You have a sample of chlorine, and you are trying to determine what fraction of your sample consists of chlorine$-35$ atoms $($with an atomic mass of about $35$ atomic mass units$)$ and what fraction is made up of chlorine$-37$ atoms $($with an atomic mass of about $37$ atomic mass units$).$ To do this, you use a mass spectrometer $($see figure$)$ that ionizes the atoms such that each atom is singly ionized, i$.$e$.$ with a charge of $+e.$ Note that $1$ atomic mass unit is equivalent to $1\mathrm{.}66{10}^{27}kg$ and that the charge of a proton is $1\mathrm{.}6{10}^{-19}C.$

The shaded region in the figure indicates where there is a magnetic field. Be careful: the B$-$field in the velocity selector might have a different magnitude and direction than the B$-$field inside the mass separator.

In the velocity selector, you use a uniform electric field of $4000\frac{N}{C}$ and a uniform magnetic field of $B=2\mathrm{.}0{10}^{-2}T.$ These two fields are mutually perpendicular and are each perpendicular to the velocity of the chlorine ions when they enter the velocity selector.

$($a$)$ What is the speed of chlorine ions that pass undeflected through the velocity selector?

$($b$)$ Assuming the chlorine$-37$ ions have practically no kinetic energy when they enter the accelerator, what voltage $V_1$ is needed to give them the necessary speed of part $($a$)?$

$($c$)$ In the accelerator part, which plate $($left or right$)$ should be negative, and which should be the positive plate $($left or right plate$)?$

$($d$)$ If the parallel plates in the velocity selector $($those with voltage $V_2)$ are separated by $10mm,$ then how large should $V_2$ be to attain an electric field of $4000\frac{N}{C}?$

$($e$)$ The undeflected ions continue from the velocity selector into the uniform magnetic field in the mass separator. If you want the ions of different masses to be separated by $1\mathrm{.}0mm$ after going through a half circle in the mass separator, to what value should you set the magnitude B of the magnetic field in the mass separator?

$($f$)$ Inside the mass separator, in which direction should the magnetic field point?

$($g$)$ Explain how you should change the machine and all its parts to do measurements with negative ions.