revert and fix: make sign consistent with the book

src/3_drude_model.md

@@ -46,11 +46,11 @@ We start from the following very reasonable assumptions about how electrons move
 
 - Electrons scatter randomly at uncorrelated times. The average time between scattering is $\tau$. Therefore, the probability of scattering in a time interval $dt$ is $dt / \tau$
 - After each scattering event, the electron's momentum randomizes with a zero average $⟨\mathbf{p}⟩=0$
-- Electrons are charged particles with chrage $e$, so that the Lorentz force $\mathbf{F}_L=e\left(\mathbf{E}+\mathbf{v}×\mathbf{B}\right)$ acts on the electrons in between the scattering events
+- Electrons are charged particles with chrage $-e$, so that the Lorentz force $\mathbf{F}_L=-e\left(\mathbf{E}+\mathbf{v}×\mathbf{B}\right)$ acts on the electrons in between the scattering events
 
 !!! warning "Electron charge"
 
-    We are using a convention where $e$ is the electron charge, so $e<0$.
+    We are using a convention where $-e$ is the electron charge, so $e>0$.
 
 The first assumption here is the least obvious: why does the time between scattering events not depend on e.g. electron velocity? There is no physical answer to this: the model is only an approximation.
 The second assumption can be justified by symmetry: since we expect the electrons to scatter equally to all directions, their average velocity will be zero right after scattering.
@@ -141,8 +141,8 @@ HTML(anim.to_html5_video())
 
 ### Equations of motion
 Our goal is finding the *electric current density* $j$.
-Each electron with charge $e$ and velocity $\mathbf{v}$ carries current $e\mathbf{v}$.
-Therefore if the electron density is $n$, the *average* current they carry is $ne⟨\mathbf{v}⟩$.
+Each electron with charge $-e$ and velocity $\mathbf{v}$ carries current $-e\mathbf{v}$.
+Therefore if the electron density is $n$, the *average* current they carry is $-ne⟨\mathbf{v}⟩$.
 Our goal is then to compute the *average* velocity.
 
 The key idea is that although the motion of an individual electron is hard to calculate, the *average* motion of the electrons is much simpler.
@@ -233,12 +233,12 @@ A magnetic field $\mathbf{B}$ acts on the wire along the z-direction, like shown
 
 Let us take a look at the equations of motion again:
 $$
-m\frac{d\mathbf{v}}{dt} = -m\frac{\mathbf{v}}{τ} + e(\mathbf{E} + \mathbf{v}\times\mathbf{B})
+m\frac{d\mathbf{v}}{dt} = -m\frac{\mathbf{v}}{τ} - e(\mathbf{E} + \mathbf{v}\times\mathbf{B})
 $$
 Once again, we consider the steady state $d\mathbf{v}/dt = 0$.
-After substituting $\mathbf{v} = \mathbf{j}/ne$, we arrive to
+After substituting $\mathbf{v} = -\mathbf{j}/ne$, we arrive to
 $$
-\mathbf{E}=\frac{m}{ne^2τ}\mathbf{j} - \frac{1}{ne}\mathbf{j}\times\mathbf{B}.
+\mathbf{E}=\frac{m}{ne^2τ}\mathbf{j} + \frac{1}{ne}\mathbf{j}\times\mathbf{B}.
 $$
 The first term is the same as before and describes the electric field parallel to the current, while the second is the electric field **perpendicular** to the current flow.
 In other words, if we send a current through a sample and apply a magnetic field, a voltage develops in the direction perpendicular to the current—this is called *Hall effect*, the voltage is called *Hall voltage*, and the proportionality coefficient $B/ne$ the *Hall resistivity*.
@@ -254,7 +254,7 @@ with $a, b ∈ \{x, y, z\}$, and $ρ$ the *resistivity matrix*.
 Its diagonal elements are $ρ_{xx}=ρ_{yy}=ρ_{zz}=m/ne^2τ$—the same as without magnetic field.
 The only nonzero off-diagonal elements when $\mathbf{B}$ points in the $z$-direction are
 $$
-ρ_{xy}=-ρ_{yx}=\frac{B}{ne}\equiv -R_\mathrm{H}B,
+ρ_{xy}=-ρ_{yx}=\frac{B}{ne}\equiv R_\mathrm{H}B,
 $$
 where $R_H=-1/ne$ is the *Hall coefficient*.
 So by measuring the Hall voltage and knowing the electron charge, we can determine the density of free electrons in a material.