Introduction to Electromagnetic Theory Study Guide

Fundamental Concepts of Electromagnetic Fields

• Electric Field vs. Magnetic Field:

  • An Electric Field ($\mathbf{E}$) is generated by static or moving electric charges and exerts forces on both stationary and moving charges.

  • A Magnetic Field ($\mathbf{B}$) is generated by moving charges (electric currents) or permanent magnets and exerts forces exclusively on moving charges and magnetic materials.

• Comparison Between Electric and Magnetic Fields:

• Electromagnetic Waves:

  • Electromagnetic waves consist of time-varying electric field vectors ($\mathbf{E}$) and magnetic field vectors ($\mathbf{B}$) oscillating in mutually perpendicular planes, propagating perpendicular to both fields.

  • In a plane EM wave traveling along the $x$-axis, if the electric field oscillates along the $y$-axis, the magnetic field oscillates along the $z$-axis.

• The Electromagnetic Spectrum:

  • Electromagnetic waves span a continuous spectrum of wavelengths ($\lambda$) and frequencies:

  • Gamma rays: $\approx 10^{-13}\,\text{cm}$

  • X-rays: $\approx 10^{-9}\,\text{cm}$

  • Ultraviolet: $\approx 10^{-6}\,\text{cm}$

  • Visible light: $\approx 10^{-4}\,\text{cm}$

  • Infrared: $\approx 10^{-2}\,\text{cm}$

  • Microwaves: $\approx 1\,\text{cm}$

  • Radio waves: $\approx 1\,\text{km}$

• Magnetic Flux:

  • Magnetic flux ($\Phi_B$) quantifies the total magnetic field passing through a given surface area $A$.

  • Formula: $\Phi_B = B A \cos(\theta)$

  • Where $B$ is the magnetic field magnitude in Tesla ($\text{T}$), $A$ is the surface area in square meters ($\text{m}^2$), and $\theta$ is the angle between the magnetic field vector $\mathbf{B}$ and the surface normal.

Faraday's Laws of Electromagnetic Induction & Lenz's Law

• Electromagnetic Induction:

  • Electromagnetic induction is the phenomenon where an electromotive force (EMF) or current is generated in a conductor by a changing magnetic flux.

• Faraday's First Law:

  • Whenever the magnetic flux linked with a closed coil or circuit changes over time, an electromotive force (EMF) is induced in it.

  • This induced EMF persists only as long as the change in magnetic flux continues.

• Faraday's Second Law:

  • The magnitude of the induced EMF ($\varepsilon$) is directly proportional to the time rate of change of magnetic flux linkage through the circuit.

  • Equation: $\varepsilon = -N \frac{d\Phi_B}{dt}$

  • Where $N$ represents the total number of turns in the coil, and $\frac{d\Phi_B}{dt}$ represents the rate of change of magnetic flux.

• Lenz's Law:

  • Statement: The direction of the induced EMF (and the resulting induced current) is always such that it opposes the change in magnetic flux that produces it.

  • The negative sign in Faraday's law formula reflects Lenz's law.

  • Physical Mechanism:

  • As a North magnetic pole approaches a coil, the magnetic flux through the coil increases. According to Lenz's law, the induced current flows in a direction that generates a North pole on the near face of the coil to repel the approaching magnet.

  • As the North pole moves away, flux decreases; the coil induces current in the opposite direction to form a South pole on the near face, attracting the receding magnet.

  • Energy Conservation Basis: Lenz's law is a direct consequence of the principle of conservation of energy. Mechanical work must be expended against the repulsive or attractive magnetic forces to maintain flux variation, which is converted into electrical energy.

• Parameters Controlling Induced Current:

  • Direction reversal occurs if:

  1. The motion of the magnet is reversed.

  2. The opposite magnetic pole (South instead of North) enters the coil first.

  • Magnitude increases by:

  1. Increasing the relative velocity/speed between magnet and coil.

  2. Increasing the magnetic field strength of the magnet.

  3. Increasing the number of turns ($N$) on the coil.

• Fleming's Right-Hand Rule:

  • Purpose: Determines the direction of induced current when a conductor moves across a magnetic field.

  • Statement: Stretch the thumb, forefinger (index finger), and middle finger of the right hand so that they are mutually perpendicular to each other:

  • Thumb: Points in the direction of the Motion of the conductor.

  • Forefinger (Index finger): Points in the direction of the Magnetic Field (from North to South).

