Maxwell’s Equations

Maxwell’s Equations#

Maxwell’s Equations are a set of four laws that completely describe classical electromagnetism and ultimately reveal the true nature of light. These are the basic equations that FDTDX like any other electromagnetic simulation software is build on.

To understand the physics of these equations, we look at the interaction of four fundamental vector fields: the Electric Field ($\mathbf{E}$), the Electric Displacement Field ($\mathbf{D}$), the Magnetic Flux Density ($\mathbf{B}$), and the Magnetic Field ($\mathbf{H}$).

Electric Connection: $\mathbf{D}$ and $\mathbf{E}$#

When you apply an external Electric Field ($\mathbf{E}$) to an insulating material (a dielectric), the atoms stretch. The negative electron clouds are pulled one way, and the positive nuclei are pushed the other. This stretching creates billions of microscopic dipoles, a phenomenon called Polarization ($\mathbf{P}$). The Electric Displacement Field ($\mathbf{D}$) was created to ignore this messy internal polarization and focus solely on the “free” charges we can control. Therefore, $\mathbf{D}$ is essentially the fundamental electric field $\mathbf{E}$, plus the material’s internal polarization response. The fundamental connection is: $$ \mathbf{D} = \varepsilon_0 \mathbf{E} + \mathbf{P} $$

However, for a lot of materials the polarization is directly proportional to the applied electric field. This allows us to drastically simplify the equation into its most famous form: $ \mathbf{D} = \varepsilon \mathbf{E} $.

$\varepsilon$ (epsilon) is the permittivity of the material. It is a single number that measures how much a material resists the formation of an electric field within it. It is calculated as $\varepsilon = \varepsilon_0 \varepsilon_r$, where $\varepsilon_0$ is the vacuum permittivity and $\varepsilon_r$ is the relative permittivity (or dielectric constant) of the specific material. In a pure vacuum, $\varepsilon_r = 1$, so $\mathbf{D} = \varepsilon_0 \mathbf{E}$.

Magnetic Connection: $\mathbf{B}$ and $\mathbf{H}$#

Just as electric fields stretch atoms, magnetic fields force the electrons orbiting within atoms to align their spins. This alignment turns the atoms into tiny microscopic magnets, creating a collective effect called Magnetization ($\mathbf{M}$). The Magnetic Flux Density ($\mathbf{B}$) is the total, fundamental magnetic reality. It accounts for both the magnetic field generated by the actual electrical current running through your wires ($\mathbf{H}$) and the magnetic field added by the aligned atoms of the material ($\mathbf{M}$). The fundamental connection is: $$ \mathbf{B} = \mu_0 (\mathbf{H} + \mathbf{M}) $$

Just like with the electric fields, for a lot of materials, the magnetization is proportional to the magnetic field intensity. This allows us to combine the terms into a much simpler, highly practical equation: $ \mathbf{B} = \mu \mathbf{H} $.

$\mu$ (mu) is the permeability of the material. It measures how easily a material can support the formation of a magnetic field within itself. It is calculated as $\mu = \mu_0 \mu_r$, where $\mu_0$ is the vacuum permeability and $\mu_r$ is the relative permeability of the material. In a pure vacuum, $\mu_r = 1$, so $\mathbf{B} = \mu_0 \mathbf{H}$. In highly magnetic materials like iron, $\mu_r$ can be in the thousands, which is why an iron core drastically magnifies the $\mathbf{B}$ field inside an electromagnet.

Gauss’s Law for Electricity#

If you have a positive charge, electric field lines radiate outward from it. If you have a negative charge, the field lines point inward. In this macroscopic form, the law specifically states that the outward flow (or flux) of the electric displacement field from a closed region is directly proportional only to the free electric charge enclosed within it, neatly hiding the complex internal polarization of the material itself. Specifically: $$ \nabla \cdot \mathbf{D} = \rho_v $$

$\nabla \cdot \mathbf{D}$ (divergence of $\mathbf{D}$) represents the outward flow of the electric displacement field. $\rho_v$ is the free volume charge density (the measurable, physical charge you can add or remove, ignoring the microscopic bound charges within the atoms of a material).

Gauss’s Law for Magnetism#

Magnetic fields do not have isolated “charges” (monopoles). While you can have an isolated positive or negative electric charge, you can never have an isolated “North” or “South” magnetic pole. If you break a bar magnet in half, you just get two smaller magnets, each with its own North and South pole. Because of this, magnetic field lines have no beginning or end; they always form continuous, closed loops. Specifically: $$ \nabla \cdot \mathbf{B} = 0 $$

$\nabla \cdot \mathbf{B}$ (divergence of $\mathbf{B}$) is exactly zero everywhere. What flows into a region must flow out, meaning there is no single “source” or “sink” for a magnetic field.

