## Transcribed Text

C- The Faraday effect
We study the influence of a static magnetic field B0 on the propagation of an
electromagnetic wave in a material medium. The medium is here an atomic gas. Let us
consider one of these atoms, whose electron with mass m and charge q is located at a position
r and has a speed V. It is described by the model of elastically bound electron.
1) The atom is submitted to a static magnetic field Bo=B0ê2 and the electric field of the
electromagnetic wave at the location of the atom, given by E(t) = ê+], where
Re denotes the real part. Show that in the permanent regime, the electron trajectory is a circle
in a plane perpendicular to the Z axis.
2) Let's denote D the electric dipole moment induced by E(t). We write
D(t)= Re[d exp(-ico 1) ê+]. Calculate the dipole amplitude d In this expression we may use the
Larmor angular frequency WB = -qB0/2m.
3) We consider the propagation through the gas of the circularly polarized wave, with electric
will neglect the effects of the magnetic field of the
wave on the electron.
Calculate the electric polarization of the medium induced by the wave in the presence of B0.
In this expression we may use the plasma angular frequency whose square is given by
= N , where N/V is the atomic density per unit of volume.
meo
Demonstrate that the circularly polarized wave propagates in the medium without change of
polarization. Deduce the susceptibility X+ of the atomic vapor for this wave.
4) What happens if the wave has a circular polarization with opposite helicity, taking
with =
We now study the propagation of a linearly polarized plane wave, whose polarization
direction is along the x axis at Z = 0. We consider the limit case where co-coo Tc1. We also
consider that the atomic density is weak and the frequency is far out of resonance, so that the
real part X of the susceptibility is extremely reduced (x' << 1) and the refractive index can be
approximated by 1+x'/2. We will assume that the field B0 is small enough to select only the
0th and 1st orders in powers of B0 in the expression of the susceptibility.
5) Calculate the difference of indices n+ and n. associated to X+ and X- in the medium. Show
that it is proportional to B0.
6) Show that the polarization of the wave remains linear during the propagation, with a
direction which changes in the presence of a magnetic field B0. This effect is called the
Faraday effect. Denoting A(Z) the angle between the polarization of the wave and the x axis in
the plane of coordinate z, show that A(Z) is proportional to both B0 and z. The proportionality
constant R = 0(z)/Boz is called the Verdet constant. Calculate R in the framework of this
classical model.
7) Indicate a few applications of the Faraday effect.

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