The causal chain of source $\longrightarrow$ field $\longrightarrow$ observable reflects the factorization of physics into three components: excitations (sources), propagation through a medium (determined by materials and boundary conditions), and measuring quantities (observations). The notes within my classical electromagnetism section will follow this structure with the first section modeling sources.
Electromagnetic fields are vector-valued functions of space and time. Using the Cartesian unit vectors $\hat{\mathbf{x}},\hat{\mathbf{y}},\hat{\mathbf{z}}$, a generic vector field is written as
\[\mathbf{V}(x,y,z,t) = V_x\,\hat{\mathbf{x}} + V_y\,\hat{\mathbf{y}} + V_z\,\hat{\mathbf{z}}.\]The electric and magnetic fields are denoted by $\mathbf{E}$ and $\mathbf{H}$, and the electric and magnetic flux densities by $\mathbf{D}$ and $\mathbf{B}$, respectively. In the presence of electric and magnetic charge and current densities $(\rho_e, \mathbf{J})$ and $(\rho_m, \mathbf{M})$, Maxwell’s equations in differential form are
\(\nabla \times \mathbf{E} = -\frac{\partial \mathbf{B}}{\partial t} - \mathbf{M}\) \(\nabla \times \mathbf{H} = \frac{\partial \mathbf{D}}{\partial t} + \mathbf{J}\) \(\nabla \cdot \mathbf{D} = \rho_e \quad \nabla \cdot \mathbf{B} = \rho_m\)
In integral form over an oriented surface $S$ with boundary curve $\partial S$, these become
\(\oint_{\partial S} \mathbf{E} \cdot d\boldsymbol{\ell} = - \int_{S} \left( \frac{\partial \mathbf{B}}{\partial t} + \mathbf{M} \right) \cdot d\mathbf{S}\) \(\oint_{\partial S} \mathbf{H} \cdot d\boldsymbol{\ell} = \phantom{-} \int_{S} \left( \frac{\partial \mathbf{D}}{\partial t} + \mathbf{J} \right) \cdot d\mathbf{S}\) \(\int_{S} \mathbf{D} \cdot d\mathbf{S} = \int_{V} \rho_e\, dV\) \(\int_{S} \mathbf{B} \cdot d\mathbf{S} = \int_{V} \rho_m\, dV\)
where $V$ is the volume enclosed by $S$. The magnetic charge densities and currents introduced above are not known to exist physically but including them now allows for symmetry within Maxwell’s equations and enables later use of duality transformations.
Maxwell’s equations in linear, isotropic media exhibit a striking symmetry. If we exchange electrical quantities with magnetic quantities (and vice versa), the form of the equations remains invariant. This allows us to solve a problem for an electric source ($\mathbf{J}$) and immediately know the solution for the corresponding magnetic source ($\mathbf{M}$) without have to solve for a completely new set of equations.
The general duality transformation is a rotation in electromagnetic space. However, two specific variations are most common: Mathematical Duality (unit-agnostic) and Scaled Duality (unit-preserving).
1. Mathematical Duality This simple substitution swaps vectors directly. It is useful for verifying symmetries but ignores units (e.g., swapping Volts/m for Amps/m).
\[\mathbf{E} \to \mathbf{H}, \quad \mathbf{H} \to -\mathbf{E}, \quad \mathbf{J} \to \mathbf{M}, \quad \mathbf{M} \to -\mathbf{J}, \quad \epsilon \leftrightarrow \mu\]2. Scaled Duality To preserve physical units (so that Volts map to Volts) and maintain power relations, we introduce the intrinsic impedance of the medium, $\eta = \sqrt{\mu/\epsilon}$. This scaling is critical when converting known antenna solutions (e.g., electric dipoles) to their dual counterparts (e.g., magnetic loops).