Faraday’s Law of Induction#

A changing magnetic field generates an electric field.This is the principle behind almost all modern power generation. If you fluctuate a magnetic field, it creates a swirling electric field in the surrounding space. If a wire is present, this field pushes electrons through it, creating an electrical current. Specifically: $$ \nabla \times \mathbf{E} = -\frac{\partial \mathbf{B}}{\partial t} $$

$\nabla \times \mathbf{E}$ (curl of $\mathbf{E}$) represents the swirling, circulating nature of the induced electric field. $-\frac{\partial \mathbf{B}}{\partial t}$ represents the rate at which the magnetic flux density is changing over time. The negative sign (Lenz’s Law) dictates that the induced field opposes the change that created it.

The Ampère-Maxwell Law#

Magnetic fields are generated by flowing electrical currents and by changing electric fields. Ampère originally discovered that a wire carrying a current generates a magnetic field around it. However, Maxwell noticed a flaw in the math when dealing with broken circuits, like charging capacitors. He realized that just as a changing magnetic field creates an electric field (Faraday’s Law), a changing electric field must create a magnetic field. Specifically: $$ \nabla \times \mathbf{H} = \mathbf{J} + \frac{\partial \mathbf{D}}{\partial t} $$

$\nabla \times \mathbf{H}$ represents the swirling magnetic field intensity. $\mathbf{J}$ is the free current density (the actual, macroscopic flow of electrical charge). $\frac{\partial \mathbf{D}}{\partial t}$ is Maxwell’s addition, known as the “displacement current.” It represents the magnetic field generated by a changing electric displacement field over time.

Electromagnetic Wave Equation#

The electromagnetic (EM) wave equation is a second-order partial differential equation derived from Maxwell’s equations that describes the propagation of electric and magnetic fields through vacuum or a medium. To derive this equation, we want to decouple $ \mathbf{E} $ and $ \mathbf{B} $ to find an equation that describes only the electric field. To do this, we take the curl ($ \nabla \times $) of both sides of Faraday’s Law:

\[ \nabla \times (\nabla \times \mathbf{E}) = \nabla \times \left(-\frac{\partial \mathbf{B}}{\partial t}\right) \]

Because space and time derivatives are independent, we can swap the order of the curl and the time derivative on the right side:

\[ \nabla \times (\nabla \times \mathbf{E}) = -\frac{\partial}{\partial t} (\nabla \times \mathbf{B}) \]

Now, we use a standard vector calculus identity for the “curl of a curl”, which states that for any vector field $ \mathbf{A} $: $$ \nabla \times (\nabla \times \mathbf{A}) = \nabla(\nabla \cdot \mathbf{A}) - \nabla^2 \mathbf{A} $$ Applying this to the left side of the previous equation for electric field gives: $$ \nabla(\nabla \cdot \mathbf{E}) - \nabla^2 \mathbf{E} = -\frac{\partial}{\partial t} (\nabla \times \mathbf{B}) $$

Now we bring in the other Maxwell equations to simplify both sides:

Left side: If we assume a vacuum (no electric charges), we know from Gauss’s Law that $\nabla \cdot \mathbf{E} = 0 $. Therefore, the term $ \nabla(\nabla \cdot \mathbf{E})$ completely vanishes.
Right side: From the Ampere-Maxwell Law, we can substitute $\nabla \times \mathbf{B} $ with $ \mu_0 \epsilon_0 \frac{\partial \mathbf{E}}{\partial t}$.

Plugging these substitutions back in, we get: $$-\nabla^2 \mathbf{E} = -\frac{\partial}{\partial t} ( \mu_0 \epsilon_0 \frac{\partial \mathbf{E}}{\partial t} )$$

Multiply both sides by -1 and pull the constants out of the derivative: $$\nabla^2 \mathbf{E} = \mu_0 \epsilon_0 \frac{\partial^2 \mathbf{E}}{\partial t^2} $$

This is the final wave equation for electric fields. Note: You can repeat this exact same process starting with Ampere’s Law (taking the curl of $ \mathbf{B} $) to find the identical wave equation for the magnetic field: $$ \nabla^2 \mathbf{B} = \mu_0 \epsilon_0 \frac{\partial^2 \mathbf{B}}{\partial t^2} $$

The equation we just found perfectly matches the standard 3D wave equation from classical physics $ \nabla^2 f = \frac{1}{v^2} \frac{\partial^2 f}{\partial t^2} $. By comparing our derived equation to the standard wave equation, we can see that electric fields propagate as waves, and their velocity $ v $ squared is given by $ \frac{1}{v^2} = \mu_0 \epsilon_0 $ with $ v = \frac{1}{\sqrt{\mu_0 \epsilon_0}} $ . When you plug in the known constants for $ \mu_0 $ and $ \epsilon_0 $, you get exactly $ c $ (the speed of light, roughly $3 \times 10^8$ m/s).