[NOTES/EM-07019] Solving for Vector Potential

Solving for vector potential

The equation for each component of the vector potential is similar to the equation \begin{equation} \nabla^2 \phi= -\frac{\rho}{\epsilon_0} \end{equation} which has solution \begin{equation}\label{EQ13}
\phi(\vec{r}) =\frac{1}{4\pi\epsilon_0}\iiint \frac{\rho(\vec{r}\Prime)}{|\vec{r}-\vec{r}\Prime|}, d^3r\Prime.
\end{equation}Therefore we can write the solution for the vector also in the form \begin{equation}\vec{A} =\frac{\mu_0}{4\pi} \iiint \frac{\vec{j}(\vec{r}\Prime)}{|\vec{r}-\vec{r}\Prime|}, d^3r\Prime. \label{EQ14} \end{equation}From the above equation we can arrive at the Biot-Savart law.

In fact computation of \(\vec{B}\) as \(\nabla \times\vec{A}\) leads to  \begin{equation}\label{EQ15} \vec{B}(\vec{r})=\frac{\mu_0}{4\pi}\iiint_V\frac{\vec{j}(\vec{r}\Prime)\times  \vec{R}}{R^3}\,d^3r\Prime, \quad \text{where}\quad \vec{R}=\vec{r}-\vec{r}\Prime.
\end{equation} The expression for the magnetic field, obtained above, of course satisfies \({\nabla}\cdot\vec{B}=0\)
The \eqRef{EQ15}, can be written down for line and and surface currents. \begin{equation}\label{EQ16}
\vec{A}(\vec{r}) = \frac{\mu_0}{4\pi}\int_\gamma\frac{I\overrightarrow{dl}}{R}, \qquad \vec{A}(\vec{r}) = \frac{\mu_0}{4\pi}\iint_S\frac{\vec{K}dS}{R}
\end{equation} where \(I\) is the current in circuit \(\gamma\) and \(\vec{K}\) is the surface current density and \(\vec{R}=\vec{r}-\vec{r}\Prime\). The line and surface integrals are over the wire and the area of surface carrying the current respectively. The integrations are over the variable \(\vec{r}\Prime\) which denotes the position of line and surface elements \(\overrightarrow{dl}\), \(dS\) in respective integrals.
The corresponding expressions for the magnetic field are \begin{equation}\label{EQ17} \vec{B}(\vec{r}) = \frac{\mu_0 I}{4\pi}\int_\gamma\frac{\overrightarrow{dl}\Prime\times \vec{R}}{R^3}, \qquad \qquad \vec{B}(\vec{r}) = \frac{\mu_0}{4\pi}\iint_S\frac{\vec{K}(\vec{r}\Prime)\times \vec{R}}{R^3}dS\Prime  \end{equation}