Study Notes on Electromagnetic Waves - Chapter 9

Chapter 9: Electromagnetic Waves

9.1 Waves in One Dimension

9.1.1 The Wave Equation

  • Definition of a Wave: A wave is a disturbance of a continuous medium that propagates with a fixed shape at constant velocity.

    • Qualifiers:

    • In the presence of absorption, the wave diminishes in size as it moves.

    • In a dispersive medium, different frequencies travel at different speeds.

    • In two or three dimensions, the wave's amplitude decreases as it spreads out.

    • Standing waves do not propagate.

  • Mathematical Representation of Wave Motion:

    • A wave can be represented with the function $f(z, t)$, which represents the displacement of a medium at position $z$ and time $t$.

    • If the wave moves to the right at velocity $u$, then each point on the wave shifts:
      f(z,t)=f(z−ut,0)=g(z−ut)f(z, t) = f(z - ut, 0) = g(z - ut)

    • This demonstrates that the displacement at point $z$ at time $t$ is equal to the displacement at $z - ut$ at time $0$:
      f(z,t)=g(z−ut).f(z, t) = g(z - ut).

  • Examples of Waves: Different types of wave functions can represent various shapes of waves:

    1. f1(z,t)=Ae−b(z−vt)2f_1(z, t) = A e^{-b(z - vt)^2}

    2. f2(z,t)=Aextsin[b(z−vt)]f_2(z, t) = A ext{sin}[b(z - vt)]

    3. f3(z,t)=Aextsech2(z−vt)f_3(z, t) = A ext{sech}^2(z - vt)

  • Wave Equation:

    • A stretched string under tension $T$ supports wave motion due to forces acting on the string segments:
      Net Transverse Force:
      extNetForce=Tracd2ydz2extforasegmentbetweenzextandz+extdz.ext{Net Force} = T rac{d^2 y}{dz^2} ext{ for a segment between } z ext{ and } z + ext{d}z.

    • Using Newton's second law, the equation of motion for small disturbances on the string leads to:
      racextd2fextdt2=racT<br>horacextd2fextdz2rac{ ext{d}^2 f}{ ext{d}t^2} = rac{T}{<br>ho} rac{ ext{d}^2 f}{ ext{d}z^2}

  • This equation is the classical wave equation, which implies that all functions of the form
    f(z,t)=g(z−vt)f(z, t) = g(z - vt) where $g$ is any differentiable function will satisfy it and represent waves propagating in the $z$ direction at speed $v$.

  • General Solution: The wave equation allows for the general solution:
    f(z,t)=g(z−vt)+h(z+vt)f(z, t) = g(z - vt) + h(z + vt)

    • This solution expresses a wave traveling to the right and another to the left.

  • Significance of the Wave Equation:

    • It is crucial in various physical contexts where oscillation or wave phenomena are present, such as acoustics, optics, and oceanography.

  • Problems:

    • Problem 9.1: Check that $f_1$, $f_2$, and $f_3$ satisfy the wave equation.

    • Problem 9.3: Show that a standing wave satisfies the wave equation and express it as the sum of waves traveling in opposite directions.

9.1.2 Sinusoidal Waves

  • Terminology:

    • The sinusoidal wave form is characterized by:
      f(z,t)=Aextcos[k(z−vt)+heta].f(z, t) = A ext{cos}[k(z - vt) + heta].

    • Components:

    • Amplitude ($A$): The peak value of the wave displacement.

    • Wave Number ($k$): Defined as k=rac2extπextλk = rac{2 ext{π}}{ ext{λ}} where $ ext{λ}$ is the wavelength.

    • Phase Constant ($ heta$): Determines the phase shift in the wave.

    • As the wave propagates, its central maximum moves to the right at speed $v$. Each point along the wave oscillates in time, completing a cycle in a period $T = rac{2 ext{π}}{ω}$, where $
      u$ is the frequency.

  • Angular Frequency:

    • Related to traditional frequency as:
      ω=2π<br>u=kv.ω = 2π<br>u = k v.

    • The sinusoidal wave can be rewritten in terms of angular frequency:
      f(z,t)=Aextcos[kz−ωt+heta].f(z, t) = A ext{cos}[kz - ωt + heta].

  • Complex Notation: Using Euler's formula:
    Aei(kz−ωt+heta)=Aextcos[kz−ωt+heta]+iAextsin[kz−ωt+heta].A e^{i(kz - ωt + heta)} = A ext{cos}[kz - ωt + heta] + iA ext{sin}[kz - ωt + heta].

    • The complex wave function simplifies calculations. The actual wave is the real part of the complex amplitude:
      f(z,t)=extRe[Aei(kz−ωt)].f(z, t) = ext{Re}[A e^{i(kz - ωt)}].

