Analysis of a Divergence-Free Vector Field

The analysis of the divergence-free vector field $v$ and its vector potential $A$ visually and mathematically demonstrates a fundamental principle of vector calculus: a field with no sources or sinks can be fully described by a second field-its vector potential-whose curl generates the first field. This shows that the rotational nature of $A$ gives rise to the outward flow of $v$, linking the two seemingly distinct properties in an elegant and consistent way.

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✍️Mathematical Proof

$\gg$Mathematical Structures Underlying Physical Laws

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Verifying the Vector Field is Divergence-Free

The core takeaway is that the vector field $v=\frac{1}{r^2} e_r$ is divergence-free $(\nabla \cdot v=0)$ for all points where $r>0$. This property is fundamental to the field's behavior. A divergence-free field is one that has no sources or sinks; in other words, what flows into a given volume must flow out, leaving no net change.

The mathematical proof relies on applying the divergence operator in spherical coordinates, which is the most natural coordinate system for this vector field due to its radial symmetry.

Finding the Vector Potential

The analysis successfully demonstrates how to find a vector potential $A$ for a divergence-free field. The existence of a vector potential is guaranteed by the property that $\nabla \cdot v=0$. The key steps are:

Solve the Curl Equation: By setting $v=\nabla \times A$, we used the given form of $A$ to establish a relationship between the components of $v$ and the unknown $A_{\varphi}$. This led to an expression for $A_{\varphi}$ in terms of $r$ and $\theta$.
Verify Divergence of the Potential: The final check confirmed that the derived vector potential, $A=-\frac{\cot \theta}{r} e_{\varphi}$, is also divergence-free ( $\nabla \cdot A=0$ ). This is an important consistency check, as it ensures the potential itself does not have any sources or sinks, a common requirement in many physical applications.

🎬Demonstration

The animation is the visual confirmation of the relationship between a divergence-free vector field and its vector potential. The radial field $v$ shows vectors decreasing in length as they move away from the center. This visualizes the concept of a source without a sink-a field that, while expanding, maintains a constant "flux" across any given surface. The animation's pulsating effect reinforces that the field is present everywhere, but its source is not a single point; rather, the "flux" is conserved. The vector potential $A$ shows vectors that are purely rotational, circling the origin. This swirling pattern is a visual representation of a field whose curl generates the radial field of $v$. The animation demonstrates that the "twisting" motion of $A$ is what gives rise to the outward "flow" of $v$. This is a crucial concept, as it shows that a divergence-free field can be described as the curl of another field, its vector potential.

the two vector fields have distinct behaviors with the rotational potential field A acting as the source for the radial field v through the curl operation