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I understand that I can define the direction of the E field by
considering a loop in the solenoid coaxial with the solenoid and then use
Lenz's Law to find the direction of the induced current in the loop and
then use this direction as the direction of the E field. I will then
have defined the direction of the induced Electric Field at a specific
point in the solenoid. My real problem is this: I can now put a metal
loop in the solenoid that passes through that point and is not coaxial to
the solenoid in such a way that the induced current in that loop
(according to Lenz's Law) is opposite to the direction predicted by my
original coaxial loop. Does that mean that the current in this new loop
is opposite to the direction of the E field?