6.2 Vector potential

Definition of vector potential

The vector potential $\vec{A}$ exists because of a fundamental identity in vector calculus: the divergence of any curl is always zero. Since Maxwell's equations require $\nabla \cdot \vec{B} = 0$ everywhere, the magnetic field can always be written as the curl of some vector field. That vector field is $\vec{A}$.

The defining relation is:

$\vec{B} = \nabla \times \vec{A}$

The units of $\vec{A}$ are Tesla-meters (T·m), or equivalently $\text{V·s/m}$ (volt-seconds per meter).

Why bother with $\vec{A}$ when you already have $\vec{B}$? Because in many problems, especially those involving radiation, quantum mechanics, or complex geometries, working with potentials turns coupled vector equations into simpler scalar ones. The vector potential also appears directly in the Lagrangian and Hamiltonian formulations of electrodynamics, making it more fundamental than $\vec{B}$ in some contexts.

Relation to magnetic field

The relation $\vec{B} = \nabla \times \vec{A}$ automatically guarantees one of Maxwell's equations:

$\nabla \cdot \vec{B} = \nabla \cdot (\nabla \times \vec{A}) = 0$

This follows from the vector identity that the divergence of any curl vanishes identically. So by constructing $\vec{B}$ from a vector potential, you never have to worry about satisfying the no-monopole condition separately.

The reverse direction matters too: given any divergence-free field $\vec{B}$, the Helmholtz decomposition theorem guarantees that at least one $\vec{A}$ exists such that $\vec{B} = \nabla \times \vec{A}$. However, that $\vec{A}$ is not unique. This non-uniqueness is the origin of gauge freedom, discussed next.

Gauge transformations

If $\vec{A}$ produces a given $\vec{B}$, then so does any transformed potential:

$\vec{A}' = \vec{A} + \nabla \lambda$

where $\lambda(\vec{r}, t)$ is an arbitrary scalar function. This works because the curl of a gradient is always zero:

$\nabla \times \vec{A}' = \nabla \times \vec{A} + \nabla \times (\nabla \lambda) = \nabla \times \vec{A} = \vec{B}$

The magnetic field is completely unchanged. In the full time-dependent theory, you must also transform the scalar potential simultaneously:

$\phi' = \phi - \frac{\partial \lambda}{\partial t}$

to keep the electric field $\vec{E} = -\nabla \phi - \frac{\partial \vec{A}}{\partial t}$ invariant as well.

This freedom to choose $\lambda$ is called gauge freedom, and picking a specific condition on $\vec{A}$ is called fixing a gauge. The two most common choices are below.

Coulomb gauge

The Coulomb gauge imposes:

$\nabla \cdot \vec{A} = 0$

This is the natural choice for magnetostatics. Under this condition, substituting $\vec{B} = \nabla \times \vec{A}$ into Ampère's law ($\nabla \times \vec{B} = \mu_0 \vec{J}$) and using the vector identity $\nabla \times (\nabla \times \vec{A}) = \nabla(\nabla \cdot \vec{A}) - \nabla^2 \vec{A}$ gives:

$\nabla^2 \vec{A} = -\mu_0 \vec{J}$

This is a vector Poisson equation, directly analogous to $\nabla^2 \phi = -\rho/\epsilon_0$ in electrostatics. Each Cartesian component of $\vec{A}$ satisfies its own scalar Poisson equation, which is a significant computational simplification.

In the Coulomb gauge, $\vec{A}$ is purely transverse: it has no longitudinal (irrotational) component. This makes it especially useful in radiation problems where you want to cleanly separate transverse radiative degrees of freedom from longitudinal Coulomb interactions.

Lorenz gauge

The Lorenz gauge imposes:

$\nabla \cdot \vec{A} + \mu_0 \epsilon_0 \frac{\partial \phi}{\partial t} = 0$

In covariant notation, this is $\partial_\mu A^\mu = 0$. The key advantage is that it decouples the equations for $\vec{A}$ and $\phi$ into separate wave equations:

$\nabla^2 \vec{A} - \mu_0 \epsilon_0 \frac{\partial^2 \vec{A}}{\partial t^2} = -\mu_0 \vec{J}$

$\nabla^2 \phi - \mu_0 \epsilon_0 \frac{\partial^2 \phi}{\partial t^2} = -\frac{\rho}{\epsilon_0}$

Both equations have the same symmetric structure (inhomogeneous wave equations), which is why the Lorenz gauge is the standard choice for electrodynamics and relativistic calculations. It also respects Lorentz covariance, meaning it holds in all inertial frames.

