If we place a wire within a uniform exterior magnetic field ($B_x$) that's perpendicular to it's length, and allowed current to flow withing the that conductive medium, there is another magnetic field generated from that wire ($B_w$). There is a Lorentz force as well, however, due to the $B_w$ existing with $B_x$ would they add or subtract from the net strength of each field?
The orange lines is of the exterior magnetic field, wanted to illustrate it in the most important views(right & top).
As a result of the magnetic field, there is a Lorentz force, would $B$ in the formula be only $B_x$? Why wouldn't it be ($B_x$ - $B_s$) or ($B_x$ + $B_s$)? Likewise, if the wire was able to accelerate in the the magnetic field, the induced motional EMF's $B$ be the same?