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
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