  • Middle finger: Points in the direction of the Induced Current (or induced EMF).

S

Electrodynamics Before Maxwell

• Historical Pre-Maxwell Equations:

  • Before James Clerk Maxwell's modifications, classical electrodynamics was summarized by four distinct experimental laws in electrostatic and magnetostatic regimes.

• Comparative Unit Formulations (SI vs. Gaussian/CGS):

• Detailed Derivation of Equation 1: Electrostatic Gauss's Law:

  • Integral Form: The net electric flux through any closed surface $S$ equals the enclosed electric charge divided by $\epsilon_0$: $\Phi_E = \oint_S \mathbf{E} \cdot d\mathbf{A} = \frac{q_{\text{enclosed}}}{\epsilon_0}$

  • Expressing enclosed charge in terms of volume charge density $\rho$: $\frac{q_{\text{enclosed}}}{\epsilon_0} = \int_V \frac{\rho}{\epsilon_0}\,dV$

  • Applying Gauss's Divergence Theorem to the surface integral of $\mathbf{E}$: $\oint_S \mathbf{E} \cdot d\mathbf{A} = \int_V (\nabla \cdot \mathbf{E})\,dV$

  • Equating the volume integrals: $\int_V (\nabla \cdot \mathbf{E})\,dV = \int_V \frac{\rho}{\epsilon_0}\,dV$

  • Since this relation holds for any arbitrary volume $V$, the integrands must be equal, yielding the Differential Form: $\nabla \cdot \mathbf{E} = \frac{\rho}{\epsilon_0}$

• Detailed Derivation of Equation 2: Magnetostatic Gauss's Law:

  • Integral Form: The net magnetic flux through any closed surface $S$ is identically zero: $\oint_S \mathbf{B} \cdot d\mathbf{S} = 0$

  • Applying Gauss's Divergence Theorem: $\int_V (\nabla \cdot \mathbf{B})\,dV = \oint_S \mathbf{B} \cdot d\mathbf{S} = 0$

  • Because the volume $V$ is arbitrary, the Differential Form is: $\nabla \cdot \mathbf{B} = 0$

  • Physical Significance:

  • Magnetic field lines do not start or terminate on isolated sources; they form continuous closed loops.

  • The magnetic field vector $\mathbf{B}$ is a solenoidal vector field.

  • Isolated magnetic monopoles (free magnetic charges) do not exist in nature, unlike free electric charges ($+q$ and $-q$) in electrostatics.

• Detailed Derivation of Equation 3: Faraday's Law of Induction:

  • Induced EMF around a closed loop $C$ is the line integral of the electric field $\mathbf{E}$: $\text{EMF} = \oint_C \mathbf{E} \cdot d\mathbf{l} = -\frac{d}{dt} \left[ \int_S \mathbf{B} \cdot \hat{n}\,dA \right]$

  • Applying Stokes' Theorem to transform the closed line integral into a surface integral: $\oint_C \mathbf{E} \cdot d\mathbf{l} = \int_S (\nabla \times \mathbf{E}) \cdot \hat{n}\,dA$

  • Equating the two surface integrals: $\int_S (\nabla \times \mathbf{E}) \cdot \hat{n}\,dA = \int_S \left(-\frac{\partial \mathbf{B}}{\partial t}\right) \cdot \hat{n}\,dA$

  • Since this equality holds for any surface $S$, the Differential Form is: $\nabla \times \mathbf{E} = -\frac{\partial \mathbf{B}}{\partial t}$

• Detailed Derivation of Equation 4: Original Ampère's Circuital Law:

  • Integral Form: The line integral of magnetic field $\mathbf{B}$ along a closed path $C$ equals $\mu_0$ times the total enclosed current $I_{\text{enclosed}}$: $\oint_C \mathbf{B} \cdot d\mathbf{l} = \mu_0 I_{\text{enclosed}}$

  • Expressing current in terms of volume current density $\mathbf{J}$ over surface $S$: $I_{\text{enclosed}} = \int_S \mathbf{J} \cdot d\mathbf{S}$

  • Applying Stokes' Theorem to the left side: $\oint_C \mathbf{B} \cdot d\mathbf{l} = \int_S (\nabla \times \mathbf{B}) \cdot d\mathbf{S}$

  • Equating integrals gives: $\int_S (\nabla \times \mathbf{B}) \cdot d\mathbf{S} = \mu_0 \int_S \mathbf{J} \cdot d\mathbf{S}$

