We have an infinitely long half-cylindrical shell of radius r and charge density σ as shown below
I am supposed to find the electric field at a point on the cylindrical axis, as seen in the diagram. Consider a coordinate system where the cylinder extends infinitely along the z-axis and is in the upper half of the x-y plane, centered at the origin. Based on the symmetry of the problem, I know that the electric field must point in the -y direction. However I can't find a good method to figure out what the magnitude of the field is.
Direct integration is a no go, because this is an infinite half cylinder.
Gauss' Law doesn't apply here.
I am not sure about the method of images, but I don't see any clear way to match boundary conditions.
Possibly Laplace's Equation would apply? But again the boundary conditions would be odd.
Thanks for any help.