In-Depth Notes on Electromagnetic Theory: Plane Waves in Lossless Media

Section 9.1: Maxwell’s Equations in Differential Phasor Form

  • Introduction to Phasor Form of Maxwell's Equations

    • Focuses on differential form of Maxwell’s equations.
    • Phasor representation simplifies analysis by converting time derivatives to multiplication by $j\omega$ (where $\omega$ is angular frequency).

  • Maxwell’s Equations in Time-Domain:

    • Gauss's Law: ∇⋅D=ρv\nabla \cdot \mathbf{D} = \rho_v (Equation 9.1)
    • Maxwell-Faraday Equation (MFE): ∇×E=−∂B∂t\nabla \times \mathbf{E} = - \frac{\partial \mathbf{B}}{\partial t} (Equation 9.2)
    • Gauss's Law for Magnetism: ∇⋅B=0\nabla \cdot \mathbf{B} = 0 (Equation 9.3)
    • Ampere’s Law: ∇×H=J+∂D∂t\nabla \times \mathbf{H} = \mathbf{J} + \frac{\partial \mathbf{D}}{\partial t} (Equation 9.4)

  • Deriving Phasor Representations:

    • Define phasor quantities:
    1. D=Re(D ejωt)\mathbf{D} = \text{Re}(\mathbf{D\text{~}}e^{j\omega t}) (Equation 9.5)
    2. ρ<em>v=Re(ρ</em>v ejωt)\rho<em>v = \text{Re}(\rho</em>v\text{~}e^{j\omega t}) (Equation 9.6)
    • Substitute into Gauss's Law:
      ∇⋅(Re(Dejωt))=Re(ρvejωt)\nabla \cdot (\text{Re}(\mathbf{D} e^{j\omega t})) = \text{Re}(\rho_v e^{j\omega t}) (Equation 9.7)
    • Apply linearity of the divergence operator:
      Re(∇⋅(Dejωt))=Re(ρvejωt)\text{Re}(\nabla \cdot (\mathbf{D} e^{j\omega t})) = \text{Re}(\rho_v e^{j\omega t}) (Equation 9.8)
    • Conclude with:
      ∇⋅D =ρv \nabla \cdot \mathbf{D}^\text{~} = \rho_v^\text{~} (Equation 9.10)

  • Maxwell-Faraday Relation:

    • Similarly derive for MFE:
      ∇×E =−jωB \nabla \times \mathbf{E}^\text{~} = -j\omega\mathbf{B}^\text{~} (Equation 9.17)
    • Validity of phasor simplifications confirms that differentiation in time is replaced by multiplication by $j\omega$ - a simplification when solving wave equations.

  • Additional equations obtained similarly:

    • ∇⋅B =0\nabla \cdot \mathbf{B}^\text{~} = 0 (Equation 9.18)
    • ∇×H =J +jωD \nabla \times \mathbf{H}^\text{~} = \mathbf{J}^\text{~} + j\omega\mathbf{D}^\text{~} (Equation 9.19)

Section 9.2: Wave Equations for Source-Free and Lossless Regions

  • Wave Equation Derivation:

    • Start with phasor forms from Section 9.1. Transition to wave equations when $\rho_v = 0$ and $\mathbf{J} = 0$ leads to:
    • ∇⋅D =0\nabla \cdot \mathbf{D}^\text{~} = 0 (Equation 9.24)
    • ∇×E =−jωB \nabla \times \mathbf{E}^\text{~} = -j\omega\mathbf{B}^\text{~} (Equation 9.25)
    • ∇⋅B =0\nabla \cdot \mathbf{B}^\text{~} = 0 (Equation 9.26)
    • ∇×H =+jωD \nabla \times \mathbf{H}^\text{~} = +j\omega\mathbf{D}^\text{~} (Equation 9.27)

  • Relation in Lossless Media:

