Problem $4$

You are given a conductor as shown in the figure. The conductor is infinite in both $\backslash ($ x $\backslash )$ and y directions. A line charge $($perpendicular to the plane$)$ with charge density of $\backslash (\backslash$lambda $\backslash )$ is placed at the coordinate $\backslash (($a$,$ b$)\backslash ).$

$1.$ Use an image method to find suitable line charges to replace the conductor. Draw the configuration of the line charges, and what are they charge densities $($There should be $3$ image line charges and one original charges.

$2.$ Check that the image charges satisfy the boundary conditions, that of the conductor $\backslash (($E$($x$,$ y$=0)=0\backslash )$ and $\backslash ($ E$($x$=0,$ y$)=0)\backslash )$

$3.$ Find the force per unit length exerted on the original line charge.

$4.$ Find the charge per unit length induced at the origin.