deletions | additions       diff --git a/For_a_circular_coil.tex b/For_a_circular_coil.tex  deleted file mode 100644  index dfff522..0000000  --- a/For_a_circular_coil.tex  +++ /dev/null 
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For a circular coil centered in (0,0,0) of linear element $\mathbf{dl}$ crossed by an electrical current $I(t)$, using Coulomb gauge (i.e., $\nabla . \mathbf{A} = 0$ where $\mathbf{A}$ is the magnetic potential vector), and negligible propagation time of electromagnetic waves, the electrical field $\mathbf{E}(\mathbf{r},t)$ along space $\mathbf{r}$ and time $t$ is equal to \cite{jackson1998classical}:  \begin{equation}  \mathbf{\mathbf{E(\mathbf{r},t)}} = - \nabla \Phi - \frac{d I}{d t} \frac{N \mu_0}{4\pi}\int{\frac{\mathbf{dl}}{r}}  \label{Equation1}  \end{equation}  where $\Phi$ is the electrostatic scalar potential, $N$ is the number of turns of the coil and $\mu_0$ is the magnetic permeability of the coil material. In an unbounded medium, $\Phi$ is only due to free charges \cite{grandori1991magnetic}, that we supposed negligible in our case. Being additive, the total electrical field created by two or more coils is simply the sum of the contribution of each coil. The induced electrical current density $\mathbf{j}$ is retrieved using the local Ohm's law $\mathbf{j}=\sigma \mathbf{E}$, where $\sigma$ is the electrical conductivity of the medium.  The body Lorentz force $\mathbf{f}$ can then be calculated using the relationship $\mathbf{f} = \mathbf{j} \times \mathbf{B}$, where $\mathbf{B}$ is the magnetic field created by the permanent magnet. Considering the tissue as an elastic, linear and isotropic solid, Navier's equation governs the displacement $\mathbf{u}$ at each point of the tissue submitted to an external body force $\mathbf{f}$ \cite{aki1980quantitative}:  \begin{equation}  \rho\frac{d^2\mathbf{u}}{dt^2} = (K + \frac{4}{3}\mu) \nabla (\nabla . \mathbf{u}) + \mu \nabla \times (\nabla \times \mathbf{u}) + \mathbf{f}  \label{Equation3}  \end{equation}  where $\rho$ is the medium density, $\mathbf{u}$ the local displacement, $K$ the bulk modulus and $\mu$ the shear modulus.  The displacement $\mathbf{u}$ can then propagates as a shear wave with a velocity $c_s = \sqrt{\mu/\rho}$ \cite{sarvazyan1998shear}.        diff --git a/layout.md b/layout.md  index 492b194..17af35d 100644  --- a/layout.md  +++ b/layout.md 
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Introduction.tex  Physical_model.tex  figures/Figure1/Figure6.png  For_a_circular_coil.tex  Material_methods.tex  Simulation_study.tex  figures/Figure3/Figure3.png