The question arises as to how the information on the presence of the magnetic field from inside the solenoid reaches the electric charge. In terms of the fields B and E the explanation is very simple: the variation of B inside the solenoid produces an electric field both inside and outside the solenoid, in the same way in which a charge distribution produces an electric field both inside and outside the distribution. In this sense the information from inside and outside is mediated by the electric field which must be continuous over all space due to the Maxwell equations and their boundary condition.