec e_x}\;dS\)

The average radiation pressure is then :

\(\left\langle p \right\rangle = {\varepsilon _0}E_0^2\)

The volumetric energy density of the incident wave is :

\({e_i} = \frac{1}{2}{\varepsilon _0}{E_i^2} + \frac{1}{2}\frac{{{B_i^2}}}{{{\mu _0}}}\)

Whose average value is :

\(\left\langle {{e_i}} \right\rangle =\frac{1}{2} {\varepsilon _0}E_0^2\)

We can deduce the radiation pressure :

\(\left\langle p \right\rangle = {\varepsilon _0}E_0^2 = 2 \left\langle {{e_i}} \right\rangle\)

We can notice that :

\(\left\langle p \right\rangle = {\varepsilon _0}E_0^2 = 2 \left\langle {{e_i}} \right\rangle = 2n^*h\nu\)

Where \(n^*\) is the density of photons of the incident wave and \(h\nu\) the energy of one photon.