Consider an ideal solenoid that is infinitely long. As the current flowing through the solenoid varies over time, I am taught that an electric field will be induced and the field lines will be circles that are concentric with the axis of the solenoid.
Why does this have to be the case? Why can't there be a component of the electric field that is parallel to the solenoid?
It would be helpful for somebody to explain this directly from Maxwell's equations.
My own guess (if you know the answer please tell me if I am right or wrong): Focusing on Maxwell's equations in vacuum. The rate of change of B-field is equal to the negative of the curl of E-field. This implies that, when the B-field's strength is changed with it's direction being kept constant (which is the case here), the E-field created must be normal to the B-field's direction. We also know that, in vacuum, the divergence of the E-field is 0. The only possibility is for the E-field to be circles that are concentric with the axis of the solenoid.