Electromagnetic Field Equations & Conservation Laws

Before the revolutionary theoretical synthesis by James Clerk Maxwell (1861–1865), classical electromagnetism was formulated through empirical laws developed primarily for static or steady-state conditions:

1. Gauss's Law for Electrostatics: $\nabla \cdot \mathbf{E} = \frac{\rho}{\epsilon_0}$ 2. Gauss's Law for Magnetism: $\nabla \cdot \mathbf{B} = 0$ 3. Faraday's Law of Electromagnetic Induction: $\nabla \times \mathbf{E} = -\frac{\partial \mathbf{B}}{\partial t}$ 4. Ampère's Circuital Law: $\nabla \times \mathbf{B} = \mu_0 \mathbf{J}$

The Fundamental Mathematical Contradiction

In vector calculus, the divergence of any curl of a vector field identically vanishes. Applying the divergence operator to both sides of Ampère's law yields: $$\nabla \cdot (\nabla \times \mathbf{B}) = \mu_0 (\nabla \cdot \mathbf{J})$$ Since $\nabla \cdot (\nabla \times \mathbf{A}) \equiv 0$ for any twice-differentiable vector field $\mathbf{A}$, the left-hand side is identically zero: $$0 = \mu_0 (\nabla \cdot \mathbf{J}) \implies \nabla \cdot \mathbf{J} = 0$$

However, the fundamental law of Conservation of Electric Charge requires that the divergence of the current density equals the negative rate of charge accumulation, as stated by the Continuity Equation: $$\nabla \cdot \mathbf{J} + \frac{\partial \rho}{\partial t} = 0 \implies \nabla \cdot \mathbf{J} = -\frac{\partial \rho}{\partial t}$$

Ampère's circuital law is therefore mathematically incompatible with the conservation of charge whenever the charge density varies with time ($\partial \rho / \partial t \neq 0$). Ampère's law holds strictly for steady currents (magnetostatics) where $\partial \rho / \partial t = 0$, but fails catastrophically for time-dependent circuits.

The Physical Gedankenexperiment: The Capacitor Paradox

Consider a circular parallel-plate capacitor being charged by a steady current $I(t)$ flowing through a connecting wire. Draw a closed loop $C$ encircling the wire. By Stokes' theorem, the line integral of $\mathbf{B}$ around $C$ equals the surface integral of current density through any open surface $S$ bounded by $C$: $$\oint_C \mathbf{B} \cdot d\mathbf{l} = \mu_0 \int_S \mathbf{J} \cdot d\mathbf{a} = \mu_0 I_{\text{enc}}$$

- Surface $S_1$ (Flat Disk): Choose a flat circular disk bounded by $C$. The conducting wire pierces $S_1$, so the enclosed current is $I_{\text{enc}} = I$, giving $\oint_C \mathbf{B} \cdot d\mathbf{l} = \mu_0 I$. - Surface $S_2$ (Balloon/Bowl Shape): Bulge the surface out like a balloon so that it passes entirely between the capacitor plates without touching the wire. Since no conduction current passes through the vacuum gap between the plates, $I_{\text{enc}} = 0$, giving $\oint_C \mathbf{B} \cdot d\mathbf{l} = 0$.

Because the line integral $\oint_C \mathbf{B} \cdot d\mathbf{l}$ around the exact same physical curve $C$ cannot depend on the arbitrary mathematical choice of surface $S$, classical electrodynamics faced an irreconcilable paradox.

The Displacement Current Derivation

To resolve the contradiction, Maxwell proposed adding a missing current density term, $\mathbf{J}_d$ (the displacement current density), to Ampère's equation: $$\nabla \times \mathbf{B} = \mu_0 (\mathbf{J} + \mathbf{J}_d)$$

Taking the divergence of both sides: $$\nabla \cdot (\nabla \times \mathbf{B}) = 0 = \mu_0 \left( \nabla \cdot \mathbf{J} + \nabla \cdot \mathbf{J}_d \right)$$ $$\nabla \cdot \mathbf{J}_d = -\nabla \cdot \mathbf{J}$$

Using the continuity equation $\nabla \cdot \mathbf{J} = -\frac{\partial \rho}{\partial t}$: $$\nabla \cdot \mathbf{J}_d = \frac{\partial \rho}{\partial t}$$