  • Linear Combinations: Any waveform can be expressed as a linear combination of sinusoidal waves.
    f(z,t)=rac12extπimesextintegralimesf(k)ei(kz−wt)dk.f(z, t) = rac{1}{2 ext{π}} imes ext{integral} imes f(k)e^{i(kz - wt)} dk.

  • Problems:

    • Problem 9.3: Use the complex representation to combine waves.

    • Problem 9.4: Derive sinusoidal solutions directly from the wave equation.

9.1.3 Boundary Conditions: Reflection and Transmission

  • Boundary Conditions Overview: The behavior of waves at boundaries is influenced by how the medium is terminated.

  • Incident Wave: A wave moving towards a boundary creates reflected and transmitted waves governed by the characteristics of the two materials (mediums).

    • Formulate the equations for an incident wave hitting a boundary (string and wave properties):

    • Ei(z,t)=A1eik1ze−iracextωc1tE_i(z, t) = A_1 e^{ik_1z} e^{-i rac{ ext{ω}}{c_1}t} for $z < 0$.

    • Reflected wave:

    • Er(z,t)=ARe−ik1ze−iracextωc1tE_r(z, t) = A_R e^{-ik_1z} e^{-i rac{ ext{ω}}{c_1}t} for $z < 0$.

    • Transmitted wave: Et(z,t)=ATeik2ze−iracωc2tE_t(z, t) = A_T e^{ik_2z} e^{-i rac{ω}{c_2}t} for $z > 0$.

  • The reflection and transmission coefficients give the proportion of energy reflected and transmitted when the wave meets a boundary.

  • Boundary Conditions:

  1. Displacement continuity:
    E(0−,t)=E(0+,t).E(0^-, t) = E(0^+, t).

  2. Slope continuity: ($ rac{d E(0^-, t)}{dz} = rac{d E(0^+, t)}{dz} $)

  • Reflection and Transmission Coefficients:

    • In terms of the amplitudes:
      R=racARAIextandT=racATAI.R = rac{A_R}{A_I} ext{ and } T = rac{A_T}{A_I}.

  • Problems:

    • Problem 9.5: Analyze incident wave reflection & transmission through boundaries.

    • Problem 9.6: Formulate appropriate boundary conditions with mass in the knot.

9.1.4 Polarization

  • Types of Waves:

    • Transverse waves cause displacement perpendicular to the direction of travel.

    • Longitudinal waves cause displacement parallel to the direction of travel.

  • In transverse waves, polarization states can be classified into two dimensions perpendicular to propagation direction:

    1. Vertical Polarization:
      E(z,t)=Aei(kz−wt)extbfy^E(z, t) = A e^{i(kz - wt)} extbf{ŷ} where displacement is in the y-direction.

    2. Horizontal Polarization:
      E(z,t)=Aei(kz−wt)extbfx˙E(z, t) = A e^{i(kz - wt)} extbf{ẋ} where displacement is in the x-direction.

    3. General Polarization:
      E(z,t)=Aei(kz−wt)extbfnE(z, t) = A e^{i(kz - wt)} extbf{n} where n is the polarization vector.

  • Linear Polarization: is a superposition of two orthogonal components. The phase difference of 90° induces circular motion of the string; this is termed circular polarization.

  • Problems:

    • Problem 9.8: Analyze the general linearly polarized wave. Find the conditions for circularly polarized waves.

9.2 Electromagnetic Waves in Vacuum

9.2.1 The Wave Equation for E and B

  • Maxwell's Equations in Free Space: In regions void of free charge or current, Maxwell's equations express relationships between electric field $ extbf{E}$ and magnetic field $ extbf{B}$:

  1. <br>abla∙extbfE=0.<br>abla \bullet extbf{E} = 0.

  2. <br>abla∙extbfB=0.<br>abla \bullet extbf{B} = 0.

  3. <br>ablaimesextbfE=−racextdBextdt.<br>abla imes extbf{E} = - rac{ ext{d}B}{ ext{d}t}.

  4. <br>ablaimesextbfB=μ0extε0racextdextbfEextdt.<br>abla imes extbf{B} = μ_0 ext{ε}_0 rac{ ext{d} extbf{E}}{ ext{d}t}.

  • Decoupling Electromagnetic Equations: Both equations can decouple into the wave equations for $ extbf{E}$ and $ extbf{B}$ fields:
    <br>abla2extbfE=extμ0extε0racextd2extbfEextdt2<br>abla^2 extbf{E} = ext{μ}_0 ext{ε}_0 rac{ ext{d}^2 extbf{E}}{ ext{d}t^2} and
    <br>abla2extbfB=extμ0extε0racextd2extbfBextdt2.<br>abla^2 extbf{B} = ext{μ}_0 ext{ε}_0 rac{ ext{d}^2 extbf{B}}{ ext{d}t^2}.