    • Define:
    • D =εE \mathbf{D}^\text{~} = \varepsilon \mathbf{E}^\text{~}
    • B =μH \mathbf{B}^\text{~} = \mu \mathbf{H}^\text{~}
    • Wave equations reduce to:
    • ∇⋅ extbfE =0\nabla \cdot \ extbf{E}^\text{~} = 0 (Equation 9.28)
    • ∇× extbfE =−jω extbfH \nabla \times \ extbf{E}^\text{~} = -j\omega\ extbf{H}^\text{~} (Equation 9.29)
    • ∇⋅ extbfH =0\nabla \cdot \ extbf{H}^\text{~} = 0 (Equation 9.30)
    • ∇× extbfH =+jωε extbfE \nabla \times \ extbf{H}^\text{~} = +j\omega\varepsilon \ extbf{E}^\text{~} (Equation 9.31)

  • Time Rate of Propagation: No conduction currents possible in source-free regions; therefore corresponding equations describe lossless scenarios.

Section 9.3: Types of Waves

  • Types of Wave Solutions:

    • Spherical Wave:
    • Phasefronts form concentric spheres; relevant for point sources, e.g. antennas.
    • Cylindrical Wave:
    • Phasefronts form concentric cylinders. Good model for line sources.
    • Plane Wave:
    • Phasefronts are parallel planes. Particularly useful when observed with the locally planar approximation.

  • Locally Planar Approximation:

    • Waves appear planar over small regions. E.g., waves from parabolic reflectors do this effectively.

Section 9.4: Uniform Plane Waves: Derivation

  • Definitions & Assumptions:

    • Uniform Plane Wave: Fields $\mathbf{E}$ and $\mathbf{H}$ have constant magnitude and phase in a given plane.
    • Derive for a scenario where these fields are consistent over a designated plane (like $z$ plane).

  • Expressions under Simplifications:

    • Apply curl and divergence to derive key equations in uniformity. Key equations retain formats essential for standardized engineering analysis.

Section 9.5: Uniform Plane Waves: Characteristics

  • Periodic Nature:

    • Wave characterized by wavelength $\lambda$ and phase velocity $v_p$.
    • λ=2πβ\lambda = \frac{2\pi}{\beta} (Equation 9.80)
    • vp=ωβv_p = \frac{\omega}{\beta} (Equation 9.81)
    • Phase velocity in free space ($c$) given by constant found in electromagnetic nature.

  • Wave Impedance:

    • Often $eta / \omega\mu$ results in quantifying impedance in free space (377 Ohm standard).
    • Characterized by:
    • η=με\eta = \sqrt{\frac{\mu}{\varepsilon}} (Equation 9.73)

  • Plane Wave Relationships:

    • H=1ηk^×E\mathbf{H} = \frac{1}{\eta} \hat{k} \times \mathbf{E} (Equation 9.89)
    • Fundamental expression to find $ extbf{H}$ given $ extbf{E}$, applicable across wave contexts.

Section 9.6: Wave Polarization

  • Definition of Polarization:

    • Orientation of electric field vector with respect to propagation direction.

  • Types of Polarization:

    • Linear Polarization: Electric field remains in fixed direction.
    • Circular Polarization: Electric vector rotates in a circle maintaining constant magnitude.
    • Elliptical Polarization: Electric vector describes an ellipse, commonly arising from mixed vectors.

Section 9.7: Wave Power in a Lossless Medium

  • Spatial Power Density:

    • Calculated from the Poynting vector, S=E×H\mathbf{S} = \mathbf{E} \times \mathbf{H} (Equation 9.106).
    • Average power density is derived to ensure practical applications for electromagnetic waves in analysis.

  • Time-Averaged Power Density Equation:

    • Sˉ<em>ave=∣E</em>0∣22η\bar{S}<em>{\text{ave}} = \frac{|E</em>0|^2}{2 \eta} (Equation 9.112)
    • Connects energy in the electric field to that in the magnetic field through Poynting's theorem which dictates conservation of energy.

  • Practical Applications of Poynting's Theorem:

    • Ensures electromagnetic field transfers maximize usable power while minimizing losses in design.