Problem $4$: Electric potential of continuous charge distributions

A uniform line charge has a charge $q=4\mathrm{.}1mC$ and a length $L=1\mathrm{.}2m.$ Point $P$ is at $d=0\mathrm{.}4m$ to the right of the line charge as shown. In this problem you will integrate to find the electric potential at point $P$ due to this line charge.

A$.$ Write an expression for the linear charge density $$ of the line charge, in terms of the given variables $($don$\text{'}$t plug in numbers yet$).$

B$.$ A small length $dx$ of the line charge is marked in dark grey. How much charge $dq$ is in the marked segment, in terms of the given variables?

C$.$ What is the distance $r$ from the marked segment to point $P,$ in terms of the given variables?

D$.$ The marked segment by itself will contribute a small amount $dV$ to the electric potential at point $P.$ Use your answers above to write an expression for $dV,$ in terms of $k_e$ and the given variables.

E$.$ Now we need to set up an integral for the total potential at point $P.$

First, what are the bounds on $x-$i$.$e$.,$ what are the smallest and largest values $x$ can have?

Set up an integral expression using your expression for $dV,$ including the bounds you found above.

Evaluate your integral, using ${\displaystyle \underset{\phantom{}}{\overset{\phantom{}}{}}}\frac{dx}{x}=lnx.$

Finally, plug in values to find the potential at point $P.$

$4$