Electromagnetic Boundary Conditions

Why This Matters

Electromagnetic boundary conditions are the rules that govern how fields behave when they encounter an interface between different materials—and this is where the real physics happens. You're being tested on your ability to apply Maxwell's equations at boundaries, which means understanding why certain field components must be continuous while others can jump. These conditions form the foundation for analyzing wave reflection, transmission, optical coatings, waveguides, and shielding—topics that appear repeatedly in both multiple-choice and free-response questions.

The key insight is that boundary conditions aren't arbitrary rules to memorize; they're direct consequences of Maxwell's equations applied to infinitesimally thin regions at interfaces. Master the tangential vs. normal distinction, understand when surface charges and currents create discontinuities, and you'll be able to tackle any boundary problem. Don't just memorize the equations—know which Maxwell equation each condition comes from and what physical situation would cause a discontinuity.

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Tangential Field Continuity

The tangential components of fields must be continuous across boundaries—this follows directly from applying Faraday's law and Ampère's law to tiny rectangular loops straddling the interface. When there's no time for flux to change through an infinitesimally thin loop, the line integrals of E and H around it must match on both sides.

Continuity of Tangential Electric Field

• $E_{1t} = E_{2t}$—the tangential electric field cannot jump across any interface, regardless of material properties
• Derived from Faraday's law applied to a rectangular loop shrunk to zero height at the boundary
• No exceptions exist for this condition; even perfect conductors enforce it (by requiring $E_t = 0$ on both sides of the surface)

Continuity of Tangential Magnetic Field (No Surface Current)

• $H_{1t} = H_{2t}$ when no surface current flows—the tangential H-field is continuous across ordinary dielectric interfaces
• Derived from Ampère's law using the same loop geometry; surface currents would contribute enclosed current
• Breaks down for perfect conductors where idealized surface currents $K_s$ can exist, creating discontinuities

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Normal Field Discontinuities

Normal components tell a different story—they can be discontinuous when surface charges or the material properties change. These conditions come from applying Gauss's laws to tiny pillbox surfaces straddling the interface.

Discontinuity of Normal Electric Displacement

• $D_{1n} - D_{2n} = \sigma_s$—the jump in normal D-field equals the free surface charge density
• Derived from Gauss's law for electric fields using a pillbox that captures surface charge
• For charge-free interfaces, $D_{1n} = D_{2n}$, which means $\epsilon_1 E_{1n} = \epsilon_2 E_{2n}$—the E-field does jump when permittivities differ

Continuity of Normal Magnetic Field

• $B_{1n} = B_{2n}$—the normal component of B is always continuous since magnetic monopoles don't exist
• Derived from Gauss's law for magnetism ($\nabla \cdot B = 0$) applied to a pillbox
• No exceptions—unlike the D-field condition, there's no "magnetic surface charge" to create discontinuities

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Surface Sources at Interfaces

When fields are discontinuous, something physical must cause it. Surface charge density and surface current density are the sources that "live" at interfaces and create the jumps in field components.

Surface Charge Density

• $\sigma_s = D_{1n} - D_{2n}$—surface charge accumulates wherever the normal D-field is discontinuous
• Appears at conductor surfaces where external fields terminate, and at dielectric interfaces with different polarizations
• Determines field behavior in capacitors, at electrode surfaces, and in layered dielectric structures

Surface Current Density

• $\vec{K}_s = \hat{n} \times (\vec{H}_1 - \vec{H}_2)$—surface current flows when tangential H-field jumps across the boundary
• Idealized concept that becomes physical in superconductors and approximates behavior in good conductors at high frequencies
• Direction matters—the cross product with the surface normal determines current flow direction, critical for shielding calculations

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Perfect Conductor Boundaries

Perfect conductors represent an idealized limit where conductivity approaches infinity, forcing specific field configurations. Inside a perfect conductor, fields must be zero—all the action happens at the surface.

Boundary Conditions for Perfect Conductors

• $E_t = 0$ at the surface—any tangential E-field would drive infinite current, so it must vanish
• $B_n = 0$ at the surface—combined with the interior condition, the normal B-field cannot penetrate
• Surface charge $\sigma_s = D_n$ and surface current $\vec{K}_s = \hat{n} \times \vec{H}$ carry all the boundary information, completely shielding the interior

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Wave Behavior at Interfaces

Boundary conditions directly determine how electromagnetic waves reflect and transmit at interfaces. The Fresnel equations and total internal reflection are consequences of enforcing these conditions on propagating waves.

Fresnel Equations for Reflection and Transmission

• Derived by matching tangential E and H at the interface for incident, reflected, and transmitted waves
• Depend on polarization—s-polarization (E perpendicular to plane of incidence) and p-polarization have different reflection coefficients
• Brewster's angle occurs for p-polarization when the reflection coefficient vanishes, a direct consequence of the boundary conditions

Total Internal Reflection

• Occurs when $\theta_i > \theta_c = \sin^{-1}(n_2/n_1)$—light traveling from higher to lower refractive index beyond the critical angle
• Boundary conditions still satisfied but the transmitted wave becomes evanescent, decaying exponentially into the second medium
• Enables fiber optics and prism-based devices—the evanescent field is key to understanding frustrated total internal reflection

Dielectric-Dielectric Interface Behavior

• All four boundary conditions apply—tangential E and H continuous, normal D and B continuous (assuming no free charges/currents)
• Wave impedance mismatch $\eta = \sqrt{\mu/\epsilon}$ determines reflection and transmission amplitudes
• Snell's law emerges from requiring phase matching of tangential fields along the interface

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Quick Reference Table

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Self-Check Questions

1. Which two boundary conditions are always true regardless of surface charges or currents, and which Maxwell equation does each come from?

2. Compare and contrast how the normal components of D and B behave at an interface—why is one always continuous while the other can be discontinuous?

3. At a perfect conductor surface, why must $E_t = 0$ while $H_t$ can be nonzero? What physical quantity accounts for the H-field discontinuity?

4. If you're given the electric field on both sides of an interface and asked to find the surface charge density, which boundary condition do you use and what's the formula?

5. Explain how the Fresnel equations and total internal reflection both emerge from applying electromagnetic boundary conditions—what's the key difference in the resulting wave behavior?