Effects on the exterior magnetic field from a field induced by a wire?

I have an experimental set-up where I place a conductive wire in a magnetic field created by an electromagnet like so: There is current flowing in a wire as well(aside from the electromagnet) displayed in this diagram: (Ignoring the Lorentz force acting on the wire and possibly the electromagnet)The magnetic field created by the electromagnet(that is composed of multiple turns and a ferromagnetic core like soft iron to increase the magnetic field) is $B$ acting on the wire that has current flowing within it. Likewise, the wire will induce it's own magnetic field $B_W$ as shown here: Usually, the $B_W$ is insignificant due to the distance and the amount of current flowing within it, However, if(neglecting resistance and fusing on the wire) the current was quite high enough to generate a magnetic field where this condition $B_W$ > $B$ is true, what would happen to $B$?

• It's all linear, so add the fields as vectors. – George Herold Apr 8 '16 at 1:37
• @GeorgeHerold Would the field of the wire affect the magnetization of the ferromagnetic core? If it is indeed was higher than B? Assume the gap was much much smaller, the field lines of the wire will be perpendicular to the applied field, and to the ferromagnetic core, possibly changing it's magnetization? Image: i.imgur.com/C970tC3.jpg – Pupil Apr 8 '16 at 2:05
• OK if the fields are near the saturation of the ferromagnetic material then indeed there can be some effect. Because of non-linearity in the magnetic material. (sorry for my quick response, with not enough thought.) – George Herold Apr 8 '16 at 12:44
• @GeorgeHerold If there is indeed a change in magnetization, is it possible that the magnetic field produced by the electromagnet is weakened? Due to the core's regional moments changing their direction? – Pupil Apr 16 '16 at 8:20

Assuming it's a 2D problem for simplicity, the net magnetic field would strengthen on one side of the coil and would weaken on the other side of the coil: - The picture above shows lines of flux adding at the top and subtracting at the bottom producing this effect (and a force): - 