FLUXGATE magnetometer - how it works

Assuming that I am looking at an 'ideal' fluxgate magnetometer.


I am feeding this transformer an ideal sine wave. The secondary winding output and the primary winding input are fed into two channels of an oscilloscope.

A magnetic field from the outside (like earth magnetic field) is applied.

Would I get a shift of the output signal in relationship to the input signal by X or by Y axis?



VS


or


Would this depend on the angle of the magnetic flux in relationship to the transformer?
Would this be different for a bar transformer vs a ring transformer.

Thank you.