Boundary Logic

Electromagnetic Interface Analysis

Physics: EM Waves

Lesson 1: Boundary Conditions

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1. The Integral Foundation

Boundary conditions are derived by applying the integral forms of Maxwell's equations to infinitesimal Gaussian surfaces and Ampèrian loops at the interface between two media.

Gauss's Law for E

\[ \oint_S \mathbf{D} \cdot d\mathbf{a} = Q_{f,enc} \]

Used for the normal component of the electric displacement field.

Gauss's Law for B

\[ \oint_S \mathbf{B} \cdot d\mathbf{a} = 0 \]

Used for the normal component of the magnetic flux density.

Faraday's Law

\[ \oint_C \mathbf{E} \cdot d\mathbf{l} = -\frac{d\Phi_B}{dt} \]

Used for the tangential component of the electric field.

Ampère-Maxwell Law

\[ \oint_C \mathbf{H} \cdot d\mathbf{l} = I_{f,enc} + \frac{d\Phi_D}{dt} \]

Used for the tangential component of the magnetic field H.

2. Derivation Analysis: Normal Components

Consider an interface between Medium 1 (\(\epsilon_1, \mu_1\)) and Medium 2 (\(\epsilon_2, \mu_2\)). A thin Gaussian pillbox of height \(h \rightarrow 0\) straddles the boundary.

Q1: Starting from Gauss's Law for B, prove that the normal component \(B^\perp\) is continuous across the boundary (\(B_1^\perp = B_2^\perp\)). Why does the flux through the sides of the pillbox vanish as \(h \rightarrow 0\)?

Q2: For the D-field, the condition is \(D_2^\perp - D_1^\perp = \sigma_f\). Under what conditions can we assume \(D^\perp\) is continuous? How does this relate to the concept of "linear, isotropic dielectrics"?

3. Tangential Components & Surface Currents

Now consider a rectangular Ampèrian loop of length \(L\) and height \(h \rightarrow 0\) oriented perpendicular to the interface.

Q3: Faraday's Law yields \(E_1^{||} = E_2^{||}\). Explain physically why the magnetic flux term \(\frac{d\Phi_B}{dt}\) does not contribute to the line integral in the limit where the loop area goes to zero.

Q4: The magnetic field H obeys \(\mathbf{n} \times (\mathbf{H}_2 - \mathbf{H}_1) = \mathbf{K}_f\). In most optical materials (non-conductors), we assume the free surface current density \(\mathbf{K}_f = 0\). What does this imply for the tangential H components? Is this also true for the tangential B components if the materials have different permeabilities?

Mastery Check

An EM wave hits a perfect conductor. Inside the conductor, \(\mathbf{E} = 0\) and \(\mathbf{B} = 0\). Use the boundary conditions to describe the fields just outside the surface. Are there surface charges or currents present?

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