 Solution

Let $B_E>0$ be the northward component of the ambient magnetic field. Define $\eta =+1$ if positive north-to-south current in A produces a northward field, and $\eta =-1$ otherwise. With opposite windings and the same signed current in both,A compass reverses only when this northward component becomes negative. For current chosen to make the coil field southward,At equality the ideal axial field is zero and the needle has no preferred axial direction. If $n_1=n_2$, the opposing coil fields cancel and no finite current reverses it in this model. A numerical threshold cannot be given without $B_E$ and the coil parameters.