Boundary Condition on P and D | Lecture Note - Edubirdie

University:
Santa Fe College
Course:
ETS 2527C | Electromechanical Components and Mechanisms
Academic year:

2024
Views:

211

Pages:

6
Author:

Jabin F.

# Boundary Cond'n on [\vec{\nabla}\cdot\vec{\epsilon}=\frac{\rho}{\epsilon_{0}}=\frac{1}{\epsilon_{0}}(\rho_{f}+\rho_{b})] If free charge on the boundary (\langle\sigma_{f}\rangle=0) [\vec{E}{\text{above}}^{\perp}-\vec{E}{\text{below}}^{\perp}=\frac{\sigma}{\epsilon_{0}}=\frac{\sigma_{b}+\sigma_{f}}{\epsilon_{0}}] [\vec{\nabla}\cdot\vec{D}=\rho_{f}\longrightarrow D_{\text{above}}^{\perp}-D_{\text{below}}^{\perp}=\sigma_{f}] [\vec{\nabla}\cdot\vec{P}=-\rho_{b}\longrightarrow P_{\text{above}}^{\perp}-P_{\text{below}}^{\perp}=-\sigma_{b}] (Dielectric-boundary) (of ≠0) [\vec{E}=\frac{\vec{D}-\vec{P}}{\epsilon_{0}}] [\vec{n}\times\vec{E}=0] Boundary Condition [\begin{aligned} D_{\text{above}}^{\perp}-D_{\text{below}}^{\perp}&=\sigma_{f}=0 \ E_{\text{above}}^{\perp}-E_{\text{below}}^{\perp}&=0 \ E_{1}\epsilon_{1}-E_{2}\epsilon_{2}&=0 \ E_{1}\epsilon_{1}\cos\theta_{1}&=E_{2}\epsilon_{2}\cos\theta_{2} \ E_{1}\sin\theta_{1}&=E_{2}\sin\theta_{2} \ \frac{\tan\theta_{1}}{\tan\theta_{2}}&=\frac{\epsilon_{1}}{\epsilon_{2}} \end{aligned}]

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