Now, substitute Gauss's law $\rho = \epsilon_0 (\nabla \cdot \mathbf{E})$: $$\nabla \cdot \mathbf{J}_d = \frac{\partial}{\partial t} \left( \epsilon_0 \nabla \cdot \mathbf{E} \right) = \nabla \cdot \left( \epsilon_0 \frac{\partial \mathbf{E}}{\partial t} \right)$$

Equating the vector terms under the divergence yields Maxwell's formulation for the displacement current density in free space: $$\mathbf{J}_d = \epsilon_0 \frac{\partial \mathbf{E}}{\partial t}$$

Total Current in the Capacitor Gap

The total displacement current $I_d$ passing through the cross-sectional area $A$ between the capacitor plates is: $$I_d = \int_A \mathbf{J}_d \cdot d\mathbf{a} = \epsilon_0 \frac{d}{dt} \int_A \mathbf{E} \cdot d\mathbf{a} = \epsilon_0 \frac{d\Phi_E}{dt}$$ where $\Phi_E$ is the electric flux. For a parallel-plate capacitor with plate charge $Q(t)$ and plate area $A$: $$E(t) = \frac{\sigma(t)}{\epsilon_0} = \frac{Q(t)}{\epsilon_0 A} \implies \Phi_E(t) = E(t) A = \frac{Q(t)}{\epsilon_0}$$ Differentiating with respect to time: $$I_d = \epsilon_0 \frac{d}{dt} \left( \frac{Q(t)}{\epsilon_0} \right) = \frac{dQ}{dt} = I_c(t)$$

Thus, the displacement current $I_d$ between the plates is identically equal to the conduction current $I_c$ flowing in the external circuit wires. The total current $I_{\text{tot}} = I_c + I_d$ is perfectly continuous everywhere in the circuit, completely eliminating the capacitor paradox.

The addition of the displacement current term completed the classical theory of electrodynamics. Maxwell's equations describe the behavior of macroscopic and microscopic electromagnetic fields in terms of charge densities $\rho$ and current densities $\mathbf{J}$.

Maxwell's Equations in Vacuum (Microscopic Form)

Maxwell's Equations in Matter (Macroscopic Form)

In material media, charges and currents are split into free charges/currents (conducted electrons, free ions: $\rho_f, \mathbf{J}_f$) and bound charges/currents arising from dielectric polarization $\mathbf{P}$ and magnetization $\mathbf{M}$: $$\rho_b = -\nabla \cdot \mathbf{P}, \quad \mathbf{J}_b = \nabla \times \mathbf{M}, \quad \mathbf{J}_p = \frac{\partial \mathbf{P}}{\partial t}$$

Defining the auxiliary fields: - Electric Displacement: $\mathbf{D} \equiv \epsilon_0 \mathbf{E} + \mathbf{P}$ - Magnetic Field Intensity: $\mathbf{H} \equiv \frac{1}{\mu_0} \mathbf{B} - \mathbf{M}$

The macroscopic Maxwell equations take the form: 1. $\nabla \cdot \mathbf{D} = \rho_f$ 2. $\nabla \cdot \mathbf{B} = 0$ 3. $\nabla \times \mathbf{E} = -\frac{\partial \mathbf{B}}{\partial t}$ 4. $\nabla \times \mathbf{H} = \mathbf{J}_f + \frac{\partial \mathbf{D}}{\partial t}$

For linear, isotropic, and homogeneous media: $$\mathbf{D} = \epsilon \mathbf{E} = \epsilon_r \epsilon_0 \mathbf{E}, \quad \mathbf{B} = \mu \mathbf{H} = \mu_r \mu_0 \mathbf{H}$$

The Dynamic Cross-Coupling of Fields

A critical insight of Maxwell's equations is the mutual interdependence of electric and magnetic fields in time-dependent situations: $$\frac{\partial \mathbf{B}}{\partial t} \neq 0 \implies \nabla \times \mathbf{E} \neq 0 \quad \text{and} \quad \frac{\partial \mathbf{E}}{\partial t} \neq 0 \implies \nabla \times \mathbf{B} \neq 0$$ Even in a pure vacuum with zero charges ($\rho = 0$) and zero currents ($\mathbf{J} = 0$), a changing magnetic field creates an electric field, and that changing electric field in turn generates a magnetic field. This self-sustaining, reciprocal generation allows electromagnetic disturbances to detach from their sources and propagate across infinite distances through empty space as electromagnetic radiation.

