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\section{Discussions}  \subsection{Practical application}  This study used an ultrasound device to image the sample and track shear waves, due to its high temporal resolution, availability and ease of use. However, for a clinical implementation such as brain elasticity imaging, MRI is more suited for tracking shear waves, as acoustic waves used in ultrasound imaging for shear wave tracking are attenuated by the skull. A clinical MRI scanner has, moreover, a five to ten times higher magnetic field than the permanent magnet used in this study. In a practical MRI implementation, no magnet would be necessary, and MRI-compatible coils should be used. Magnetic Resonance Elastography is  usually use employing  continuous shear waves; but induction of a continuous electrical current by the coil could affect MRI measurements, so "repetitive transient" excitations, which would lead to a continuous wave, could be used. \subsection{Displacement amplitude}  In our numerical study, Lorentz force magnitude reached about 120 90  N.m$^{-3}$ for a 0.2 0.15  T permanent magnetic field and a 5 S.m$^{-1}$ medium. Numerous measurements of grey and white matter electrical conductivity have been performed, and results vary from 0.02 to 2 S.m$^{-1}$ \cite{19636081}. By assuming a an average  value of 0.1 to 0.3 S.m$^{-1}$, in a 1.5 T MRI system, the Lorentz force would reach a magnitude of about 18 to 54 N.m$^{-3}$. This is comparable to the magnitude of the acoustic radiation force used for shear wave elastography: this force is force,  calculated with the equation $f = 2 \alpha I \Delta t / c$, with $\alpha$ attenuation of the medium, $I$ ultrasound intensity, $\Delta t$ duration of force application and $c$ speed of sound. Using sound, is about 80 N.m$^{-3}$ (using  Nightingale's parameters \cite{Nightingale_2002} ($\alpha$ \cite{Nightingale_2002}: $\alpha$  = 0.4 Np.cm$^{-1}$, $I$ = 1000 W.cm$^{-2}$, $\Delta t$ = 0.7 ms, $c$ = 1540 m.s${-1}$), this force of about 80 N.m$^{-3}$: it is the same order of magnitude as the previously calculated Lorentz force. m.s${-1}$).  We found in our numerical study a displacement slightly lower than the experimental value in the phantom. This was expected, as various factors impacting the displacement have not been included, especially viscosity and border effects. Moreover, there are uncertainties about electrical current amplitude and shape in the coil, as constructor values were used.