Hall effect. Consider a rectangular conductor of length $L$ along the $x$$-$axis, width $w$ along $y$$,$ and very small thickness $$\backslash$Delta$$.($i$)$ If the volume density of carriers is $n$$,$ and each carrier has charge $q$>0$$ and drift velocity $$\backslash$mathbf$\{$i$\}$ v$\_$x$$,$ what is $I$\_$x$$,$ the current along $x$ $?($ii$)$ If one turns on a magnetic field $$\backslash$mathbf$\{$B$\}=\backslash$mathbf$\{$k$\}$ B$\_$z$ perpendicular to the sample, what additional force will each of the charges initially feel? $($iii$)$ What will they do in response to this and what will happen when they reach equilibrium? $($iv$)$ Find the electric field $$\backslash$mathbf$\{$E$\}=\backslash$mathbf$\{$j$\}$ E$\_$y$ due to accumulated charges that balance the magnetic force, so that in equilibrium the current flows along $x$$.$ The Hall conductance is defined as the ratio $$\backslash$sigma$\_\{$x y$\}=\backslash$frac$\{$I$\_$x$\}\{$V$\_$y$\}$$$.($v$)$ Show that $$\backslash$sigma$\_\{$x y$\}=\backslash$frac$\{$q n $\backslash$Delta$\}\{$B$\_$z$\}=\backslash$frac$\{$q $\backslash$tilde$\{$n$\}\}\{$B$\_$z$\}$$ where $$\backslash$tilde$\{$n$\}=$n $\backslash$Delta$ is the charge per unit area. Find the Hall voltage $V$\_$y$ in a sample that has $n$=6\backslash$cdot $10^\{28\}$$ electrons per $m$^3$$$,$ and thickness $$\backslash$Delta$=10^\{-4\}$ m$$,$ for the case $I$\_$x$=2$ A$,$ B$\_$z$=1\mathrm{.}6\backslash$mathrm$\{$~T$\}$$$.$