  • Differential Form (Original Ampère's Law): $\nabla \times \mathbf{B} = \mu_0 \mathbf{J}$

• Mathematical Inconsistency of Original Ampère's Law:

  • Taking the vector divergence of both sides of Faraday's Law: $\nabla \cdot (\nabla \times \mathbf{E}) = \nabla \cdot \left(-\frac{\partial \mathbf{B}}{\partial t}\right) = -\frac{\partial}{\partial t}(\nabla \cdot \mathbf{B})$

  • Since the divergence of any curl vector is mathematically zero ($\nabla \cdot (\nabla \times \mathbf{A}) \equiv 0$) and $\nabla \cdot \mathbf{B} = 0$, both sides equal zero ($0 = 0$), confirming mathematical consistency.

  • Taking the vector divergence of both sides of original Ampère's Law: $\nabla \cdot (\nabla \times \mathbf{B}) = \mu_0 (\nabla \cdot \mathbf{J})$

  • Since $\nabla \cdot (\nabla \times \mathbf{B}) = 0$, this requires: $\mu_0 (\nabla \cdot \mathbf{J}) = 0 \implies \nabla \cdot \mathbf{J} = 0$

  • While $\nabla \cdot \mathbf{J} = 0$ holds true for steady, direct currents (magnetostatics), it fails for time-varying currents and non-steady fields where charge density changes in time.

  • Consequently, original Ampère's law is incomplete and incorrect beyond steady-state magnetostatics.

Volume Current Density and the Continuity Equation

• Volume Current Density ($\mathbf{J}$):

  • When electric charge flows through a three-dimensional region, current distribution is described by the volume current density vector $\mathbf{J}$.

  • Definition: Consider a tube of infinitesimal cross-sectional area $da_\perp$ aligned parallel to charge flow carrying current $dI$: $\mathbf{J} \equiv \frac{dI}{da_\perp}$

  • Physical Meaning: $\mathbf{J}$ represents the electric current passing per unit cross-sectional area perpendicular to the flow.

  • Total Current Equation: The net current $I$ crossing a macroscopic surface $S$ is: $I = \int_S J\,da_\perp = \int_S \mathbf{J} \cdot d\mathbf{a}$

  • Microscopic Charge Relation: If $\rho$ is the mobile volume charge density and $\mathbf{v}$ is charge velocity: $\mathbf{J} = \rho \mathbf{v}$

• Derivation of the Continuity Equation (Local Charge Conservation):

  • The total electric charge leaving a volume $V$ bounded by surface $S$ per unit time is: $\oint_S \mathbf{J} \cdot d\mathbf{a} = \oint_S \rho \mathbf{v} \cdot d\mathbf{a}$

  • By the principle of conservation of charge, charge cannot be created or destroyed. Any outward flow of charge through surface $S$ must decrease the total remaining charge inside volume $V$: $\oint_S \mathbf{J} \cdot d\mathbf{a} = -\frac{d}{dt} \left[ \int_V \rho\,d\tau \right] = -\int_V \left(\frac{\partial \rho}{\partial t}\right)\,d\tau$

  • Applying Gauss's Divergence Theorem to the surface integral of $\mathbf{J}$: $\oint_S \mathbf{J} \cdot d\mathbf{a} = \int_V (\nabla \cdot \mathbf{J})\,d\tau$

  • Equating both volume integrals: $\int_V (\nabla \cdot \mathbf{J})\,d\tau = -\int_V \left(\frac{\partial \rho}{\partial t}\right)\,d\tau$

  • Since this relation holds for any arbitrary volume $V$, the integrands must be identical: $\nabla \cdot \mathbf{J} = -\frac{\partial \rho}{\partial t}$

  • Rearranging yields the standard Continuity Equation: $\nabla \cdot \mathbf{J} + \frac{\partial \rho}{\partial t} = 0$

  • Statement: The continuity equation is the precise mathematical statement of local charge conservation.

Maxwell's Modification of Ampère's Law & Displacement Current

• The Capacitor Paradox:

  • Consider a circuit containing a battery charging a parallel-plate capacitor with conduction current $I$ flowing in the connecting wires.

  • Apply original Ampère's Law $\oint_C \mathbf{B} \cdot d\mathbf{l} = \mu_0 I_{\text{enclosed}}$ to an Amperian loop $C$ enclosing the wire:

  1. Flat Surface Option: If a flat disk surface bounded by loop $C$ is selected, the wire penetrates the surface carrying current $I$. Thus $I_{\text{enclosed}} = I$, and $\oint_C \mathbf{B} \cdot d\mathbf{l} = \mu_0 I$.

