You are developing an electromagnetic blood flow meter system for use in an artery with a diameter $($ D $)$ of $2$ cm $.$ The blood flow $\backslash ((\backslash$vec$\{$v$\})\backslash )$ is in the $(+$x $)$ direction, and electrodes are positioned along the $($ y $)-$axis, as depicted in the figure. The left electrode is grounded and the voltage at the right electrode undergoes amplification through an inverting amplifier and subsequently passes through a level shifter. The level shifter output is then directed to a $12-$bit ADC capable of sampling analog signals between $2$ V and $3$ V $.$ The magnitude of the magnetic field, $($ vec$\{$B$\}),$ is $(|\backslash$vec$\{$B$\}|=1$ T $).$ Identify:

a$.$ The direction of the magnetic field, $($ vec$\{$B$\}),$ if the digital values increase with blood flow.

b$.$ The $($ R $)$ resistance value such that a $(20$ um$/$s$)$ increase in blood flow corresponds to a binary ADC output code increment of $1.($In other words, the system resolution is $(20$ um$/$s$0$