Electromagnetic potentials

The electromagnetic potentials $(V,\mathbf{A})$ are a scalar field and a vector field that together generate the Electric field and Magnetic field in the time-varying case:

$\mathbf{E}=-\nabla V-\frac{\partial \mathbf{A}}{\partial t},\mathbf{B}=\nabla \times \mathbf{A}.$

$V$ is the Electric potential (volts), $\mathbf{A}$ is the Vector magnetic potential (Wb/m). In statics, these reduce to $\mathbf{E}=-\nabla V$ and $\mathbf{B}=\nabla \times \mathbf{A}$ — the familiar separate descriptions. In dynamics, $V$ alone doesn’t capture $\mathbf{E}$, because Faraday’s law produces a curling (non-conservative) electric field that’s not the gradient of any scalar.

Why the extra ${\partial }_t\mathbf{A}$ term

In the static case, $\nabla \times \mathbf{E}=0$ lets you write $\mathbf{E}=-\nabla V$. In the dynamic case, $\nabla \times \mathbf{E}=-{\partial }_t\mathbf{B}$, which is generally nonzero. Substituting $\mathbf{B}=\nabla \times \mathbf{A}$:

$\nabla \times \mathbf{E}=-\frac{\partial }{\partial t}(\nabla \times \mathbf{A})=-\nabla \times \frac{\partial \mathbf{A}}{\partial t}.$

So $\nabla \times (\mathbf{E}+{\partial }_t\mathbf{A})=0$ — the combination $\mathbf{E}+{\partial }_t\mathbf{A}$ is curl-free, hence the gradient of some scalar (by the curl-of-gradient identity’s converse). Defining that scalar as $-V$:

$\mathbf{E}+\frac{\partial \mathbf{A}}{\partial t}=-\nabla V⟹\mathbf{E}=-\nabla V-\frac{\partial \mathbf{A}}{\partial t}.$

The $-\nabla V$ piece is the “static-like” Coulombic part; the $-{\partial }_t\mathbf{A}$ piece is the “induced” part, the EMF-source piece of Faraday’s law.

Why use potentials at all

Two reasons:

1. Four scalars instead of six. $\mathbf{E}$ and $\mathbf{B}$ together have six components. $(V,\mathbf{A})$ has four (after gauge-fixing — see below). Working with potentials is less redundant.

2. Two of Maxwell’s equations come for free.

   • $\nabla \cdot \mathbf{B}=0$ is automatic from $\mathbf{B}=\nabla \times \mathbf{A}$ (since $\nabla \cdot (\nabla \times \mathbf{A})=0$).
   • $\nabla \times \mathbf{E}=-{\partial }_t\mathbf{B}$ is automatic from the potential definitions (substitute and check).

   The remaining two — Gauss’s law and Ampère’s law — become differential equations for $V$ and $\mathbf{A}$ themselves, sourced by ${\rho }_v$ and $\mathbf{J}$.

3. Wave equations in Lorenz gauge. With the Lorenz gauge $\nabla \cdot \mathbf{A}+\mu ϵ{\partial }_{t}V=0$, the source equations reduce to symmetric wave equations:

${\nabla }^{2}V-\mu ϵ\frac{{\partial }^{2}V}{\partial t^2}=-\frac{{\rho }_v}{ϵ},{\nabla }^2\mathbf{A}-\mu ϵ\frac{{\partial }^2\mathbf{A}}{\partial t^2}=-\mu \mathbf{J}.$

Each component of $(V,\mathbf{A})$ obeys the same wave equation, with a source on the right. This is the cleanest formulation of dynamical electromagnetism. Solutions are retarded potentials — the field at $(\mathbf{r},t)$ depends on $({\rho }_v,\mathbf{J})$ at earlier times, with delays $|\mathbf{r}-{\mathbf{r}}^{\prime }|/c$ for propagation across space.

Gauge freedom

$\mathbf{E}$ and $\mathbf{B}$ don’t uniquely determine $(V,\mathbf{A})$. The substitution

$\mathbf{A}\to \mathbf{A}+\nabla \psi ,V\to V-\frac{\partial \psi }{\partial t},$

for any scalar $\psi (\mathbf{r},t)$ leaves $\mathbf{E}$ and $\mathbf{B}$ unchanged. Check: $\nabla \times (\mathbf{A}+\nabla \psi )=\nabla \times \mathbf{A}$ since $\nabla \times \nabla \psi =0$ (Curl of gradient identity); the ${\partial }_t\mathbf{A}$ shift in $\mathbf{E}$ is exactly cancelled by the $-\nabla V$ shift. This is gauge invariance.

Two common choices:

• Lorenz gauge: $\nabla \cdot \mathbf{A}+\mu ϵ{\partial }_{t}V=0$. Produces symmetric wave equations. Used in radiation, antenna, and relativistic problems.
• Coulomb gauge: $\nabla \cdot \mathbf{A}=0$. Makes $V$ obey the instantaneous Poisson equation, ${\nabla }^{2}V=-{\rho }_v/ϵ$, with $\mathbf{A}$ carrying all the dynamics. Used in magnetostatics and quantum field theory.

The physics doesn’t depend on the choice — only computational convenience does.

The four-potential

In relativistic notation, $V$ and $\mathbf{A}$ combine into a single four-component object $A^{\mu }=(V/c,\mathbf{A})$ that transforms as a four-vector under Lorentz boosts. Maxwell’s equations become a single covariant equation ${\partial }_{\mu }{F}^{\mu \nu }={\mu }_0{J}^{\nu }$ for the field tensor $F^{\mu \nu }={\partial }^{\mu }{A}^{\nu }-{\partial }^{\nu }{A}^{\mu }$. This is the natural language for electromagnetism in special relativity and the entry point to gauge field theory in particle physics.

In Electromagnetics

For coursework, the takeaway is: in time-varying problems, the electric field has two contributions — a Coulombic $-\nabla V$ and an inductive $-{\partial }_t\mathbf{A}$. Forgetting the second term is a common mistake in transformer-EMF and radiation problems.