The electrostatic field of the Faraday cage

[Xainnia]

I have a function of the cartesian coordinates x, y, z such that:

f = a * ln|sin(px + iqy)| + by + c

This equation should be describing the electric field nearby the wall of the Faraday cage. I solved the Laplace equation Δf = 0 and got : p = +/- q, so p = q. I should choose the constants a, b, c, p, q such that my given function f is describing the homogenous electrostatic field E in the half-space y∠ 0 where the field is shielded by the conductig grounded rods. The distance between the rods is L. The diameter of the rods is d and this diameter is esentially smaller than L. But I am stuck here, I have no idea how to find the constants. I need to satisfy that my field is homogenous. I really don't know how to move forward with this. Thank you for helping me.

 

[member 553083]

In order to find the constants, you need to use the boundary conditions of the electrostatic field. The boundary conditions will tell you what the value of the field should be at the boundaries of the domain. For example, if the domain is a region bounded by two grounded rods, then the field should be zero on the boundaries of the domain (i.e. the rods). You can then use these boundary conditions to solve for the constants a, b, c, p, q. For example, if the boundary conditions are that the electric field should be zero on the boundaries of the domain (i.e. the rods), then you can solve for the constants by setting f(x,y,z) = 0 at the boundaries of the domain. From this, you can solve for the constants a, b, c, p, q. Additionally, you can also use other physical constraints to solve for the constants. For example, you may want the electric field to decay exponentially away from the rods. In this case, you can set up an equation with the constants a, b, c, p, q that describes this exponential decay, and then use this equation to solve for the constants. I hope this helps!