The Field on the Axis of a Current Loop

Picture a circular loop of wire carrying current I. Every tiny segment of that loop creates a small magnetic field given by the Biot-Savart law, and each segment is the same distance from any point on the axis. But these little field vectors do not point straight along the axis — they point off at an angle. Only the component parallel to the axis survives when you add up contributions from the whole loop, because for every segment, there is one directly opposite it whose sideways (radial) component cancels. The axial components all add, since they all point the same way.

This gives the general result B = mu0 N I R² / (2(x² + R²)^3/2), where x is the distance along the axis from the center, R is the loop radius, and N is the number of turns. At the center, x = 0, so the formula collapses to B = mu0 N I / (2R) — the field is strongest there because every segment is as close as it can be to that point, and the cancellation of radial components is total and clean. Move away from center and each segment gets farther, and more of its field tilts away from the axis, so B drops.

Far from the loop, when x is much bigger than R, the formula behaves like B proportional to 1/x³ — the loop starts looking like a simple magnetic dipole, the same falloff you see for the electric field far from a dipole. This sets up the next idea: treating current loops as dipoles with a magnetic moment, which is exactly how bar magnets and atomic currents are modeled.