I have been looking at some derivations for the electric field inside a solenoid. I know how to find it, but I don't get the symmetry argument used. This is often of the form:
Since if we choose a circular path (coaxial to the solenoid) every point on the path is equivalent the electric field must be tangential .
This, to me, does not seem to be sound reasoning. There are many other directions which the electric field could be in and still have each point on the circle be equivalent. So why, (preferable) using a symmetry argument, can we assume that the electric field at each point is tangential to the circular path (shown below)?