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.

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✍️Mathematical Proof

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Product Rule for Curl

The analysis demonstrates the crucial role of the vector calculus identity for the curl of a cross product. This product rule is the primary tool used to compute $\nabla \times A$ and is essential for working with complex vector fields like the one for a magnetic dipole.

Mathematical Simplification

A key takeaway is the importance of understanding properties of the vectors involved. The analysis shows how knowing that $m$ is a constant vector ( $\nabla \cdot m=0$ ) and that $\nabla \cdot\left(x / r^3\right)=0$ for $r>0$ drastically simplifies the complex product rule identity, leading to a manageable calculation.

Physical Interpretation of the Result

The final result that $\nabla \times B=0$ for $r>0$ is a significant physical insight. This confirms that the magnetic field outside a magnetic dipole is solenoidal, meaning it has no sources or sinks. This is a fundamental principle of magnetostatics, indicating that magnetic field lines form closed loops and do not originate from a point source. The only location where $\nabla \times B$ would be non-zero is at the origin ( $r=0$ ), where the dipole source resides.

🎬Demonstration

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.

the magnetic field and vector potential of a magnetic dipole