I know the formula of the electric field, however:
suppose that we put two infinitely long and thin, straight wires symetrically into the coordinate system, so that y axis is between them.
Now consider that we have wires charged with charge $q_1=q$ and $q_2= -q$.
If we sum the contributions to electric field from both wires with vector summation (note that the field has only radial component) in point T (T lies anywhere on the X=0 plane), we find out that the distance from both wires to the point T where we want to calculate the electric field, is the same.
But due to the fact that we have a positive and negative charge, i get:
(this is the part of the equation)
$(\frac{1}{r} - \frac{1}{r}) = 0$; where $r_1=r_2=r$
$$E_1 = \frac{q}{2\pi\varepsilon_0\rho_1}$$
$$E_2 = \frac{-q}{2\pi\varepsilon_0\rho_2}$$
which means that electric field on plane x=0 is zero, but I know that it isn't, because it is perpendicular to the equipotential V=0.
So where do I go wrong in calculating the field in point T?
In the end I would like to show that the electric field is indeed perpendicular to the equipotential V=0 which is on x=0, but i can't even start doing that when I get that electric field is 0 there....