  2. Balloon Surface Option: If a balloon-shaped surface bulging between the capacitor plates is selected, no conduction wire crosses this surface ($I_{\text{enclosed}} = 0$). Thus $\oint_C \mathbf{B} \cdot d\mathbf{l} = 0$

  • Contradiction: Two different surfaces sharing the exact same boundary curve $C$ give conflicting magnetic field predictions. Ampère's law fails in the gap between capacitor plates.

• Maxwell's Resolution & Derivation of Displacement Current:

  • The uniform electric field $E$ between parallel capacitor plates with charge $Q$ and surface area $A$ is: $E = \frac{\sigma}{\epsilon_0} = \frac{Q}{\epsilon_0 A}$

  • Differentiating with respect to time $t$: $\frac{\partial E}{\partial t} = \frac{1}{\epsilon_0 A} \frac{dQ}{dt} = \frac{I}{\epsilon_0 A}$

  • Expressing conduction current $I$ in terms of rate of change of electric field: $I = \epsilon_0 A \frac{\partial E}{\partial t} = \epsilon_0 \int_S \left(\frac{\partial \mathbf{E}}{\partial t}\right) \cdot d\mathbf{a}$

  • Maxwell defined this time-varying electric field term as the Displacement Current ($I_D$): $I_D \equiv \epsilon_0 \frac{\partial \Phi_E}{\partial t} = \epsilon_0 \int_S \left(\frac{\partial \mathbf{E}}{\partial t}\right) \cdot d\mathbf{a}$

  • Definition of Displacement Current Density ($\mathbf{J}_D$): $\mathbf{J}_D = \epsilon_0 \frac{\partial \mathbf{E}}{\partial t} = \frac{\partial \mathbf{D}}{\partial t}$

  • Where $\mathbf{D} = \epsilon_0 \mathbf{E}$ is the electric displacement field.

• Modified Ampère's Law (Ampère-Maxwell Law):

  • Maxwell added displacement current to conduction current, defining total current $I_{\text{total}} = I_C + I_D$

  • Integral Form: $\oint_C \mathbf{B} \cdot d\mathbf{l} = \mu_0 I_{\text{enclosed}} + \mu_0 \epsilon_0 \int_S \left(\frac{\partial \mathbf{E}}{\partial t}\right) \cdot d\mathbf{a}$

  • Differential Form: $\nabla \times \mathbf{B} = \mu_0 \mathbf{J}_C + \mu_0 \epsilon_0 \frac{\partial \mathbf{E}}{\partial t}$

• Proof of Consistency with Charge Conservation:

  • Taking divergence of both sides of the Ampère-Maxwell Law: $\nabla \cdot (\nabla \times \mathbf{B}) = \mu_0 (\nabla \cdot \mathbf{J}_C) + \mu_0 \epsilon_0 \nabla \cdot \left(\frac{\partial \mathbf{E}}{\partial t}\right)$

  • Substituting $\nabla \cdot (\nabla \times \mathbf{B}) = 0$: $0 = \mu_0 (\nabla \cdot \mathbf{J}_C) + \mu_0 \epsilon_0 \frac{\partial}{\partial t}(\nabla \cdot \mathbf{E})$

  • Substitute Electrostatic Gauss's Law ($\nabla \cdot \mathbf{E} = \frac{\rho}{\epsilon_0}$): $\mu_0 (\nabla \cdot \mathbf{J}_C) + \mu_0 \epsilon_0 \frac{\partial}{\partial t}\left(\frac{\rho}{\epsilon_0}\right) = 0$ $\mu_0 \left(\nabla \cdot \mathbf{J}_C + \frac{\partial \rho}{\partial t}\right) = 0 \implies \nabla \cdot \mathbf{J}_C + \frac{\partial \rho}{\partial t} = 0$

  • The modified law perfectly recovers the continuity equation, resolving the mathematical contradiction.