  • Speed of Light: This shows electromagnetic waves travel at the speed of light given by
    c=rac1ext√μ0extε0=3.00imes108m/s.c = rac{1}{ ext{√μ}_0 ext{ε}_0} = 3.00 imes 10^8 m/s.

  • Electric and Magnetic Fields:

    • The components of $ extbf{E}$ and $ extbf{B}$ must satisfy the condition of being perpendicular to the propagation direction
      (transverse nature of electromagnetic waves).

  • Problems:

    • Problem 9.9: Determine the electromagnetic fields for polarizations in different directions.

9.2.2 Monochromatic Plane Waves

  • Monochromatic Waves: Examined in regards to specific frequency $w$, propagation in the z direction, independent of x or y.

  • Uniform Electric and Magnetic Fields: The form is given as  extbfE(z,t)=E0ei(kz−wt)\ extbf{E}(z, t) = E_0 e^{i(kz - wt)} while for magnetic field extbfB(z,t)=B0ei(kz−wt)extbf{B}(z,t) = B_0 e^{i(kz - wt)}

    • upon separating dimensions, shown as accompanying electric and magnetic amplitudes.

  • Relation of Fields: Must maintain:
    racB0E0=racE0crac{B_0}{E_0} = rac{E_0}{c} where they remain in-phase.

  • Problems:

    • Problem 9.10: Write and sketch electromagnetic equations expressed in terms of complex numbers.

9.2.3 Energy and Momentum in Electromagnetic Waves

  • Energy Density (u): The expression for energy density stored in electromagnetic fields:
    u=rac12extε0E2+rac12racB2extμ0.u = rac{1}{2} ext{ε}_0 E^2 + rac{1}{2} rac{B^2}{ ext{μ}_0}.

  • Average Energy Density: For average values we use $ar{E} = ext{E}_0 ext{cos}^2$.

  • Poynting Vector: Defined as the vector representing the flow of energy per unit area carried by the wave given by
    extbfS=rac1extμ0(extbfEimesextbfB).extbf{S} = rac{1}{ ext{μ}_0} ( extbf{E} imes extbf{B}).

  • Wave Momentum Density: In electromagnetic waves, momentum density:
    p=rac1cE2extsin2(wt)extexp(−2z/λ).p = rac{1}{c} E^2 ext{sin}^2 (w t) ext{exp( -2z/λ)}.

  • Reflection Pressure: The intensity and resulting force are inferred from energy density:
    F/A=rac1cextIntensityF/A = rac{1}{c} ext{Intensity} where T denotes perfect absorbers.

  • Problems:

    • Problem 9.11: Consider momentum changes when light reflects on absorbing surface versus perfect reflectors.

9.3 Electromagnetic Waves in Matter
9.3.1 Propagation in Linear Media

  • variations and behavior under external electric fields.

  • Problems:

    • Problem 9.19: Investigate the implications of increasing frequency on skin depth and electromagnetic wave transparency mechanisms.

9.5 Guided Waves

  • Waveguides Overview: Waves confined within a structure can form modes depending on geometry. Maxwell's equations dictate existence and propagation characteristics within transmissioMaxwell's Equations Adjustment for Media: Under linear, homogenous circumstances, Maxwell's equations change to capture free charge effects.
    extD=extε0E+P.ext{D} = ext{ε}_0 E + P.

  • Wave Equation in a Medium: Solutions consist of waves that integrate these dielectric principles and modify speed and amplitude based on medium constants.

  • Problems:

    • Problem 9.12: Derive relationships between propagation constants inside waveguides.

9.4 Absorption and Dispersion

  • Absorption in Conductors: Electric field associations operate under conditions where conduction loss manifests.

    • Maxwell's equations simplified yield:
      <br>abla2extE−extεextω2extE=iextωμJ<br>abla^2 ext{E} - ext{ε} ext{ω}^2 ext{E} = i ext{ωμJ} where attenuation for conducting mediums can be modeled.

    • The absorption coefficient reflects this, defining the skin depth and extent of penetration without energy loss.

  • Complex Susceptibility: Through harmonic systems, we analyze dispersion properties governed by frequency n media.

    • Boundary Conditions Influence: Must govern electric potential at waveguide surfaces and facilitate nonzero magnetic conditions leading to propagative behavior.

  • TE and TM Modes: Analysis yields distinct wave characteristics based on dimensional properties and resultant wave numbers applied with conditions at boundaries—a unique functional form arises for known geometries.

  • Coaxial Transmission Lines: Here a central conductive core allows TEM wave modes, affecting charge density and magnetostatics.

  • Problems:

    • Problem 9.30: Delve into how different configurations affect modes and frequencies within a coaxial setup and reflection modulus from exterior surfaces.