Electromagnetics: Maxwell's Equations and Waves

Electromagnetics becomes truly unified when you stop treating electricity and magnetism as separate topics and start treating them as a coupled field theory. Maxwell’s equations are the compact statement of that theory. They explain how charges and currents create electric and magnetic fields, how time-varying fields generate each other, and why electromagnetic waves can propagate through space. From there, the same framework connects to practical engineering tools like reflection at boundaries and transmission line analysis.

Maxwell’s Equations: The Core Laws

Maxwell’s equations are typically presented in differential form because it directly shows how fields vary in space and time. In linear, isotropic media, the field quantities are the electric field $\mathbf{E}$, magnetic field $\mathbf{H}$, electric flux density $\mathbf{D}$, and magnetic flux density $\mathbf{B}$, along with charge density $\rho$ and current density $\mathbf{J}$.

Gauss’s Law for Electricity

Gauss’s law relates electric fields to charge:

$$\nabla \cdot \mathbf{D}=\rho$$

In a simple material, $\mathbf{D}=\epsilon \mathbf{E}$, where $\epsilon$ is the permittivity. This equation encodes the idea that electric flux “diverges” outward from positive charge and converges into negative charge.

Gauss’s Law for Magnetism

Magnetic monopoles do not appear in classical electromagnetics, which is expressed as:

$$\nabla \cdot \mathbf{B}=0$$

With $\mathbf{B}=\mu \mathbf{H}$ in a simple medium, this means magnetic field lines form continuous loops rather than beginning or ending at isolated sources.

Faraday’s Law of Induction

A time-varying magnetic field creates a circulating electric field:

$$\nabla \times \mathbf{E}=-\frac{\partial \mathbf{B}}{\partial t}$$

This is the field-level statement behind induced voltage in a loop and transformer action. It also becomes one half of the wave mechanism: changing magnetic fields drive electric fields.

Ampère-Maxwell Law

Currents create magnetic fields, and crucially, time-varying electric fields do too:

$$\nabla \times \mathbf{H}=\mathbf{J}+\frac{\partial \mathbf{D}}{\partial t}$$

The additional term $\frac{\partial \mathbf{D}}{\partial t}$ is Maxwell’s displacement current. Without it, the equations would fail for time-varying situations such as a charging capacitor. With it, the system becomes self-consistent and predicts electromagnetic waves.

Time-Varying Fields and the Electromagnetic Wave Equation

Maxwell’s equations are coupled: curls link $\mathbf{E}$ and $\mathbf{H}$ through time derivatives. In a source-free region (no free charge and no conduction current), $\rho =0$ and $\mathbf{J}=0$. Assuming uniform material properties, you can combine Faraday’s law and the Ampère-Maxwell law to obtain a wave equation for the electric field:

$${\nabla }^2\mathbf{E}=\mu \epsilon \frac{{\partial }^2\mathbf{E}}{\partial t^2}$$

A similar equation holds for $\mathbf{H}$. The key result is the wave speed:

$$v=\frac{1}{\sqrt{\mu \epsilon }}$$

In free space, $\mu ={\mu }_0$ and $\epsilon ={\epsilon }_0$, giving the speed of light $c=1/\sqrt{{\mu }_0{\epsilon }_0}$. This is the physical bridge between electromagnetics and optics: light is an electromagnetic wave.

Plane Waves: The Simplest Propagating Solution

A plane wave traveling in the $+z$ direction can be written in a simple sinusoidal form, for example:

• $\mathbf{E}$ oscillates in one transverse direction (say $\hat{x}$)
• $\mathbf{H}$ oscillates in a perpendicular transverse direction (say $\hat{y}$)
• the wave propagates along $\hat{z}$

In lossless media, $\mathbf{E}$ and $\mathbf{H}$ are in phase, and their magnitudes are related by the intrinsic impedance of the medium:

$$\eta =\sqrt{\frac{\mu }{\epsilon }}$$

In free space, ${\eta }_0\approx 377\Omega$. This impedance is not a circuit resistance; it is the ratio of field strengths in a propagating wave: $|\mathbf{E}|/|\mathbf{H}|=\eta$.

