The electric field of an uniformly charged infinite surface plane is totally independent of how far away the test charge is. But isn't electric field depends upon distance by the relation $1/r^2$? I understand that if the plane is truly infinite then it does'nt matter how far away you are. But there is no such thing as infinite plane. So, is the independency of distance works in real or practical life?
1/r² (inverse square) is valid for a point charge. Infinite wires have 1/r dependency.
It is all consistent.
In fact, the electric field of an infinite plane cannot vary with with the inverse square law. If you apply Gauss theorem, you find that the electric field of that plane depends only on the surface charge distribution
The independency of distance works in real life, if you have a charge or a charged conductor that is small enough and close enough to the plane, it sees the electric field lines parallel to each other.
On the other hand, if you get close to the edge, you will get edge effects and the electric field lines will not be parallel.
In a real life example: when you calculate the electric acceleration of a charged particle through a synchrotron, like the LHC, you usually assume the particle to be small enough and the plates to be big enough so that no edge effects apply. It makes calculations easier, and it gives a good first approximation before getting in the real equations.