• Comparison: Conduction Current vs. Displacement Current:

Complete Formulation and Properties of Maxwell's Equations

• Complete Set of Maxwell's Equations:

• In SI Units:

  1. Gauss's Law for Electrostatics: $\nabla \cdot \mathbf{E} = \frac{\rho}{\epsilon_0}$

  2. Gauss's Law for Magnetostatics: $\nabla \cdot \mathbf{B} = 0$

  3. Faraday's Law of Electromagnetic Induction: $\nabla \times \mathbf{E} = -\frac{\partial \mathbf{B}}{\partial t}$

  4. Ampère-Maxwell Law: $\nabla \times \mathbf{B} = \mu_0 \mathbf{J} + \mu_0 \epsilon_0 \frac{\partial \mathbf{E}}{\partial t}$

• In Gaussian (CGS) Units:

  1. $\nabla \cdot \mathbf{E} = 4\pi\rho$

  2. $\nabla \cdot \mathbf{B} = 0$

  3. $\nabla \times \mathbf{E} = -\frac{1}{c}\frac{\partial \mathbf{B}}{\partial t}$

  4. $\nabla \times \mathbf{B} = \frac{4\pi}{c}\mathbf{J} + \frac{1}{c}\frac{\partial \mathbf{E}}{\partial t}$

• Lorentz Force Law Integration:

  • Maxwell's equations together with the Lorentz force equation summarize the entire classical electrodynamics theory: $\mathbf{F} = q \left( \mathbf{E} + \mathbf{v} \times \mathbf{B} \right)$

• Core Properties of Maxwell's Equations:

  1. Linearity: Maxwell's equations are linear differential equations. This property underpins the principle of superposition: if two electric/magnetic fields satisfy Maxwell's equations, their vector sum also satisfies them.

  2. Implicit Charge Conservation: Divergence of the 4th equation automatically yields the continuity equation $\nabla \cdot \mathbf{J} + \frac{\partial \rho}{\partial t} = 0$, enforcing global and local charge conservation.

  3. Field Mutual Interconversion:

  • A time-varying magnetic field generates a space-varying electric field ($\nabla \times \mathbf{E} = -\frac{\partial \mathbf{B}}{\partial t}$).

  • A time-varying electric field generates a space-varying magnetic field ($\nabla \times \mathbf{B} = \mu_0 \epsilon_0 \frac{\partial \mathbf{E}}{\partial t}$).

  • This continuous interconversion allows self-sustaining electromagnetic wave propagation through space without physical media.

Electromagnetic Wave Equation in Free Space

• Maxwell's Equations in Free Space (Vacuum):

  • In vacuum, there are no free electric charges ($\rho = 0$) and no electric conduction currents ($\mathbf{J} = 0$).

  • Expressing fields in terms of $\mathbf{E}$ and $\mathbf{H}$ (where $\mathbf{B} = \mu_0 \mathbf{H}$ and $\mathbf{D} = \epsilon_0 \mathbf{E}$): $\nabla \cdot \mathbf{E} = 0 \quad \text{--- (1)}$ $\nabla \cdot \mathbf{H} = 0 \quad \text{--- (2)}$ $\nabla \times \mathbf{E} = -\mu_0 \frac{\partial \mathbf{H}}{\partial t} \quad \text{--- (3)}$ $\nabla \times \mathbf{H} = \epsilon_0 \frac{\partial \mathbf{E}}{\partial t} \quad \text{--- (4)}$

• Derivation of Wave Equation for Electric Field ($\mathbf{E}$):

  • Take curl of equation (3): $\nabla \times (\nabla \times \mathbf{E}) = \nabla \times \left(-\mu_0 \frac{\partial \mathbf{H}}{\partial t}\right)$

  • Apply vector identity $\nabla \times (\nabla \times \mathbf{E}) = \nabla(\nabla \cdot \mathbf{E}) - \nabla^2 \mathbf{E}$: $\nabla(\nabla \cdot \mathbf{E}) - \nabla^2 \mathbf{E} = -\mu_0 \frac{\partial}{\partial t}(\nabla \times \mathbf{H})$

  • Substitute $\nabla \cdot \mathbf{E} = 0$ from (1) and $\nabla \times \mathbf{H} = \epsilon_0 \frac{\partial \mathbf{E}}{\partial t}$ from (4): $-\nabla^2 \mathbf{E} = -\mu_0 \frac{\partial}{\partial t}\left(\epsilon_0 \frac{\partial \mathbf{E}}{\partial t}\right)$ $\nabla^2 \mathbf{E} = \mu_0 \epsilon_0 \frac{\partial^2 \mathbf{E}}{\partial t^2}$

