Maxwell's Equations Complete
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You should be comfortable with vector calculus identities, specifically the divergence and curl theorems, and have a working knowledge of the continuity equation for charge conservation.
Why this matters
Maxwell's equations represent the first complete unification of electricity and magnetism into a single, self-consistent field theory.
For centuries, electricity and magnetism were treated as separate phenomena. The discovery of induction and the subsequent correction of Ampère's law revealed that time-varying fields are inextricably linked. By assembling these four laws into a single set, we obtain a system that not only predicts the behaviour of charges and currents but also implies the existence of electromagnetic waves traveling at the speed of light. This lecture consolidates the static and dynamic laws into their final differential form, demonstrating that the equations are internally consistent and uniquely determine the fields given the sources.
The Unified Framework
Imagine the electromagnetic field as a single entity with two visible aspects: the electric field and the magnetic field. Previously, we studied them in isolation or as static snapshots. Now, we view them as a dynamic, coupled system. The electric field is sourced by charge, while the magnetic field is sourced by moving charge (current) and changing electric fields. Crucially, a changing magnetic field creates an electric field, and a changing electric field creates a magnetic field. This mutual generation allows the fields to sustain themselves in empty space, propagating as waves. The four equations we are about to assemble describe exactly how these fields interact with matter and with each other, forming a closed logical system where no external assumptions are needed to predict the field evolution once the sources are known.
Assembling the Four Laws
Maxwell's equations in their differential form are the local, point-by-point statements of the field behaviour. They are derived from the integral laws using the divergence and curl theorems, assuming the fields are smooth and differentiable. The first two equations describe the sources of the fields, while the latter two describe how the fields change in time and space.
Gauss's law for the electric field states that electric charge is the source of the electric field. In differential form, the divergence of the electric field is proportional to the charge density \(\rho\). This is a direct consequence of the inverse-square nature of the electrostatic force.
Gauss's law for magnetism states that there are no magnetic monopoles. The net magnetic flux through any closed surface is zero, meaning magnetic field lines always form closed loops. In differential form, the divergence of the magnetic field is identically zero.
Faraday's law of induction captures the essence of electromagnetic induction. A time-varying magnetic field induces a circulating electric field. This is the principle behind generators and transformers. The curl of the electric field is equal to the negative rate of change of the magnetic field.
Finally, the Ampère–Maxwell law generalizes the original Ampère's law by including the displacement current. This term, \(\epsilon_0 \frac{\partial \mathbf{E}}{\partial t}\), ensures that the law remains consistent with the conservation of charge in time-dependent situations. The curl of the magnetic field is proportional to the total current density, which includes both conduction current \(\mathbf{J}\) and displacement current.
Consistency and Charge Conservation
A critical test of any physical theory is internal consistency. Maxwell's equations must be compatible with the continuity equation, which expresses the local conservation of electric charge. If we take the divergence of the Ampère–Maxwell law, we can verify this consistency directly.
Applying the divergence operator to both sides of the Ampère–Maxwell equation:
Using the vector identity \(\nabla \cdot (\nabla \times \mathbf{B}) = 0\) and substituting Gauss's law \(\nabla \cdot \mathbf{E} = \rho / \epsilon_0\), the equation simplifies to:
Dividing by \(\mu_0\) and rearranging terms, we recover the continuity equation:
This result confirms that the inclusion of the displacement current term is not arbitrary but is mathematically required to maintain charge conservation. Without this term, the divergence of the Ampère law would imply \(\nabla \cdot \mathbf{J} = 0\), which is only true for steady currents and fails for time-dependent charge distributions.
Uniqueness and Boundary Conditions
Maxwell's equations form a coupled set of first-order partial differential equations. Given the charge density \(\rho(\mathbf{r}, t)\) and current density \(\mathbf{J}(\mathbf{r}, t)\), these equations uniquely determine the electric and magnetic fields \(\mathbf{E}(\mathbf{r}, t)\) and \(\mathbf{B}(\mathbf{r}, t)\), provided appropriate initial and boundary conditions are specified.
The equations are hyperbolic in nature, meaning they support wave-like solutions. The coupling between \(\mathbf{E}\) and \(\mathbf{B}\) through the time derivatives ensures that changes in the sources propagate through space at a finite speed. This finite propagation speed is a direct consequence of the displacement current term, which introduces the factor \(1/c^2\) (where \(c = 1/\sqrt{\mu_0 \epsilon_0}\)) into the wave equation for the fields.
