Laplace Equation (Electrostatic Field)
Laplace Equation (Electrostatic Field): Theoretical Foundations
Laplace Equation
Professor, how is the Laplace equation different from the Poisson equation?
In regions without charge ($\rho_v = 0$), the Poisson equation becomes the Laplace equation:
The electric potential distribution in spaces without free charge, such as inside insulators between electrodes or inside dielectrics, is determined by this.
So, high-voltage equipment insulation design mostly uses the Laplace equation?
Yes. The potential distribution is determined solely by the electrode potentials (Dirichlet BC). Calculating the electric field distribution in insulators and equipotential lines falls under the domain of the Laplace equation.
Maximum Principle
An important property of the Laplace equation: The maximum and minimum values of the potential always lie on the boundary (not inside the domain).
So, if there are no extreme values inside, it would be strange if an FEM result showed a potential peak inside the domain, right?
Correct. This can be used for validating result plausibility. If there is a peak inside, it's likely a mesh issue or a mistake in setting up the charge source.
Summary
- $\nabla^2 \phi = 0$ — Charge-free region
- Maximum Principle — Extreme values exist only on the boundary
- Fundamental equation for insulation design — Electric field distribution determined solely by electrode potentials
"No maximum/minimum values in Laplace equation solutions" — Impact on practical work
The "maximum principle" possessed by solutions of the Laplace equation is a subtly important property in CAE practice. In spaces where no charge exists (regions where $\nabla^2\phi = 0$ holds), the maximum and minimum values of the potential always appear on the boundary. This means a "local maximum point of potential" cannot suddenly appear inside the domain. This is useful for insulation design, as it confirms that "the location of strongest electric field = on the boundary (electrode or dielectric interface)". Therefore, you don't need to check the entire interior in detail; just increase accuracy near the boundary. In high-voltage equipment design, techniques leveraging this principle to concentrate mesh refinement in specific areas are commonly used in the field.
Computational Methods for Laplace Equation (Electrostatic Field)
FEM Solution
The FEM for the Laplace equation is the Poisson equation with its right-hand side set to zero. $[K]\{\phi\} = \{0\}$ (no charge) + boundary conditions.
The solution method is exactly the same. The difference is that the right-hand side vector contains only terms originating from the Dirichlet conditions of the electrodes.
Comparison with Analytical Solutions
| Problem | Analytical Solution | Application |
|---|---|---|
| Parallel plates | $\phi = V_0 (1 - x/d)$ | Simplest verification |
| Coaxial cylinders | $\phi = V_0 \ln(r/b)/\ln(a/b)$ | Cable verification |
| Concentric spheres | $\phi = V_0 (1/r - 1/b)/(1/a - 1/b)$ | 3D spherical symmetry verification |
Summary
- Solution method identical to Poisson equation — Right-hand side zero
- Analytical solutions effective for FEM verification — Parallel plates, coaxial cylinders
"Solving Laplace equation in fluids too" — Universality of harmonic functions
The Laplace equation $\nabla^2\phi = 0$ governs not only electrostatic potential but also a surprisingly wide variety of physical phenomena: velocity potential in incompressible potential flow, temperature distribution in steady-state heat conduction, pressure field in porous media, etc. This means "even with different boundary conditions, it can be solved with the same solver code." In practice, people who have written electrostatic field analysis code sometimes experience that "replacing permittivity with thermal conductivity turned it into a thermal analysis." This mathematical commonality is called "analogy," and in the past, experiments were conducted to measure fluid pressure fields using electrical analog circuits. The Laplace equation is a "common language" of nature.