  • Rearranging gives the 3D Wave Equation for $\mathbf{E}$: $\nabla^2 \mathbf{E} - \mu_0 \epsilon_0 \frac{\partial^2 \mathbf{E}}{\partial t^2} = 0$

• Derivation of Wave Equation for Magnetic Field ($\mathbf{H}$):

  • Take curl of equation (4): $\nabla \times (\nabla \times \mathbf{H}) = \nabla \times \left(\epsilon_0 \frac{\partial \mathbf{E}}{\partial t}\right)$

  • Apply vector identity $\nabla \times (\nabla \times \mathbf{H}) = \nabla(\nabla \cdot \mathbf{H}) - \nabla^2 \mathbf{H}$: $\nabla(\nabla \cdot \mathbf{H}) - \nabla^2 \mathbf{H} = \epsilon_0 \frac{\partial}{\partial t}(\nabla \times \mathbf{E})$

  • Substitute $\nabla \cdot \mathbf{H} = 0$ from (2) and $\nabla \times \mathbf{E} = -\mu_0 \frac{\partial \mathbf{H}}{\partial t}$ from (3): $-\nabla^2 \mathbf{H} = \epsilon_0 \frac{\partial}{\partial t}\left(-\mu_0 \frac{\partial \mathbf{H}}{\partial t}\right)$ $\nabla^2 \mathbf{H} = \mu_0 \epsilon_0 \frac{\partial^2 \mathbf{H}}{\partial t^2}$

  • Rearranging gives the 3D Wave Equation for $\mathbf{H}$: $\nabla^2 \mathbf{H} - \mu_0 \epsilon_0 \frac{\partial^2 \mathbf{H}}{\partial t^2} = 0$

• Velocity of Propagation & Speed of Light:

  • Standard 3D wave equation for a scalar wave function $f$: $\nabla^2 f = \frac{1}{v^2} \frac{\partial^2 f}{\partial t^2}$

  • Comparing $\nabla^2 \mathbf{E} = \mu_0 \epsilon_0 \frac{\partial^2 \mathbf{E}}{\partial t^2}$ with standard form: $\frac{1}{v^2} = \mu_0 \epsilon_0 \implies v = \frac{1}{\sqrt{\mu_0 \epsilon_0}}$

  • Substituting fundamental physical constants:

  • Permeability of free space: $\mu_0 = 4\pi \times 10^{-7}\,\text{N/A}^2$

  • Permittivity of free space: $\epsilon_0 \approx 8.854 \times 10^{-12}\,\text{F/m}$ $v = \frac{1}{\sqrt{\mu_0 \epsilon_0}} = c = 3 \times 10^8\,\text{m/s}$

  • Conclusion: Electromagnetic waves travel in free space at speed $c = 3 \times 10^8\,\text{m/s}$, establishing theoretical proof that light is an electromagnetic wave.

Transverse Nature of Electromagnetic Waves

• Plane Wave Solutions:

  • Plane harmonic wave solutions propagating in direction of wave vector $\mathbf{k}$: $\mathbf{E} = \mathbf{E}_0 \exp\left[i(\mathbf{k} \cdot \mathbf{r} - \omega t)\right]$ $\mathbf{H} = \mathbf{H}_0 \exp\left[i(\mathbf{k} \cdot \mathbf{r} - \omega t)\right]$

  • Where $\mathbf{E}_0$ and $\mathbf{H}_0$ are constant complex amplitude vectors, $\mathbf{k}$ is the wave vector, $\mathbf{r}$ is position vector, and $\omega$ is angular frequency.

• Mathematical Proof of Transversality:

  • Substituting wave solutions into free-space Gauss's divergence equations ($\nabla \cdot \mathbf{E} = 0$ and $\nabla \cdot \mathbf{H} = 0$): $\nabla \cdot \mathbf{E} = i \mathbf{k} \cdot \mathbf{E} = 0 \implies \mathbf{k} \cdot \mathbf{E} = 0$ $\nabla \cdot \mathbf{H} = i \mathbf{k} \cdot \mathbf{H} = 0 \implies \mathbf{k} \cdot \mathbf{H} = 0$

  • Significance: The zero dot products $\mathbf{k} \cdot \mathbf{E} = 0$ and $\mathbf{k} \cdot \mathbf{H} = 0$ mathematically prove that both electric field vector $\mathbf{E}$ and magnetic field vector $\mathbf{H}$ are perpendicular to propagation wave vector $\mathbf{k}$.

  • Therefore, electromagnetic waves in free space are strictly transverse waves.