Physics

The key insight is that magnetic field contributions are vectors that add linearly—each turn of the coil produces its own field at the center, and since all turns are identical and coaxial, their fields point in the same direction and simply add together. Rather than deriving the full formula, focus on how the $N$ turns stack: if one loop creates field $B_1$, then $N$ identical loops each carrying the same current $I$ create a total field $NB_1$, making the superposition argument the heart of the proof.

The key insight is that magnetic field contributions are vectors that add linearly—each turn of the coil produces its own field at the center, and since all turns are identical and coaxial, their fields point in the same direction and simply add together. Rather than deriving the full formula, focus on how the $N$ turns stack: if one loop creates field $B_1$, then $N$ identical loops each carrying the same current $I$ create a total field $NB_1$, making the superposition argument the heart of the proof.