B2 - potentials and source terms

\begin{matrix} \vec{\nabla} \times \vec{B} - \frac{1}{c^{2}} \frac{\partial \vec{E}}{\partial t} = μ_{0} \vec{j} \\ \vec{\nabla} \times \vec{\nabla} \times \vec{A} - \frac{1}{c^{2}} \frac{\partial}{\partial t} (- \vec{\nabla} ϕ - \frac{\partial \vec{B}}{\partial t}) = μ_{0} \vec{j} \\ \vec{\nabla} (\vec{\nabla} \cdot \vec{A}) - \nabla^{2} \vec{A} + \frac{1}{c^{2}} \vec{\nabla} (\frac{\partial ϕ}{\partial t}) + \frac{1}{c^{2}} \frac{\partial^{2} \vec{A}}{\partial t^{2}} = μ_{0} \vec{j} \\ \frac{1}{c^{2}} \frac{\partial^{2} \vec{A}}{\partial t^{2}} - \nabla^{2} \vec{A} = μ_{0} \vec{j} \end{matrix}

\frac{1}{c^{2}} \frac{\partial^{2}}{\partial t^{2}} - \nabla^{2} = ◻

◻ \vec{A} = μ_{0} \vec{j}

\begin{matrix} \vec{\nabla} \cdot \vec{E} = \frac{ρ}{ε_{0}} \\ - \vec{\nabla} \cdot (\vec{\nabla} ϕ + \frac{\partial \vec{A}}{\partial t}) = \frac{ρ}{ε_{0}} \\ - \nabla^{2} ϕ + \frac{\partial}{\partial t} (\vec{\nabla} \cdot \vec{A}) = \frac{ρ}{ε_{0}} \\ - \nabla^{2} ϕ + \frac{1}{c^{2}} \frac{\partial^{2} ϕ}{\partial t^{2}} = \frac{ρ}{ε_{0}} \\ ◻ ϕ = \frac{ρ}{ε_{0}} \end{matrix}

\begin{matrix} ◻ \vec{E} = 0 \\ ◻ \vec{B} = 0 \\ ◻ \vec{A} = μ_{o} \vec{j} \\ ◻ ϕ = \frac{ρ}{ε_{0}} \end{matrix}