Maxwell's Equations
$$\require{physics}$$
Maxwell's Equations:
$$ \divergence{ \vb{E} } = \frac{\rho}{\epsilon_0} $$
$$ \curl{ \vb{E} } = - \pdv{ \vb{B} }{t} $$
$$ \curl{ \vb{E} } = - i \omega \vb{B} $$
$$ \divergence{ \vb{B} } = 0 $$
$$ \curl{ \vb{B} } = \mu_0 \epsilon_0 \pdv{ \vb{E} }{t} + \mu_0 \vb{J} $$
$$ \curl{ \vb{B} } = \frac{1}{c^2} \pdv{ \vb{E} }{t} + \mu_0 \vb{J} $$
$$ \curl{ \vb{B} } = \frac{1}{c^2} ( \pdv{ \vb{E} }{t} + \frac{\vb{J}}{\epsilon_0} ) $$
E and B fields defined in terms of Magnetic Vector Potential, A, and Electric Scalar Potential, V
$$ \vb{E} = - \grad{V} - \pdv{\vb{A}}{t} $$
$$ \vb{B} = \curl{\vb{A}} $$
It has been suggested by van Vlaenderen [1] that it is meaningful to speak of a third, scalar field, if we consider the case where:
$$
\divergence{\vb{A}} + \frac{1}{c^2} \pdv{V}{t} \ne 0
$$
That is, by not choosing the Lorenz Gauge. We'll call this the AT Field. It is defined as:
$$
\mathbf{T} = - \lambda_0 \epsilon_0 \mu_0 \pdv{V}{t} - \divergence{\vb{A}}
$$