deletions | additions       diff --git a/Simulation_study.tex b/Simulation_study.tex  index ab8c53a..937d0d9 100644  --- a/Simulation_study.tex  +++ b/Simulation_study.tex 
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Additionally to the experiments, a three dimensional simulation of the experiments was performed using Matlab (Matlab 2010, The MathWorks, Natick, MA, USA). The simulation was performed by (1) calculating the electrical current induced by the coil, (2) simulating the magnetic field created by the permanent magnet, (3) calculating the resulting Lorentz force inside the medium, and finally (4) calculating the displacement along space and time due to the Lorentz force.   Using Equation \ref{Equation1} with two 75 mm diameter coils crossed by a 159.10$^6$ A.s$-1$ A.s$^{-1}$  electrical current, representing the TMS coil used in the experiment, electrical field $E$ was calculated in a 5x10x10 cm$^2$ volume (see \cite{Grandori_1991} for details on mathematical solving). Using Ohm's law, the electrical current $\mathbf{j}$ was estimated assuming an electrical conductivity $\sigma$ = 5 S.m$^{-1}$ inside a a 4x8x8 cm$^2$ medium and 0 outside. No border effects have been taken into account. Induced electrical current in a XY plane at a depth of 2 cm with 2x2 mm$^2$ pixels is illustrated in Figure \ref{Figure3}-(A), with colors indicating the absolute magnitude and arrows the direction. A finite element software (Finite Element Magnetic Method \cite{FEMM}) was used to produce a two dimensional simulation of the magnetic field $\mathbf{B}$. Magnetostatic problem was solved from equations $\nabla \times H = \nabla \times M$, $\nabla B = 0$ and $B=\mu H$, with $H$ magnetic field intensity, $M$ magnetization of the medium, $B$ magnetic flux density and $\mu$ medium permeability. Medium was considered as linear, and space was meshed with approximately 0.5 cm$^2$ triangles. The magnetic field was supposed to be approximately constant in the sample along the Y axis. The software simulated a N48 NdFeB permanent magnet of 5x5 cm$^2$ placed in a 19x27 cm$^2$ surface of air. Resulting magnetic field is illustrated in Figure \ref{Figure3}-(B), with colors indicating the absolute magnitude and arrows the direction.