Energy Flow and the Poynting Vector

Electromagnetic waves carry energy, and the rate of energy flow per unit area is described by the Poynting vector:

$$\mathbf{S}=\mathbf{E}\times \mathbf{H}$$

Its direction matches the direction of propagation for a plane wave. This concept becomes practical when evaluating power transmitted down a waveguide or a transmission line, or when estimating received power in antenna problems.

Reflection and Transmission at Boundaries

When a wave encounters a boundary between two media, it generally splits into reflected and transmitted waves. The reason is straightforward: the fields must satisfy boundary conditions, and one traveling-wave solution is rarely sufficient to satisfy constraints on both sides.

Boundary Conditions: What Must Match

Maxwell’s equations imply continuity relations at an interface. For many practical cases without surface charge or surface current:

• Tangential components of $\mathbf{E}$ are continuous across the boundary.
• Tangential components of $\mathbf{H}$ are continuous across the boundary.

These conditions force specific amplitude ratios for reflected and transmitted waves.

Impedance Mismatch and the Reflection Coefficient

For normal incidence in lossless media, the reflection coefficient for the electric field depends on the intrinsic impedances:

$$\Gamma =\frac{{\eta }_2-{\eta }_1}{{\eta }_2+{\eta }_1}$$

If ${\eta }_1={\eta }_2$, then $\Gamma =0$ and there is no reflection. If the impedances differ significantly, reflection can be substantial. This same “mismatch” idea reappears in transmission lines, where the relevant impedance is the line’s characteristic impedance rather than a bulk material impedance.

What Happens to Power

Power reflection is proportional to $|\Gamma {|}^2$ in simple lossless cases. The remainder is transmitted, though in real materials there may also be absorption. Understanding how much power reflects versus transmits matters in RF systems, radar, fiber-optic analogs in guided structures, and EMC considerations.

Transmission Lines: Guided Waves in Circuits

At low frequencies and short distances, circuits can be modeled with lumped elements. But when the physical length of an interconnect becomes a significant fraction of the wavelength, voltages and currents vary along the conductor. That is when transmission line theory becomes essential.

Distributed Parameters and the Telegrapher’s Equations

A transmission line is modeled with per-unit-length parameters:

• $R$ (series resistance)
• $L$ (series inductance)
• $G$ (shunt conductance)
• $C$ (shunt capacitance)

These lead to coupled wave equations for voltage $V(z,t)$ and current $I(z,t)$. In the common low-loss approximation, the line supports forward and backward traveling waves, with a phase velocity set by $L$ and $C$.

Characteristic Impedance and Traveling Waves

The characteristic impedance $Z_0$ relates voltage and current for a single traveling wave on the line. For a lossless line:

$$Z_0=\sqrt{\frac{L}{C}}$$

If the load impedance $Z_L$ equals $Z_0$, the wave is absorbed without reflection. If not, reflections occur and create standing waves along the line.

The reflection coefficient at the load is:

$${\Gamma }_L=\frac{Z_L-Z_0}{Z_L+Z_0}$$

This is the same structure as wave reflection at a material boundary, which is not a coincidence. Both are manifestations of Maxwell’s equations applied in different geometries.

Practical Consequences: Why Engineers Care

Reflection on transmission lines causes measurable problems:

• Signal integrity issues on high-speed digital traces, where edges reflect and distort logic thresholds.
• Reduced power transfer in RF systems, where a mismatch sends power back toward the source.
• Voltage stress at specific points due to standing-wave maxima, which can exceed expected levels.

Mitigation strategies follow directly from the theory: impedance matching, proper termination (series or parallel, depending on the signaling scheme), controlled-impedance routing, and minimizing discontinuities such as stubs or abrupt geometry changes.

Pulling It Together: One Theory, Many Applications

Maxwell’s equations are not just a set of abstract identities. They explain, with a single consistent structure, how time-varying fields propagate as waves, how those waves reflect and transmit at boundaries, and how guided electromagnetic waves appear in transmission lines that we treat as “wires” in everyday engineering. Once you see the continuity between free-space propagation and guided-wave behavior, concepts like impedance, reflection coefficient, and power flow stop feeling like separate formulas and start reading as the natural language of electromagnetic fields.