The transformation from an Electric System ($e$) to a Magnetic System ($m$) is:
\[\mathbf{E}_m = -\eta \mathbf{H}_e \qquad \mathbf{H}_m = \frac{1}{\eta} \mathbf{E}_e\] \[\mathbf{M}_m = \eta \mathbf{J}_e \qquad \rho_{mm} = \eta \rho_{ee}\]Maxwell’s equations couple to matter through material laws
\[\mathbf{D} = \epsilon\,\mathbf{E}, \qquad \mathbf{B} = \mu\,\mathbf{H}, \qquad \mathbf{J} = \sigma\,\mathbf{E}.\]where $\epsilon$, $\mu$, and $\sigma$ are the permittivity, permeability, and conductivity of the material and define how a medium responds to electormagnetic fields. Media are categorized based on how these parameters behave relative to space, time, direction, and field strength. In the most general case, material parameters possess the following characteristics:
Furthermore, we can make these additional categorizations
| Category | Definition | Constitutive Relation |
|---|---|---|
| Isotropic | Response is independent of field orientation. | $\mathbf{D} = \epsilon \mathbf{E}$ (Scalar $\epsilon$) |
| Anisotropic | Response depends on direction (e.g., crystals). | $\mathbf{D} = \bar{\bar{\epsilon}} \cdot \mathbf{E}$ (Tensor $\bar{\bar{\epsilon}}$) |
| Nonlinear | Response depends on field magnitude. | $\mathbf{P} = \epsilon_0(\chi^{(1)}\mathbf{E} + \chi^{(2)}\mathbf{E}^2 + \dots)$ |
| Bianisotropic | Electric and magnetic fields are coupled. | $\mathbf{D} = \epsilon \mathbf{E} + \xi \mathbf{H}$ |
In optics and nanophotonics, it is often more convenient to describe a medium using the dimensionless refractive index $n$. For a linear, isotropic, and non-magnetic ($\mu = \mu_0$) medium, $n$ is defined as:
\[n = \sqrt{\frac{\epsilon}{\epsilon_0}} = \sqrt{\epsilon_r}\]To account for both phase shift and attenuation (absorption), we use the complex refractive index $\tilde{n}$:
\[\tilde{n} = n + j\kappa\]
This simple example highlights two complementary routes for solving Maxwell’s equations: using the integral form together with symmetry arguments, or using the differential form and solving the resulting partial differential equations (here, Poisson’s equation for $\phi$).
Charge conservation imposes a local constraint on admissible electromagnetic fields. In differential form, the conservation of electric (and magnetic) charge is expressed through the continuity equations
\[\nabla \cdot \mathbf{J} = -\frac{\partial \rho_e}{\partial t}.\]Integrating over a fixed volume (V),
\[\frac{d}{dt} \int_V \rho_e\,dV = -\oint_{\partial V} \mathbf{J}\cdot d\mathbf{S}.\]Thus the charge in a region can only change via current crossing its boundary.
Charge conservation is not independent of Maxwell’s equations. Taking $\nabla\cdot$ of Ampère–Maxwell and using $\nabla\cdot\nabla\times!\bullet=0$ plus Gauss’s law recovers the continuity equation.
At a more abstract level, conservation laws arise from symmetries. In electromagnetism, global gauge invariance implies conservation of electric charge, whose local expression is precisely the continuity equation. The continuity equation in turn constrains the permissible electromagnetic fields and sources.
