Question $2(1.$

: You are a chemical engineer working for a mining company extracting an

iron$-$based magnetic ore critical to produce high$-$tech coatings for satellites. The magnetic ore is extracted

in a liquid matrix $($aqueous solution$)$ and you are asked to develop a separation system using a coaxial

solenoid around the extraction pipe. The solenoid generates a magnetic field at its centre, the direction of

which is opposite to the flow of the magnetic ore stream, as shown in the schematic below.

Your team performed a preliminary investigation, and you know the following findings:

Finding $1$: Magnetic susceptibility $({}_{m-or})$ of the iron$-$based ore solution is ${}_{m-\text{o}\text{r}\text{e}}=1{10}^2$ and magnetic

permeability of free space ${}_0=4{10}^{-7}T*\frac{m}{A}.$

Finding $2$: The solenoid is to be made of cylindrical titanium wire with a diameter of $0\mathrm{.}5cm($electrical

conductivity of titanium $\{$:${}_{titanium}=2\mathrm{.}38{10}^6(*m{)}^{-1}).$

Finding $3$: The pipe has a diameter of $20cm$ and the solenoid will be fitted perfectly around it$,$ so you

can assume that the diameter of the solenoid is $d_{\text{s}\text{o}\text{l}\text{e}\text{n}\text{o}\text{i}\text{d}}=20cm$ too. The length of the solenoid is $2m,$

with $N=400$ turns in total $($NB: you can calculate the total length of the wire for the solenoid as $I_{\text{s}\text{t}\text{a}\text{i}\text{n}\text{l}\text{e}\text{s}\text{s}}$

steel $\{$:$=*d_{\text{s}\text{o}\text{l}\text{e}\text{n}\text{o}\text{i}\text{d}}*N).$

Finding $4$: To achieve an efficient separation of the iron$-$based magnetic ore from the liquid stream

flowing through the pipeline, the solenoid will have to generate at least $4T$ at its centre.

Calculate:

$($a$)$ The necessary voltage that you will have to apply to the solenoid to efficiently separate the iron$-$based

magnetic ore from the liquid stream flowing through the pipeline $(1$

e$).$

$($b$)$ Unfortunately, a geologist in your team has predicted a variation of the magnetic susceptibility of the

iron$-$based ore stream as the extraction progress continues throughout time. Her model predicts that ${}_{m-\text{o}\text{r}\text{e}}$

will vary with the extraction time $($textraction in days$)$ according to the following expression: ${}_{m-\text{o}\text{r}\text{e}}=1{10}^2-$

$0\mathrm{.}5t_{\text{e}\text{x}\text{t}\text{r}\text{a}\text{c}\text{t}\text{i}\text{o}\text{n}}.$ You must obtain an ANALYTICAL EXPRESSION of the voltage that you will have to apply to the

solenoid as a function of textraction, and prepare a GRAPHICAL REPRESENTATION of that voltage profile for $20$

days $($i$.$e$.t_{\text{e}\text{x}\text{t}\text{r}\text{a}\text{c}\text{t}\text{i}\text{o}\text{n}}$ from $0$ to $20$ days with $t_{\text{e}\text{x}\text{t}\text{r}\text{a}\text{c}\text{t}\text{i}\text{o}\text{n}}=1$ day$)($NB: you can use Excel for your calculations$)(19$