Fill in the blanks. [12 pts]
Maxwell developed the time-varying form of Ampere's law by adding the ____ term, such that
$\nabla \times \vec{B} = \mu \vec{J} +$ ____
Addition of this term makes the differential form of Ampere's law consistent with the equation of ____, which is given as
$\nabla \cdot \vec{J} =$ ____
To simplify Maxwell's equations, we use the potential functions $\vec{A}$ and $V$, and the fact that the potential function for a given magnetic or electric field is not unique. In time-varying field, using the Faraday's law, the electric field can be expressed in terms of $\vec{A}$ and $V$ as,
$\vec{E} =$ ____
Applying the above equation to the Ampere's law, with the choice of divergence of $\vec{A}$ as
$\nabla \cdot \vec{A} =$ ____
we can uncouple the wave equations for $\vec{A}$ and $V$ from the Maxwell's equations.