Asymmetry and Magnetic Monopoles

Notice the apparent asymmetry between electricity and magnetism: $$\nabla \cdot \mathbf{E} = \frac{\rho_e}{\epsilon_0}, \quad \nabla \cdot \mathbf{B} = 0$$ $$\nabla \times \mathbf{E} = -\frac{\partial \mathbf{B}}{\partial t}, \quad \nabla \times \mathbf{B} = \mu_0 \mathbf{J}_e + \frac{1}{c^2} \frac{\partial \mathbf{E}}{\partial t}$$

If magnetic monopoles existed in nature with magnetic charge density $\rho_m$ and magnetic current density $\mathbf{J}_m$, Maxwell's equations would achieve perfect dual symmetry: $$\nabla \cdot \mathbf{E} = \frac{\rho_e}{\epsilon_0}, \quad \nabla \cdot \mathbf{B} = \mu_0 \rho_m$$ $$\nabla \times \mathbf{E} = -\mu_0 \mathbf{J}_m - \frac{\partial \mathbf{B}}{\partial t}, \quad \nabla \times \mathbf{B} = \mu_0 \mathbf{J}_e + \frac{1}{c^2} \frac{\partial \mathbf{E}}{\partial t}$$

Despite extensive experimental searches in particle colliders, cosmic rays, and lunar rock samples, no isolated magnetic monopole has ever been conclusively observed. Paul Dirac proved in 1931 that the existence of even a single magnetic monopole anywhere in the universe would explain the quantization of electric charge: $$q_e q_m = 2\pi n \hbar, \quad n \in \mathbb{Z}$$

Step-by-Step Mathematical Derivation of Poynting's Theorem

The work done by electromagnetic forces on a distribution of charges in volume $V$ is governed by the Lorentz force law. The mechanical work per unit time (power) delivered to the charges is: $$\frac{dW_{\text{mech}}}{dt} = \int_V (\mathbf{F} \cdot \mathbf{v}) dq = \int_V \mathbf{E} \cdot \mathbf{J} \, d\tau$$ where $d\tau$ is the differential volume element.

To express $\mathbf{E} \cdot \mathbf{J}$ in terms of field quantities alone, solve for $\mathbf{J}$ using the Ampère-Maxwell law: $$\mathbf{J} = \frac{1}{\mu_0} (\nabla \times \mathbf{B}) - \epsilon_0 \frac{\partial \mathbf{E}}{\partial t}$$

Dot both sides with $\mathbf{E}$: $$\mathbf{E} \cdot \mathbf{J} = \frac{1}{\mu_0} \mathbf{E} \cdot (\nabla \times \mathbf{B}) - \epsilon_0 \mathbf{E} \cdot \frac{\partial \mathbf{E}}{\partial t}$$

Recall the fundamental vector calculus product rule: $$\nabla \cdot (\mathbf{E} \times \mathbf{B}) = \mathbf{B} \cdot (\nabla \times \mathbf{E}) - \mathbf{E} \cdot (\nabla \times \mathbf{B})$$ $$\mathbf{E} \cdot (\nabla \times \mathbf{B}) = \mathbf{B} \cdot (\nabla \times \mathbf{E}) - \nabla \cdot (\mathbf{E} \times \mathbf{B})$$

Substitute Faraday's law $\nabla \times \mathbf{E} = -\frac{\partial \mathbf{B}}{\partial t}$: $$\mathbf{E} \cdot (\nabla \times \mathbf{B}) = -\mathbf{B} \cdot \frac{\partial \mathbf{B}}{\partial t} - \nabla \cdot (\mathbf{E} \times \mathbf{B})$$

Now, observe that: $$\mathbf{E} \cdot \frac{\partial \mathbf{E}}{\partial t} = \frac{1}{2} \frac{\partial}{\partial t}(E^2), \quad \mathbf{B} \cdot \frac{\partial \mathbf{B}}{\partial t} = \frac{1}{2} \frac{\partial}{\partial t}(B^2)$$

Substituting these identities back into the expression for $\mathbf{E} \cdot \mathbf{J}$: $$\mathbf{E} \cdot \mathbf{J} = -\frac{1}{2} \frac{\partial}{\partial t} \left( \epsilon_0 E^2 + \frac{1}{\mu_0} B^2 \right) - \frac{1}{\mu_0} \nabla \cdot (\mathbf{E} \times \mathbf{B})$$

