Principles of Electromagnetic Waves and Materials, 2nd Edition

6Laplace Equation: Static and Low-Frequency Approximations*

In an ideal transmission line, the voltage phasor $\tilde{V}$ satisfies the ordinary differential equation (2.54):

$$\frac{{\partial }^2\tilde{V}}{\partial z^2}+{\mathrm{\beta }}^2\tilde{V}=0,$$

where

$${\mathrm{\beta }}^2={\mathrm{\omega }}^2{L}^{\prime }{C}^{\prime }=\frac{{\mathrm{\omega }}^2}{v^2}.$$

The time-harmonic equation in three dimensions (Helmholtz equation) is given by

$${\nabla }^2\tilde{E}+k^2\tilde{E}=0,$$

where

$$k^2=\frac{{\mathrm{\omega }}^2}{v^2}={\mathrm{\omega }}^2\mathrm{\mu }\mathrm{\epsilon }$$

and ∇² is the Laplacian operator.

If the frequency is zero (f = 0) or low such that β² = k² ≈ 0, the Helmholtz equation can be approximated by the Laplace equation.

Static or low-frequency problems satisfy the Laplace equation

(6.1)$${\nabla }^2\mathrm{\Phi }=0,$$

where Φ is called the potential. In the first undergraduate course in electromagnetics, the one-dimensional solution of the Laplace equation in various coordinate systems is discussed and ...