4.2 Electrical displacement and electric field equations

Electric fields and displacement are crucial concepts in piezoelectric energy harvesting. They describe how charges interact and move within materials, forming the basis for understanding piezoelectric effects.

These concepts connect to the broader study of piezoelectric constitutive equations by explaining the fundamental relationships between electric fields, charge distribution, and material properties. This knowledge is essential for modeling and optimizing piezoelectric devices.

Electric Fields and Displacement

Fundamental Concepts of Electric Fields

• Electric field intensity represents the force exerted on a unit positive charge in an electric field
• Electric field intensity measured in units of newtons per coulomb (N/C) or volts per meter (V/m)
• Electric displacement field describes the distribution of electric charge in a dielectric material
• Electric displacement field measured in units of coulombs per square meter (C/m²)
• Dielectric permittivity characterizes a material's ability to store electrical energy in an electric field
• Dielectric permittivity consists of two components
  • Vacuum permittivity (permittivity of free space)
  • Relative permittivity (dielectric constant)
• Dielectric constant represents the ratio of a material's permittivity to the permittivity of free space
• Dielectric constant influences the strength of electric fields within materials (glass, ceramics)

Relationships Between Electric Field Components

• Electric field intensity (E) relates to electric displacement (D) through the equation $D = εE$
• ε represents the dielectric permittivity of the material
• Dielectric permittivity (ε) calculated as the product of vacuum permittivity (ε₀) and relative permittivity (εᵣ)
• Equation for dielectric permittivity: $ε = ε₀εᵣ$
• Vacuum permittivity (ε₀) approximately equals 8.85 × 10⁻¹² F/m
• Relative permittivity (εᵣ) varies depending on the material (air ≈ 1, water ≈ 80)
• Electric field intensity inversely proportional to dielectric constant in a given medium
• Higher dielectric constant materials (water) reduce electric field strength compared to lower dielectric constant materials (air)

Polarization and Maxwell's Equations

Polarization in Dielectric Materials

• Polarization describes the separation of positive and negative charges in a material under an applied electric field
• Polarization vector (P) represents the dipole moment per unit volume in a material
• Relationship between polarization and electric displacement: $D = ε₀E + P$
• Types of polarization mechanisms in materials
  • Electronic polarization (displacement of electron cloud relative to nucleus)
  • Ionic polarization (displacement of ions in ionic crystals)
  • Orientational polarization (alignment of permanent dipoles)
• Polarization contributes to the overall electric displacement field in a material
• Polarization affects the dielectric constant and permittivity of materials

Maxwell's Equations and Electromagnetic Theory

• Maxwell's equations form the foundation of classical electromagnetism
• Four fundamental equations describe the behavior of electric and magnetic fields
  • Gauss's law for electricity
  • Gauss's law for magnetism
  • Faraday's law of induction
  • Ampère's law with Maxwell's correction
• Gauss's law relates electric field to charge distribution
• Gauss's law states the electric flux through a closed surface proportional to the enclosed charge
• Mathematical form of Gauss's law: $∮ E · dA = Q / ε₀$
• E represents the electric field vector
• dA represents the differential area vector
• Q represents the total enclosed charge
• ε₀ represents the permittivity of free space
• Gauss's law applies to both point charges and continuous charge distributions
• Applications of Gauss's law include calculating electric fields for symmetric charge distributions (spherical, cylindrical)