Definitions of the Field Quantities

1. Electromagnetic Energy Density ($u$): $$u \equiv \frac{1}{2} \left( \epsilon_0 E^2 + \frac{1}{\mu_0} B^2 \right) \quad [\text{Joules} / \text{m}^3]$$

2. The Poynting Vector ($\mathbf{S}$): $$\mathbf{S} \equiv \frac{1}{\mu_0} (\mathbf{E} \times \mathbf{B}) \quad [\text{Watts} / \text{m}^2]$$ The Poynting vector represents the directional rate of electromagnetic energy transfer per unit area.

Differential Form of Poynting's Theorem

$$-\frac{\partial u}{\partial t} = \nabla \cdot \mathbf{S} + \mathbf{E} \cdot \mathbf{J}$$ or in conservation law form: $$\frac{\partial u}{\partial t} + \nabla \cdot \mathbf{S} = -\mathbf{E} \cdot \mathbf{J}$$

Integral Form of Poynting's Theorem

Integrating over a finite volume $V$ bounded by closed surface $S$ and applying Gauss's divergence theorem: $$-\frac{d}{dt} \int_V u \, d\tau = \oint_S \mathbf{S} \cdot d\mathbf{a} + \int_V (\mathbf{E} \cdot \mathbf{J}) \, d\tau$$

Physical Interpretation: The time rate of decrease of electromagnetic energy stored in volume $V$ equals the total power radiated outward across the boundary surface $S$ plus the rate of work done on the charges inside $V$ (such as Ohmic Joule heating $\mathbf{J} \cdot \mathbf{E} = \sigma E^2$).

Electromagnetic Momentum Density

Electromagnetic fields carry not only energy, but also linear momentum. By applying the Lorentz force equation to volume charges, one derives the total momentum balance: $$\frac{d}{dt} (\mathbf{p}_{\text{mech}} + \mathbf{p}_{\text{field}}) = 0$$

The electromagnetic momentum density $\mathbf{g}$ stored in the fields is directly proportional to the Poynting vector: $$\mathbf{g} = \epsilon_0 (\mathbf{E} \times \mathbf{B}) = \frac{\mathbf{S}}{c^2} \quad [\text{kg} / (\text{m}^2 \cdot \text{s})]$$

The Maxwell Stress Tensor

The spatial flow of momentum is described by the Maxwell stress tensor $\overleftrightarrow{\mathbf{T}}$, a rank-2 symmetric tensor with components: $$T_{ij} \equiv \epsilon_0 \left( E_i E_j - \frac{1}{2} \delta_{ij} E^2 \right) + \frac{1}{\mu_0} \left( B_i B_j - \frac{1}{2} \delta_{ij} B^2 \right)$$ where $\delta_{ij}$ is the Kronecker delta. The electromagnetic force per unit volume is given by: $$\mathbf{f} = \nabla \cdot \overleftrightarrow{\mathbf{T}} - \epsilon_0 \mu_0 \frac{\partial \mathbf{S}}{\partial t}$$

Radiation Pressure

When an electromagnetic wave impinges on a physical surface, it transfers linear momentum to the matter, exerting a mechanical pressure known as radiation pressure ($P_{\text{rad}}$).

Let the incident wave carry intensity $I = \langle S \rangle = c \langle u \rangle$:

1. Total Absorption (Perfect Blackbody Surface): All incident momentum is transferred to the surface: $$P_{\text{rad}} = \frac{\langle S \rangle}{c} = \langle u \rangle = \frac{I}{c}$$

2. Total Reflection (Ideal Conducting Mirror): The reflected wave has reversed momentum ($\Delta p = p_{\text{final}} - p_{\text{initial}} = -p - p = -2p$). By Newton's third law, the impulse delivered to the surface is doubled: $$P_{\text{rad}} = \frac{2 \langle S \rangle}{c} = 2 \langle u \rangle = \frac{2I}{c}$$

3. Oblique Incidence at Angle $\theta$: $$P_{\text{rad}} = \frac{I}{c} (1 + R) \cos^2\theta$$ where $R$ is the surface reflection coefficient ($R=0$ for complete absorption, $R=1$ for total reflection).