Electric field of a cross-like conductor

Question

Suppose we have two thin, very long, perpendicular wires soldered to each other so they form a cross. What would electric field lines look like in the plane that wires form, and would it be possible to calculate electric field at a point at distance $d$ from the wire crossing and on one of the $90°$ angle bisectors? Would it be easier to calculate electric field on the axis perpendicular to the wires plane and going trough wire crossing?

This question is purely a product of imagination and not a real task. I am not even aware of the complexity of the problem but would like to know if it's possible to do such calculations.

UPDATE: Both wires are, of course, carrying charge of linear charge density $\alpha$.

Answer 1

I'm pretty rusty in electrostatics (and everything), so please let me know if I mess up the answer.

The first step is to realize that the two conductors can be treated as separate problems. So, we will find the electric field for the two conductors separately, and them sum them together to find the total electric field from the conductors.

The second step is to steal someone else's work. I would have used Gauss's Law directly, but my rustiness combined with the fact that I can't do this for the rest of the day has lead to my lazy approach.

The result is that the electric field from the single conductor is $ E(R) = {\rho\over 2 \pi \epsilon_0 R} $, where $ R $ is the radius or distance from the conductor.

Let's use that expression to calculate the electric field for both wires.

$ E_1(R_1) = {\rho\over 2 \pi \epsilon_0 R_1} $

$ E_2(R_2) = {\rho\over 2 \pi \epsilon_0 R_2} $

And then sum them together, $ E(R_1,R_2) = E_1 + E_2 = {\rho\over 2 \pi \epsilon_0 R_1} + {\rho\over 2 \pi \epsilon_0 R_2} = {\rho\over 2 \pi \epsilon_0} ({1 \over R_1} + {1 \over R_2}) $

So, the answer is $ E(R_1,R_2) ={\rho\over 2 \pi \epsilon_0} ({1 \over R_1} + {1 \over R_2}) $. Though it would be a trick to convert the function inputs into conventional coordinates, with some imagination, I think one could picture how the electric field would look.

Consider, that for each pair of radii $ R_1,R_2 $ that you actually have a family of points that (I assume, imagine) would form an ellipse around the cross. So, every point along each point for each ellipse would have a constant electric field.