Why is the electric field inside a solenoid tangential?

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)?

• I think you misunderstand the geometry described. If you change your conductors to go circularly around the rod (solenoid outline) you have drawn, it will make sense. Then the electric field accelerates (or decelerates) electrons in the direction of the conductor, which is exactly what you want/need for an electromotoric force (EMF) or induced voltage. – pyramids Apr 3 '15 at 7:35
• It would make more sense if the statement was "We have found the E field is tangential at one point. Since every point on the circle is equivalent, it must be tangential at all points on the circle." – mmesser314 Apr 3 '15 at 13:28