🧄Computing the Magnetic Field and its Curl from a Dipole Vector Potential

The complex, swirling vector potential ( $A$) of a magnetic dipole is not a mere mathematical construct but the direct source of the familiar, closed-loop magnetic field ( $B$ ). This relationship is governed by the curl operator, $B= \nabla \times A$, which mathematically proves that a field with rotation can produce a field with no sources or sinks. The demonstration of $\nabla \times B=0$ for $r>0$ further highlights a fundamental principle of magnetostatics: magnetic fields are solenoidal and form closed loops everywhere except at their source.

🎬the magnetic field and vector potential of a magnetic dipole

The demo is that the vector potential ( A ) and the magnetic field ( B ) of a dipole are visually and fundamentally different, yet mathematically connected. The demo illustrates that the swirling, rotational nature of the vector potential (the yellow vectors) directly generates the closed, looping magnetic field lines (the blue vectors). This is a physical demonstration of the relationship $B=\nabla \times A$, confirming that the curl operator transforms a "swirling" field into a "looping" field.

🖊️Mathematical Proof

Computing the Magnetic Field and its Curl from a Dipole Vector Potential | Proof and Derivationviadean on Notion