Charge Distribution Analysis
Charge Distribution: Theoretical Foundations
Charge Distribution
Professor, how is the charge distribution on a conductor determined?
A conductor is an equipotential body. Charge distributes only on the surface, and the electric field inside is zero. The surface charge density $\sigma_s$ is:
$E_n$: Electric field normal to the surface. Areas with high curvature (sharp points) have higher charge density and electric field.
That's why charge concentrates at the tip of a lightning rod.
Exactly. The surface electric field of a spherical conductor with radius $r$ is $E = V/r$, so a smaller radius results in a stronger electric field.
Space Charge
Charge can also exist inside insulators or gases (space charge):
- Semiconductor doping → Fixed charge from donors/acceptors
- Gas discharge → Mobile charge from ions/electrons
- Trapped charge in insulators → Cause of degradation
Summary
- Charge concentration on conductor surface — High curvature → High charge density
- $\sigma_s = \varepsilon_0 E_n$ — Relationship between surface charge density and normal electric field
- Space charge — Semiconductors, discharge, insulation degradation
Free Charge and Bound Charge—Gauss's Law as a Technique to "See Inside"
Gauss's law ∇·D=ρf, the fundamental equation of electrostatic analysis, is a simple relational expression stating that "the total amount of electric flux D (electric displacement) passing through a closed surface is equal to the free charge ρf inside." Polarization charge (bound charge) inside a dielectric does not appear in the divergence of D; by describing it with electric flux D, the influence of the dielectric can be absorbed into the material constant (relative permittivity ε_r). Thanks to this formulation, computational uniformity is achieved, allowing electrostatic fields to be solved within the same FEM framework for both dielectrics and metals.
Computational Methods for Charge Distribution
Charge Calculation in FEM
After solving for electric potential $\phi$ with FEM, surface charge density on conductors is calculated in post-processing:
Total charge is calculated by surface integration using Gauss's law. BEM (Boundary Element Method) directly treats surface charge density as an unknown, making it particularly suitable for calculating charge distribution.
Summary
FEM for Point Effect—Predicting Corona Discharge Inception Voltage
At electrode edges (sharp points) in high-voltage equipment, electric field concentration is significant, leading to local corona discharge. To predict this corona discharge inception voltage with FEM, extremely fine meshes must be placed at the tip to obtain the maximum electric field strength (Emax) with high accuracy. BEM (Boundary Element Method) automatically satisfies boundary conditions at infinity, so in some cases, such as transmission line electrode design, more accurate electric field calculations can be obtained with BEM than with cylindrical FEM. Accurate calculation of the "electric field concentration factor Kt=Emax/E_mean" is the starting point for corona countermeasure design.
Charge Distribution in Practice
Practical Applications
Main applications: Electrostatic countermeasures (ESD), anti-static design, semiconductor doping distribution.