It is important to note that the equations determine the fields up to gauge transformations in the potential formulation, but the physical fields \(\mathbf{E}\) and \(\mathbf{B}\) themselves are uniquely determined by the sources and boundary conditions. This uniqueness is a cornerstone of classical electrodynamics, ensuring that predictions are deterministic and reproducible.
The derivations
Assembly of Maxwell's Equations
Taking the four independently derived field laws — Gauss's law for the electric field, Gauss's law for magnetism, Faraday's law of induction, and the Ampere–Maxwell law with displacement current — we collect them into a single coupled set, and show that each law's integral form and differential form are exactly equivalent (via the divergence and curl theorems) on smooth field configurations. The assembled set closes on itself: given the sources ρ and J, the equations determine E and B up to boundary and initial data, and they are internally consistent with charge conservation.
The Displacement Current from Charge Conservation
The magnetostatic Ampère law ∇×B = μ0J forces ∇·J = 0, which contradicts the continuity equation ∇·J + ∂ρ/∂t = 0 whenever the charge density is time-dependent. Requiring consistency with continuity and with Gauss's law ∇·E = ρ/ε0 forces the addition of exactly one extra term, Maxwell's displacement current ε0 ∂E/∂t, giving the corrected law ∇×B = μ0(J + ε0 ∂E/∂t).
A common misconception is that the displacement current is a "real" current of moving charges. It is not. It is a term in the field equation that accounts for the changing electric field. While it produces a magnetic field just like a conduction current, it does not involve the flow of charge carriers. Confusing the two can lead to errors in calculating total current in capacitors or transmission lines. C
Worked example
Problem. How important is the displacement current inside a good conductor? Compare the amplitudes of the displacement and conduction current densities in copper (\(\sigma = 5.96 \times 10^{7}\ \mathrm{S\,m^{-1}}\), permittivity \(\approx \varepsilon_0\)) carrying a sinusoidal field \(E = E_0 \cos\omega t\) at (a) 50 Hz and (b) 1 GHz.
- Conduction current density: \(J_c = \sigma E\), amplitude \(\sigma E_0\).
- Displacement current density: \(J_d = \varepsilon_0\, \partial E/\partial t = -\varepsilon_0 \omega E_0 \sin\omega t\), amplitude \(\varepsilon_0 \omega E_0\).
- Ratio of amplitudes: \(J_d/J_c = \varepsilon_0\omega/\sigma\), independent of \(E_0\).
- (a) 50 Hz: \(\varepsilon_0\omega/\sigma = 8.854 \times 10^{-12} \times 2\pi \times 50 / 5.96 \times 10^{7} = 4.7 \times 10^{-17}\). (b) 1 GHz: \(8.854 \times 10^{-12} \times 2\pi \times 10^{9}/5.96 \times 10^{7} = 9.3 \times 10^{-10}\).
- Even at microwave frequencies the displacement current in a metal is a billionth of the conduction current; it dominates only in insulators and in vacuum, which is exactly where electromagnetic waves travel.
Units check. \(\varepsilon_0\omega\) is in \(\mathrm{F\,m^{-1}\,s^{-1}} = \mathrm{S\,m^{-1}}\), the same units as \(\sigma\), so the ratio is dimensionless.
Three quick tests
- Why is the term \(\nabla \cdot (\nabla \times \mathbf{B})\) zero in the consistency check?
Answer
It is a vector calculus identity: the divergence of the curl of any sufficiently smooth vector field is always zero. - If the charge density \(\rho\) is constant in time, what does the continuity equation imply about \(\nabla \cdot \mathbf{J}\)?
Answer
It implies \(\nabla \cdot \mathbf{J} = 0\), meaning the current is steady and has no sources or sinks. - Does the displacement current contribute to the magnetic field in the same way as conduction current?
Answer
Yes, in the Ampère–Maxwell law, both terms contribute to the curl of \(\mathbf{B}\) and thus generate the magnetic field.
Problem set for PU-204 →
Each derivation in this unit ends with five graded problems with full solutions.
Lecture notes checked line by line by the editor; every number in the worked example was recomputed.
Derivations cited: D-162 · D-161 · Threads: force, energy, light