Maxwell’s equations and gauge potentials

One consistent field theory

A changing magnetic field produces electric circulation. A changing charge distribution cannot be described by a stationary current law alone. These two observations require electricity and magnetism to be solved together. Maxwell’s equations tell us which spatial and temporal changes can coexist, while the Lorentz force tells us how the resulting fields act on a charged particle.

The central achievement is consistency. Charge continuity, the compatibility of two surfaces spanning one loop, propagation and electromagnetic energy transport follow from a common system. A formula obtained for a stationary source remains useful in its stated regime. It must be reconsidered when its neglected derivatives become relevant; substituting a time-dependent current into every static formula does not generally produce a Maxwell solution.

Potentials make this system easier to solve and connect it to mechanics and quantum theory. They introduce a deliberate redundancy: several pairs of scalar and vector potentials describe the same electric and magnetic fields. Learning to use that freedom requires keeping a gauge convention separate from a physical field, a boundary condition and an approximation.

Choose a first and an advanced path

Begin with the four laws, derive charge continuity, and understand why the displacement term is needed. Then verify that B = ∇ × A and E = −∇φ −∂A/∂t satisfy the two homogeneous laws. Follow one explicit gauge transformation all the way through the cancellation in E and B. These steps establish the framework before any sophisticated solution method is introduced.

The advanced path derives the coupled potential equations, separates Coulomb and Lorenz gauges, and checks a complete original local Maxwell solution in both descriptions. Return to constraint propagation, causal initial and boundary data, constitutive response, energy balance, topology and the action. These details explain why an attractive-looking potential or a locally curl-free field can still be an incomplete answer.

This lesson uses smooth classical fields where derivatives are written pointwise, with singular sources treated through integral or distributional laws. Its numerical illustrations are original analytic examples, rather than measurements of a capacitor or magnet. Detailed wave polarization, radiation patterns and relativistic field transformations follow in the next lesson. The present focus is the structure that makes those calculations possible.

Declare fields, sources and SI units

In vacuum-form equations, ρ is total charge density in C/m³ and J is total electric current density in A/m². Electric field E is in V/m, magnetic flux density B in tesla, scalar potential φ in volt, and vector potential A in V·s/m, equivalently T·m. A scalar gauge function χ has units V·s. Its spatial gradient can be added to A, and its time derivative can be subtracted from φ.

Write c₀² = 1/(ε₀μ₀), with c₀ the vacuum light speed and ε₀, μ₀ the vacuum constants in modern SI. Do not replace c₀ by a material’s phase velocity in the vacuum equations. In material models, the response of polarization and magnetization changes the relation between macroscopic sources and fields; it does not change the definitions of the vacuum constants.

The term ε₀∂E/∂t has units A/m² and can be added to J in the Ampère–Maxwell equation. Calling it displacement current density does not mean that ordinary charge carriers cross an insulating gap with that density. The name denotes its role in the field equation. A polarizable medium can also carry a time-dependent polarization contribution, which must be distinguished from both vacuum displacement and conduction.

Put the four laws together

The differential vacuum-form system is ∇·E = ρ/ε₀, ∇·B = 0, ∇ × E = −∂B/∂t and ∇ × B = μ₀J + μ₀ε₀∂E/∂t. The divergence equations constrain flux and the curl equations relate circulation to sources and time evolution. An electric field can have nonzero curl even when its divergence is zero, so a charge-density calculation by itself does not determine a time-dependent E.

For a fixed oriented volume and its outward boundary, the flux laws are ∮E·n̂ dA = Q/ε₀ and ∮B·n̂ dA = 0. For a fixed oriented contour C bounding S, Faraday gives ∮_C E·dl = −d(∫_S B·n̂ dA)/dt. Ampère–Maxwell gives ∮_C B·dl = μ₀∫_S J·n̂ dA + μ₀ε₀ d(∫_S E·n̂ dA)/dt. A single right-hand convention links each contour to its surface normal.

The word fixed matters in these integral statements. A moving material circuit has a motional contribution to its emf, as derived in the preceding lesson. Applying a transport theorem to an appropriate moving surface connects that circuit result to Faraday’s local law. It does not change the local Maxwell equation into a new law for each moving apparatus.

