Final answer:
The electric field at a point (x,y,0) due to a point charge Q near conducting half-planes is found by the method of images, which includes considering the contributions of both the actual charge and its mirror images across the planes.
Step-by-step explanation:
The electric field at a point due to a point charge can be determined using Coulomb's law. However, in this scenario, the presence of conducting half-planes modifies the electric field due to the method of images. To solve this problem, we mirror the point charge Q across the conducting planes to determine the positions of image charges. The half-plane at x=0 will add an image charge of -Q at point (-a,b,0), while the half-plane at y=0 will add an image charge of -Q at point (a,-b,0). Additionally, a second set of image charges occurs by mirroring these first image charges across the opposite half-plane resulting in another pair of point charges of +Q at (-a,-b,0) and another of -Q at (-a,-b,0). The electric field at a point in the region where x>0 and y>0 can be calculated by summing the contributions of the original charge and all of the image charges. The superposition principle allows for this summation of fields due to individual charges.
To determine the electric field at a specific point (x,y,0), vector addition is used to sum the electric fields produced by the actual charge and its image charges. The components of the electric field in the x and y directions need to be calculated separately and then combined to determine the total electric field at the point of interest.