Whew. So, getting back to basics. What IS an electric field or an E-vector? Well, its a mathematical construct which enables us to predict how things in the real world work. But consider measurement. One way to measure the field is measure the force at a "point". Ok, so we can measure magnitude and direction. For a wire (which unless you've good reasons not to, should be assumed to have "nearly zero" diameter - this is a convention for beginning Physics problems, not a "written in stone" definition) We can use x,y vector components. Its really a bad idea to talk about x,y and in the same thread vertical and horizontal. Make up your mind and stick to one or the other, don't waste time labeling them so that we can convert. A wire is 3 dimensional (in three dimensional space) but (see above) has negligible lateral cross-section. The field, of course will be symmetrical in planes perpendicular to the long axis...call it x-axis. This 'fact' suddenly reduces the important parameter to be r, the distance from the wire. Talking about surface effects with a wire means that you are no longer considering some "near zero radius" conductor of long length. So again, you have to make up your mind. At a 'sufficient' distance from said wire (said surface) the wire's effects can be considered as being from the center axis. Its not a bad question to ask "at what distance should we abandon the "near zero, no surface effects" paradigm. You can actually do some math and see how the assumption gets to be a worse and worse approximation as the distance, call it r/R° where R° is the radius of the wire, nears 1. A ratio of 10 will give 'not terrible' solutions but 100X would be better. So say at 50X, the force vectors break down to x,y,z components, but since (y,z) symmetry prevails, we just consider the distance, r, to the wire and can consider that the problem has been reduced to 2 dimensions, one of which is a "real" dimension (along the length of the wire) but the other dimension is a projection from the (yz) plane to a abstract dimension. In this scenario, for an infinite linear uniform charge distribution (a wire) every point charge to the left oh, sorry, to the minus x distance from any point will have an identical charge to the plus x distance from that point (don't have to call that point 0 or (0,0) or (0,0,0) could be ( 1, -3) in the reduced dimensions, or maybe (1, 3/√2,-3/√2) in real space (distance to x=0 is 3 in either case). This means for every force component → at any point (far enough from the wire) there is a ← component which cancels it from a charge at exactly the same distance in the opposite x direction. So,"by symmetry" ALL of these components cancel. Leaving only the r component of force. Hence the e field only 'depends' on this, "by symmetry". If you know something about antennas, you know that the end of a wire is where emr fields get nasty. Correctly terminating a wire requires care and is expensive in the real world. So, for beginning physics, we use infinite wires, no nasty terminal field effects, symmetrical, it makes problems so much easier. And can be a good approximation when assumptions are met. Note that lots of real world problems can't be 'reduced' by symmetry. Often these problems can only be solved by numerical (approximate) methods - there just aren't any simple solutions. In fact, most real world problems aren't "analytically tractable". There's no analytical solution to designing the "optimum" antenna, for instance.
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