diff options
Diffstat (limited to 'ElMag.tex')
| -rw-r--r-- | ElMag.tex | 44 |
1 files changed, 42 insertions, 2 deletions
@@ -98,7 +98,7 @@ \partial_{v_i} f(\vec{v}) % = f_{v_i}(\vec{v}) = \frac{\partial f}{\partial v_i} - = \lim_{h\to 0} \frac{f(\vec{v} + h\vec{e}_i) - f(\vec{v})}{h} + = \lim_{h\to 0} \frac{f(\vec{v} + h\uvec{e}_i) - f(\vec{v})}{h} \] \end{definition} @@ -126,7 +126,7 @@ If there is a relation \(x(y)\) or \(y(x)\) the above does not hold. column vector containing the partial derivatives in each direction. \[ - \grad f (\vec{x}) = \sum_{i=1}^m \partial_{x_i} f(\vec{x}) \vec{e}_i + \grad f (\vec{x}) = \sum_{i=1}^m \partial_{x_i} f(\vec{x}) \uvec{e}_i = \begin{pmatrix} \partial_{x_1} f(\vec{x}) \\ \vdots \\ @@ -375,6 +375,46 @@ Laplacian operator. We will study equations with of form in \S \ref{sec:poisson} % \subsection{Energy density} +\section{Boundary value problems} + +\subsection{Steady-state flow analysis} +The equatation for the steady-state analysis is +\begin{equation} + \laplacian \varphi = 0 \quad\text{for}\quad \vec{r} \in \Omega, +\end{equation} +with its boundary conditions: +\( + \varphi = 0 \text{ for } \vec{r}\in\Gamma_e, + \varphi = U \text{ for } \vec{r}\in\Gamma_b, + \nabla_\uvec{n} \varphi = 0 \text{ for } \vec{r}\in\Gamma_s. +\) + +\subsection{Magnetostatic analysis} +The equation for the magnetostatic analysis is +\begin{equation} + \vlaplacian \vec{A} = -\mu_0 \vec{J} \quad\text{for}\quad \vec{r} \in \Omega. +\end{equation} + +\subsection{Magnetoquasistatic analysis} +The equation for the magnetoquasistatic analysis is +\begin{equation} + \vlaplacian \vec{A} - \mu_0 \sigma \partial_t \vec{A} + = -\mu_0 \vec{J}_q \quad\text{for}\quad \vec{r} \in \Omega. +\end{equation} + +\subsection{Electrodynamic analysis} +The equationsfor the electrodynamic analysis are +\begin{subequations} + \begin{gather} + \vlaplacian \vec{E} + - \mu \sigma \partial_t \vec{E} + - \mu \epsilon \partial^2_t \vec{E} = \vec{0}, \\ + \vlaplacian \vec{H} + - \mu \sigma \partial_t \vec{H} + - \mu \epsilon \partial^2_t \vec{H} = \vec{0}. + \end{gather} +\end{subequations} + \section{Laplace and Poisson's equations} \label{sec:poisson} The so called \emph{Poisson's equation} has the form |