Problem $3$: $(30$ points total$)$

Consider a vortex filament of strength $\backslash (\backslash$Gamma $\backslash )$ in the shape of a closed circular loop of radius R$.$

$($a$)$ Obtain an expression for the velocity induced at the center of the loop in terms of $\backslash (\backslash$Gamma $\backslash )$ and R $.$

$($b$)$ Consider also a straight line through the center of the loop, perpendicular to the plane of the loop. Let A be the distance along this line, measured from the plane of the loop. Obtain an expression for the velocity at distance A on the line, as induced by the vortex filament.