Derive the displacement-term consistency

Take the divergence of Ampère–Maxwell. The divergence of a curl vanishes, so 0 = μ₀∇·J + μ₀ε₀∂(∇·E)/∂t. Using Gauss’s law gives ∂ρ/∂t + ∇·J = 0. Thus charge continuity is a compatibility condition of the complete Maxwell system. It allows charge to move and accumulate, while prohibiting an arbitrary local creation term in an otherwise unchanged set of equations.

The stationary equation ∇ × B = μ₀J alone would require ∇·J = 0. This is appropriate for a compatible steady current, but fails where the charge density changes. Adding ε₀∂E/∂t repairs that conflict because the divergence of the added contribution is ∂ρ/∂t. Continuity motivates the needed divergence; the full form and coefficient of the Maxwell equation are physical content, not a claim that continuity uniquely excludes every possible mathematical modification.

In a fixed volume, dQ/dt = −∮J·n̂ dA. Positive outward conventional current decreases the enclosed charge. The sign does not depend on whether the mobile carriers are positive or negative; their charge is already included in J. An assumed source profile must satisfy this relation before a potential solver can represent it as a Maxwell source.

An isolated prescribed point charge whose magnitude changes without an accompanying compatible current is consequently not a complete example. Nor is a wire segment with current disappearing at one stationary endpoint. Reservoirs, surface charge, external ports or other current paths must be included when they are what makes a physical source description consistent.

Compare two spanning surfaces

A loop beside a charging capacitor can be spanned by a surface cutting a feed wire or by one that passes through the insulating gap. In a suitable quasistatic plate model, the first includes conduction current I. In the second, the changing electric flux contributes ε₀dΦ_E/dt = dQ/dt = I, even though no conduction current crosses the ideal gap. Using conduction alone would make the answer depend on the chosen surface.

More generally, the divergence of J + ε₀∂E/∂t is zero when continuity and Gauss’s law hold. The flux of this total Ampère–Maxwell source through two suitable surfaces with a common boundary therefore agrees. That is the surface consistency required by one circulation of B. It remains true when conduction and displacement both contribute on each surface.

The familiar parallel-plate picture usually neglects fringing, propagation, finite wire fields and parts of the instrument geometry. It illustrates the source accounting; it does not supply an exact field everywhere around a finite charging capacitor. Our later cylindrical example instead declares a local exact vacuum solution and deliberately makes no abrupt join to an empty exterior.

Propagate the divergence constraints

The curl equations can be viewed as evolution equations: ∂B/∂t = −∇ × E and ∂E/∂t = c₀²∇ × B −J/ε₀. They evolve fields from initial data in a domain supplied with compatible boundary and source information. The divergence equations restrict the initial data rather than providing independent formulas for every later derivative.

Taking a divergence of the first evolution equation gives ∂(∇·B)/∂t = 0. Taking a divergence of the second and using charge continuity gives ∂(∇·E −ρ/ε₀)/∂t = 0. Consequently, if both constraints hold initially and the equations and sources remain compatible, the evolution preserves them. Evolution does not automatically repair an initially nonzero constraint error.

This distinction matters in analytical and numerical work. Choosing arbitrary initial E, B, ρ and J can violate Gauss’s law or continuity even when a program integrates the curl equations accurately. A discretization must preserve or control the corresponding discrete constraints. A small energy error alone is not proof that charge and magnetic flux constraints have been respected.

Introduce potentials locally

On a suitable local domain, ∇·B = 0 permits a vector potential B = ∇ × A. Substitute it into Faraday’s law: ∇ × (E + ∂A/∂t) = 0. On a simply connected patch with sufficient smoothness, a curl-free field has a scalar potential, so E + ∂A/∂t = −∇φ. This yields E = −∇φ −∂A/∂t.

These formulas satisfy ∇·B = 0 and Faraday automatically, because divergence of a curl and curl of a gradient vanish and mixed derivatives commute. Gauss and Ampère–Maxwell still constrain the potentials through the specified sources. Introducing potentials does not mean that every arbitrary φ and A solves all four laws for a source distribution selected independently afterward.

The local qualification prevents a global topological mistake. A curl-free field in a punctured region can have a nonzero closed-loop circulation. A potential may need patches, a specified circulation or additional global information. The exterior of an excluded magnetic-flux tube provides a concrete example later in this lesson.

