Maxwell's Equations & Boundary Conditions

Maxwell's equations are not just abstract math—they are the "operating system" of the universe for wireless signals. Every antenna, filter, and transmission line is simply a device designed to exploit a specific solution to these four equations.

Note: Why this matters for RF: You cannot understand impedance matching, skin depth, or antenna polarization without understanding how E and H fields interact at boundaries and in materials.

1. The Language of Fields (Prerequisites)

Before diving into the laws, we must agree on what the symbols mean physically.

The Del Operator ($\nabla$)

Think of $\nabla$ as a scanner that looks at a field at a specific point and asks "How is this changing?"

2. The Four Governing Laws

We use the differential form because it describes what is happening at a specific point in space.

I. Gauss's Law (Electric charge is the source)

$\nabla \cdot \vec{D} = \rho_v$

• Math: The divergence of the Electric Flux Density ($\vec{D}$) is equal to the volumetric charge density ($\rho_v$).
• Intuition: Electric field lines must start on positive charges and end on negative charges. They cannot just appear from nowhere.
• RF Implication: In a coax cable, field lines radiate radially from the center conductor (+) to the shield (-).

II. Gauss's Law for Magnetism (No Magnetic Monopoles)

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

• Math: The divergence of Magnetic Flux Density ($\vec{B}$) is always zero.
• Intuition: There are no "magnetic charges." Magnetic field lines form closed loops. They never start or stop.
• RF Implication: This is why inductive coupling works—the magnetic loops from one coil must close, often passing through a nearby coil.

III. Faraday's Law (Changing Magnetic Fields Create Electric Fields)

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

• Math: The curl of $\vec{E}$ is equal to the negative rate of change of $\vec{B}$.
• Intuition: If you shake a magnetic field, you create an electric swirl around it. This is induction.
• RF Implication: This is the mechanism of transformers, loop antennas, and how waves sustain themselves (the changing B creates the E).

IV. Ampere-Maxwell Law (Current AND Changing E-fields Create B-fields)

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

• Math: The curl of $\vec{H}$ is created by conductive current ($\vec{J}$) AND displacement current ($\frac{\partial \vec{D}}{\partial t}$).
• Intuition: A magnetic field swirls around a current. BUT, Maxwell added the second term: a changing Electric field acts exactly like a current!
• RF Implication: Displacement Current is the most important concept in RF. It explains:
  1. How current flows "through" a capacitor (it's actually changing E-field).
  2. How waves travel in vacuum (Changing E creates B, Changing B creates E).

3. The Constitutive Relations (Materials)

Maxwell's equations are generic. "Constitutive Relations" describe how fields behave inside specific materials (Air, Teflon, Copper, FR4).

$\vec{D} = \epsilon \vec{E}$ $\vec{B} = \mu \vec{H}$ $\vec{J} = \sigma \vec{E}$

• Permittivity ($\epsilon$): How much a material "resists" internal electric fields (stores energy). High $\epsilon_r$ means smaller wavelengths.
• Permeability ($\mu$): How much a material supports magnetic fields. (Almost $\mu_0$ for all non-ferrous RF materials).
• Conductivity ($\sigma$): How easily electrons move. High $\sigma$ means the material is a conductor (short circuit). $\sigma = 0$ is a perfect dielectric.

4. Boundary Conditions (The Interface)

What happens when a wave hits the boundary between two different materials (e.g., Air to Copper, or Air to Water)? Derived directly from the integral forms of Maxwell's laws:

The Tangential Rules

1. $E_{tan}$ is Continuous: The electric field parallel to the surface cannot change instantly.
   • $E_{1t} = E_{2t}$
   • Result: If a conductor effectively forces $E=0$ inside it, then right at the surface, the tangential E-field must also be zero. This is why metal shields reflection.
2. $H_{tan}$ is Discontinuous by Surface Current ($J_s$):
   • $H_{1t} - H_{2t} = J_s$
   • Result: RF currents flow on the surface of conductors (Skin Effect) to satisfy this boundary condition.

The Normal Rules

1. $D_{normal}$ is Discontinuous by Surface Charge ($\rho_s$):
   • $D_{1n} - D_{2n} = \rho_s$
2. $B_{normal}$ is Continuous:
   • $B_{1n} = B_{2n}$

Important: Engineering Takeaway: When an electromagnetic wave hits a perfect conductor:

1. The Tangential E-field must go to zero (Short Circuit).
2. This forces the wave to reflect completely (Phase flip).
3. This is the fundamental mechanism of Reflections ($\Gamma = -1$).

5. From Equations to Waves

If you take the curl of Faraday's Law and substitute Ampere's Law, you get the Wave Equation:

$\nabla^2 \vec{E} - \mu\epsilon \frac{\partial^2 \vec{E}}{\partial t^2} = 0$

This equation admits solutions of the form $E(x,t) = E_0 \cos(\omega t - \beta z)$. This proves that electromagnetic fields don't just sit there; they travel at a speed of $v = \frac{1}{\sqrt{\mu\epsilon}}$.

• In Vacuum: $v = c \approx 3 \times 10^8$ m/s.
• In Cable (Teflon): $v \approx 0.7c$.