电磁场与电磁波-2-泊松方程、拉普拉斯方程的推导

Derivation of poisson equation and Laplace equation

List of used equations

D = ε E (1) \tag{1}D=\varepsilon E D=εE(1)
E = − ∇ φ (2) \tag{2}E=-\nabla\varphi E=−∇φ(2)
∇ ⋅ D = ρ (3) \tag{3}\nabla \cdot D=\rho ∇⋅D=ρ(3)
∇ ⋅ ∇ = ∇ 2 (4) \tag{4}\nabla \cdot \nabla=\nabla^2 ∇⋅∇=∇2(4)

Poisson equation

  • put the third equation into the first ∇ ⋅ ε E = ρ \nabla \cdot \varepsilon E=\rho ∇⋅εE=ρ
  • put the second equation into previous equation − ∇ ⋅ ( ε ∇ φ ) = ρ -\nabla \cdot (\varepsilon \nabla \varphi)=\rho −∇⋅(ε∇φ)=ρ
  • put the fourth equation into the previous equation − ε ∇ 2 φ = ρ -\varepsilon \nabla^2 \varphi=\rho −ε∇2φ=ρ
  • simplify it, hence we get poisson equation ∇ 2 φ = − ρ ε \nabla^2\varphi=-\dfrac{\rho}{\varepsilon} ∇2φ=−ερ​

Laplace equation

  • while ρ = 0 \rho=0 ρ=0 ∇ 2 φ = 0 \nabla^2\varphi=0 ∇2φ=0
  • this’s Laplace equation
posted @ 2022-03-23 17:27  tariya  阅读(71)  评论(0)    收藏  举报  来源