Verify a gauge transformation

Let χ(r,t) be a sufficiently smooth, single-valued gauge function on the domain under discussion. Define A′ = A + ∇χ and φ′ = φ −∂χ/∂t. Then B′ = ∇ × A′ = B, since ∇ × ∇χ = 0. For the electric field, E′ = −∇φ + ∇(∂χ/∂t) −∂A/∂t −∂(∇χ)/∂t = E. The two added terms cancel rather than separately vanishing.

Changing only A while leaving φ fixed generally changes E when χ depends on time. Changing only a spatially varying φ generally changes E as well. A gauge transformation is the specified paired operation. It is not a license to remove a term from a force calculation because that term looks inconvenient.

The values of φ and A depend on the gauge, but the reconstructed E, B, Lorentz force and vacuum field-energy balance do not. In a time-dependent problem, a difference of φ values by itself is not automatically the complete reading of an arbitrary voltmeter lead arrangement. The instrument and its path couple to the physical electromagnetic field; induction cannot be discarded by relabeling potentials.

Gauge freedom is also distinct from changing the laboratory frame. A gauge transformation describes the same fields in the same coordinates. A Lorentz transformation mixes the components observed in different inertial frames. Lorenz gauge has one name; Lorentz transformations have another. Their connection is useful, but their operations should not be conflated.

Declare the original polynomial patch

Choose positive T₀ and V₀, set L₀ = c₀T₀, and use u = t/T₀, X = x/L₀ and Y = y/L₀. Define E₀ = V₀/L₀, B₀ = E₀/c₀, A₀ = V₀/c₀, ρ₀ = ε₀E₀/L₀ and J₀ = ε₀E₀/T₀. Because J₀ = ρ₀L₀/T₀, the dimensionless charge-continuity equation has unit coefficients. The gauge-function scale is χ₀ = V₀T₀.

The original scalar potential is φ/V₀ = (a/2)X² + bXY, with a = 1 + u²/2 and b = 1/3. The vector potential is A/A₀ = (0,0,cXY), with c = u²/2. Here c is a dimensionless polynomial coefficient, whereas c₀ is the vacuum speed. All fields are independent of Z = z/L₀. The displayed sampling box has |X|,|Y| ≤ 1 and 0 ≤ u ≤ 2.

This is a local analytic solution with prescribed sources. The polynomials extend smoothly beyond the sampled box, so a plotted edge is not a material interface. They do not decay at spatial infinity and are not presented as a finite-energy isolated charge distribution or the retarded field of a finite apparatus. Boundary data for a larger problem would have to match the local solution or replace it.

The model is useful precisely because every derivative, source, gauge cancellation and energy term can be checked exactly as a rational function. It illustrates consistency without asking a static approximation to pretend that it includes all time-dependent effects.

Differentiate both potential contributions

Differentiating the scalar potential gives −∇φ/E₀ = (−aX −bY, −bX, 0). Differentiating the vector potential in time gives −(∂A/∂t)/E₀ = (0,0,−uXY). Therefore E/E₀ = (−aX −bY, −bX, −uXY). The spatial curl of A gives B/B₀ = (cX,−cY,0). The two electric contributions have been computed using the common spatial and temporal scales rather than added with incompatible units.

The electric divergence is −a in units E₀/L₀, so ρ/ρ₀ = −a. The magnetic divergence is c −c = 0. The electric curl is (−uX,uY,0) in units E₀/L₀, which equals the negative time derivative of B because B₀/T₀ = E₀/L₀. Thus the homogeneous laws hold, including the spatially varying longitudinal E_z contribution.

The magnetic curl vanishes for this particular B, even though B itself is nonzero. Ampère–Maxwell then requires J/J₀ = −∂(E/E₀)/∂u = (uX,0,XY). This current is part of the prescribed local source. It is not claimed to obey an Ohmic law J = σE. A field solution and a material transport law would have to be solved consistently if a specific conductor were introduced.

At u = 1, X = 1/2, Y = −1/2, the model gives E/E₀ = (−7/12,−1/6,1/4), B/B₀ = (1/4,1/4,0), ρ/ρ₀ = −3/2 and J/J₀ = (1/2,0,−1/4). These signed components are a compact check on the derivatives and on the difference between local field magnitude and field circulation.