Current loop as a magnetic dipole

Far along the axis (x ≫ R), the loop's field μ₀IR²/2(x² + R²)^(3/2) becomes B ≈ μ₀IR²/2x³ = μ₀m/2πx³ = (μ₀/4π)(2m/x³), with m = IπR².

This matches the axial field of an electric dipole, (1/4πε₀)(2p/x³), if p → m, E → B and 1/ε₀ → μ₀.

In the plane of the loop, far away, B ≈ (μ₀/4π)(m/x³), matching the equatorial field of an electric dipole. Both results become exact for a point magnetic dipole.

Any flat current loop is equivalent to a magnetic dipole of moment m = IA.

An electric dipole is made of two monopoles (charges). A magnetic dipole, a current loop, is itself the simplest unit: magnetic monopoles have never been found.

Because a loop makes a dipole field and feels torque like a compass needle, Ampere proposed that all magnetism comes from circulating currents. Electrons and protons, however, also carry an intrinsic magnetic moment not explained by any circulating current.