Q:

Solenoid and induced electric fields

Let us suppose there is a solenoid which has $n$ turns per unit length the current is varying with time as $I =kt$ where k is a constant if the current is flowing then there must be induced electric field inside and outside the solenoid since $$\oint E_{induced}.dl= -\frac{d\phi}{dt}$$. now my question is if I place a small charge anywhere inside or outside the solenoid will it move with the induced electric field in a circle?
The induced electric field is in blue.please ignore the current equation in the figure.

A:

Yes, the charge will move under the influence of the induced E-field, but it will not move in a circle: this would require a centripetal force, but there is none. Instead, the (positive) charge will start to move in the direction of the E-field, then spiral outward.

