Biot savart law maxwell equations biography

Starting form Maxwell's equations for magnetostatics, vector potential is introduced and the Biot Savart Law is derived

1. Biot Savart Law from Maxwell's equations

For magnetostatics we have the Maxwell's equation $$ \vec{\nabla}\cdot\vec{B}=0~,~~{\nabla}\times\vec{B}=\mu_0\vec{J}\,.

$$ The equation ${\nabla}\cdot\vec{B}=0$ implies that there exists a vector field $\vec{A}(x)$ such that $$ \bar{B}=\nabla\times\bar{A} $$ Given $\vec{B},\vec{A}$ is not unique.

12.1 The Biot-Savart Law - University Physics Volume 2 - OpenStax

This can be seen as follows consider $\vec{A}$ and $\vec{A}^\prime=\vec{A}+{\nabla}\Gamma$ where $\Lambda(x)$ is an arbitrary function of $\vec{x}$. Then \begin{align*} \nabla\times\bar{A}^\prime = & \nabla\times A+\nabla\times (\nabla\Gamma)\\ = & \nabla\times\bar{A} \end{align*} Thus given magnetic field $\vec{B}$, there are solutions $\vec{A}^\prime$ related by ${\nabla}\Lambda$.

To make $\vec{A}$ unique we have to supply a condition on $\vec{A}$, cal Deriving Biot-Savart Law from Maxwell's Equations QABI