Calculation methods

From current distributions

When you know the current density $\vec{J}(\vec{r}')$, you can compute $\vec{A}$ directly:

$\vec{A}(\vec{r}) = \frac{\mu_0}{4\pi} \int \frac{\vec{J}(\vec{r}')}{|\vec{r} - \vec{r}'|} \, d^3r'$

This is the magnetostatic integral for the vector potential, analogous to the Coulomb integral for $\phi$. For a line current $I$ along a wire, it reduces to:

$\vec{A}(\vec{r}) = \frac{\mu_0 I}{4\pi} \int \frac{d\vec{l}'}{|\vec{r} - \vec{r}'|}$

Steps to use this method:

1. Identify the current distribution $\vec{J}(\vec{r}')$ (or $I \, d\vec{l}'$ for a wire).

2. Write out the separation vector $|\vec{r} - \vec{r}'|$ in your chosen coordinate system.

3. Evaluate the integral over the source region. Exploit symmetry to eliminate components where possible.

4. Once you have $\vec{A}$, take $\vec{B} = \nabla \times \vec{A}$.

This approach is most practical when the geometry has enough symmetry to make the integral tractable (infinite wires, solenoids, circular loops at special points, etc.).

From boundary conditions

When the current distribution isn't known directly, you solve the relevant differential equation with boundary conditions instead:

• In magnetostatics (Coulomb gauge): solve $\nabla^2 \vec{A} = -\mu_0 \vec{J}$ with appropriate boundary conditions.
• In electrodynamics (Lorenz gauge): solve the inhomogeneous wave equation for $\vec{A}$.

Typical boundary conditions include:

• The tangential component of $\vec{A}$ is continuous across an interface.
• The normal component of $\vec{B} = \nabla \times \vec{A}$ is continuous across an interface (from $\nabla \cdot \vec{B} = 0$).
• The discontinuity in the tangential component of $\vec{B}$ across a surface is related to the surface current density $\vec{K}$.

This method is common in waveguide problems, cavity problems, and situations with piecewise-constant material properties.

Properties of vector potential

Non-uniqueness

The vector potential for a given $\vec{B}$ is never unique. You can always add $\nabla \lambda$ for any scalar $\lambda$ without changing the physics. The curl operation "kills" gradient terms, so all the information in $\nabla \lambda$ is invisible to $\vec{B}$.

Fixing a gauge (Coulomb, Lorenz, or another) pins down $\vec{A}$ more tightly, but even within a given gauge, global boundary conditions may still leave residual freedom. The physical content is always in the fields $\vec{E}$ and $\vec{B}$, or more precisely, in gauge-invariant quantities.

Gauge invariance

All measurable electromagnetic quantities are gauge-invariant. The fields $\vec{E}$ and $\vec{B}$, the Poynting vector, the energy density, and the Lorentz force on a charge are all unchanged by gauge transformations.

This principle extends into quantum mechanics. The Schrödinger equation for a charged particle in an electromagnetic field involves $\vec{A}$ and $\phi$ explicitly, but all observable predictions (probabilities, expectation values) remain gauge-invariant. The wavefunction picks up a local phase factor under a gauge transformation, but this phase cancels out of any physical observable.

The Aharonov-Bohm effect provides a striking illustration: a charged particle traveling through a region where $\vec{B} = 0$ can still be affected by $\vec{A}$ if it encircles a region of nonzero magnetic flux. The observable quantity is the line integral $\oint \vec{A} \cdot d\vec{l} = \Phi_B$ (the enclosed magnetic flux), which is gauge-invariant even though $\vec{A}$ itself is not.

Applications

Magnetostatics

In magnetostatics, $\vec{A}$ provides a systematic route to $\vec{B}$ for any steady current distribution. You compute $\vec{A}$ from the integral formula, then take the curl. This is often easier than computing $\vec{B}$ directly from the Biot-Savart law for $\vec{B}$, because the vector potential integral doesn't involve a cross product.

Common examples: the vector potential of an infinite solenoid ($\vec{A}$ is nonzero outside even though $\vec{B} = 0$ there), current loops (leading to the magnetic dipole expansion at large distances), and current-carrying wires.

Electrodynamics

For time-dependent problems, the potentials $\vec{A}$ and $\phi$ satisfy wave equations (in the Lorenz gauge) whose solutions are the retarded potentials:

$\vec{A}(\vec{r}, t) = \frac{\mu_0}{4\pi} \int \frac{\vec{J}(\vec{r}', t_r)}{|\vec{r} - \vec{r}'|} \, d^3r'$

where $t_r = t - |\vec{r} - \vec{r}'|/c$ is the retarded time. These retarded potentials are the starting point for deriving radiation fields from accelerating charges (Lienard-Wiechert potentials) and antenna theory.

