A: As usual, we need to solve Maxwell's equations with the appropriate boundary conditions, in this case Faraday's law. As we discussed, a solution for a uniform B field is \vec{E} = (a/2)s \hat{phi}, where s is the radial distance in cylindrical coordinates, and phi the angular one. You justifiably ask, but what's the center of the coordinate system, and the answer is, one can define the z axis to lie anywhere in space: all such solutions solve the differential equation. This is math; physics requires asking what are the sources of the B field that caused the trouble? If for example we imagine a huge solenoid which has a diameter of a light year, the field will be uniform over a huge volume. However the presence of the solenoid, even if very far away, selects among the various solutions to Faraday's law inside it. The boundary condition on the E field at the surface of the solenoid forbids a perpendicular component; thus the axis of the solution must be the axis of the solenoid. Note that outside the solenoid the solution is different, of course: the E field is still azimuthal, but since there is no curl anymore (since there's no field), the magnitude of E falls off like 1/s.