Maxwell’s equations in differential form assume the field vectors are differentiable. However, at the interface between two different media, the material parameters ($\epsilon, \mu, \sigma$) often change abruptly. To handle this (alongside the mathematical necessity of coupled PDEs requiriing certain conditions for a unique solution), we introduce boundary conditions
Let an interface separate Medium 1 from Medium 2. We define the unit normal vector $\hat{\mathbf{n}}$ to point from Medium 2 into Medium 1. \(\hat{\mathbf{n}} = \hat{\mathbf{n}}_{2 \to 1}\)
| Component | Boundary Condition | Physical Implication |
|---|---|---|
| Tangential E | $\hat{\mathbf{n}} \times (\mathbf{E}_1 - \mathbf{E}_2) = -\mathbf{M}_s$ | $\mathbf{E}_{tan}$ is continuous unless magnetic current exists. |
| Tangential H | $\hat{\mathbf{n}} \times (\mathbf{H}_1 - \mathbf{H}_2) = \mathbf{J}_s$ | $\mathbf{H}_{tan}$ is continuous unless electric current exists. |
| Normal D | $\hat{\mathbf{n}} \cdot (\mathbf{D}1 - \mathbf{D}_2) = \rho{es}$ | $\mathbf{D}_{norm}$ jumps by the surface charge density. |
| Normal B | $\hat{\mathbf{n}} \cdot (\mathbf{B}1 - \mathbf{B}_2) = \rho{ms}$ | $\mathbf{B}_{norm}$ is continuous (unless magnetic monopoles exist). |
The following example applies the boundary condition approach. It solves for the radiation from a magnetic current sheet, demonstrating how a non-zero source ($\mathbf{M}_s$) forces a discontinuity in the tangential electric field and the absence of a source ($\mathbf{J}_s = 0$) preserves continuity in the tangential magnetic field.
Sources enter Maxwell’s theory in two distinct ways: explicitly as volumetric densities ($\rho, \mathbf{J}$) driving the differential equations, or implicitly as surface singularities ($\rho_s, \mathbf{J}_s$) enforcing discontinuities at boundaries.
In many practical electromagnetic problems, explicit volumetric sources are absent from the region of interest. Instead, the physics is driven by equivalent surface charges or currents defined at interfaces. A key skill in modeling is recognizing that these descriptions are often physically equivalent, differing only in mathematical convenience.
Consider a bounded volume $V$ containing “electromagnetic junk” – complex sources, nonlinear media, or intricate geometries. Let us introduce a fictitious surface enclosing the “junk” called $S$. To examine the fields ($\mathbf{E}, \mathbf{H}$) outside the volume, we can use the Equivalence Principle.
According to the Uniqueness Theorem, the field solution in the outside region is uniquely determined if we know the tangential fields on the boundary $S$. Therefore, we can perform this:
Using the boundary conditions derived in Section 1.5 (where $\hat{\mathbf{n}}$ points outward from the zero-field region into the solution region), these currents are:
\[\mathbf{J}_s = \hat{\mathbf{n}} \times (\mathbf{H} - 0) = \hat{\mathbf{n}} \times \mathbf{H}\] \[\mathbf{M}_s = -\hat{\mathbf{n}} \times (\mathbf{E} - 0) = \mathbf{E} \times \hat{\mathbf{n}}\]These fictitious currents, radiating in free space, produce the exact same fields outside $V$ as the original complex sources.
The Equivalence Principle allows us to replace a volume with surface currents, but Image Theory allows us to replace a conductive boundary with virtual sources. This is particularly useful when solving for radiation near ground planes or simplifying the equivalent currents derived in the previous section.
To apply image theory, we idealize materials into two theoretical limits where fields cannot exist.
When a source is placed near a PEC or PMC, the boundary conditions can be satisfied by removing the conductor and placing an “image source” in the region previously occupied by the conductor.
The orientation of the image source depends on whether the source vector is parallel (tangential) or perpendicular (normal) to the boundary.
Case A: Electric Source ($\mathbf{J}$) over PEC
Case B: Magnetic Source ($\mathbf{M}$) over PEC Because magnetic currents are the mathematical dual of electric currents, the rules are reversed.
| Source Type | Orientation | PEC Image | PMC Image |
|---|---|---|---|
| Electric ($\mathbf{J}$) | Vertical ($\perp$) | Same Direction ($+$) | Opposite ($-$) |
| Electric ($\mathbf{J}$) | Horizontal ($\parallel$) | Opposite ($-$) | Same Direction ($+$) |
| Magnetic ($\mathbf{M}$) | Vertical ($\perp$) | Opposite ($-$) | Same Direction ($+$) |
| Magnetic ($\mathbf{M}$) | Horizontal ($\parallel$) | Same Direction ($+$) | Opposite ($-$) |