Quantum mechanics

In quantum mechanics, the Hamiltonian for a particle of charge $q$ in an electromagnetic field is:

$H = \frac{(\vec{p} - q\vec{A})^2}{2m} + q\phi$

The canonical momentum $\vec{p}$ differs from the kinetic momentum $m\vec{v}$ by $q\vec{A}$. This coupling between $\vec{A}$ and the wavefunction is what produces the Aharonov-Bohm effect and underlies the gauge structure of quantum electrodynamics (QED).

Comparison to scalar potential

Role in electromagnetic waves

In the Lorenz gauge, both potentials satisfy inhomogeneous wave equations with source terms $\rho$ and $\vec{J}$. In free space (no sources), these become homogeneous wave equations, and the solutions are plane waves propagating at speed $c$.

The vector potential is particularly useful for describing the polarization of electromagnetic waves. For a plane wave propagating in the $\hat{z}$ direction, $\vec{A}$ lies in the transverse plane (the $xy$-plane in the Coulomb gauge). The direction of $\vec{A}$ directly encodes the polarization state: linear, circular, or elliptical.

In the Coulomb gauge with no free charges, $\phi = 0$ and the fields are determined entirely by $\vec{A}$:

$\vec{E} = -\frac{\partial \vec{A}}{\partial t}, \qquad \vec{B} = \nabla \times \vec{A}$

This makes $\vec{A}$ the single dynamical variable of the free electromagnetic field, which is why it serves as the fundamental field in the quantization of electrodynamics.

Analogy to fluid dynamics

Velocity potential

In fluid dynamics, an irrotational flow ($\nabla \times \vec{v} = 0$) can be written as $\vec{v} = \nabla \Phi$, where $\Phi$ is the velocity potential. This parallels how $\vec{E} = -\nabla \phi$ in electrostatics (where $\nabla \times \vec{E} = 0$). Both cases exploit the fact that a curl-free field can be written as a gradient.

Stream function

For two-dimensional incompressible flow ($\nabla \cdot \vec{v} = 0$), you can define a stream function $\psi$ such that the velocity components are $v_x = \partial \psi / \partial y$ and $v_y = -\partial \psi / \partial x$. This is analogous to the vector potential in 2D magnetostatics: the condition $\nabla \cdot \vec{B} = 0$ is automatically satisfied by writing $\vec{B} = \nabla \times \vec{A}$, and in two dimensions $\vec{A}$ reduces to a single scalar component that plays the same mathematical role as $\psi$.

Advanced topics

Aharonov-Bohm effect

The Aharonov-Bohm effect shows that $\vec{A}$ has physical consequences beyond determining $\vec{B}$. In the classic setup, electrons travel along two paths around a solenoid. Outside the solenoid, $\vec{B} = 0$, but $\vec{A} \neq 0$. The two paths pick up different phases proportional to $\oint \vec{A} \cdot d\vec{l}$, and the resulting interference pattern shifts depending on the enclosed flux $\Phi_B$.

The observable is the enclosed flux, which is gauge-invariant. So while $\vec{A}$ itself is gauge-dependent, its line integral around a closed loop enclosing flux is not. This effect has been confirmed experimentally and highlights the topological nature of gauge fields: the result depends on the winding of the path around the flux tube, not on local field values.

Magnetic monopoles

If magnetic monopoles existed, $\nabla \cdot \vec{B} = 0$ would no longer hold everywhere, and you could not globally define $\vec{A}$ such that $\vec{B} = \nabla \times \vec{A}$. Dirac showed that a consistent quantum theory of a monopole requires the introduction of a "Dirac string" (a line singularity in $\vec{A}$) and leads to the quantization condition:

$qg = \frac{n\hbar}{2}$

where $q$ is the electric charge, $g$ is the monopole strength, and $n$ is an integer. This would explain why electric charge is quantized. No magnetic monopoles have been observed to date, but they appear naturally in many grand unified theories.

Topological considerations

The Aharonov-Bohm effect is one instance of a broader theme: the topology of the space through which $\vec{A}$ is defined matters. When the domain is not simply connected (e.g., it has a hole, like the region outside a solenoid), gauge-inequivalent potentials can exist that give the same local $\vec{B}$ but differ in their global properties (specifically, in the holonomy $\oint \vec{A} \cdot d\vec{l}$).

These ideas connect to fiber bundle theory in mathematics and underpin modern gauge theories in particle physics. The Berry phase in quantum mechanics is another manifestation of the same geometric structure: a system's wavefunction can acquire a measurable phase from the topology of its parameter space, analogous to how $\vec{A}$ encodes topological information about magnetic flux.