Electric field around tape
I think you must use the superposition principle of the field in this case, and consider the problem as a distribution of charge along a line from a to -a, because the contribution to the field of the charge along the z axe is 0, the field in this case is always parallel to the surface xy. The problem is to build the right integral considering a dE and a dL (along the y axe).


Note: there is a little mistake, I put r=x^2+y^2 instead r^2=x^2+y^2, I have forgot to put the ^2 on the r, but I have taken into account in the rest of the calculation.