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author committer Naoki Pross 2021-10-10 20:06:00 +0200 Naoki Pross 2021-10-10 20:06:00 +0200 82b40cff6f9aaf18a3c14b7cdf9cb83d8112517c (patch) 0fc75185f14bba6a883e9b8663a4f7ef8b758bac Partial derivatives (diff) ElMag-82b40cff6f9aaf18a3c14b7cdf9cb83d8112517c.tar.gzElMag-82b40cff6f9aaf18a3c14b7cdf9cb83d8112517c.zip
Solutions to simple geometries
-rw-r--r--ElMag.tex72
1 files changed, 63 insertions, 9 deletions
 diff --git a/ElMag.tex b/ElMag.texindex 8c21412..f628fea 100644--- a/ElMag.tex+++ b/ElMag.tex@@ -88,6 +88,8 @@ \section{Vector Analysis Recap} +\subsection{Partial derivatives}+ \begin{definition}[Partial derivative] A vector valued function $$f: \mathbb{R}^m\to\mathbb{R}$$, with $$\vec{v}\in\mathbb{R}^m$$, has a partial derivative with respect to $$v_i$$@@ -117,6 +119,8 @@ To illustrate the previous theorem, in a simpler case with $$f(x,y)$$, we get Beware that this is valid only if $$x$$ and $$y$$ are indipendent. If there is a relation $$x(y)$$ or $$y(x)$$ the above does not hold. +\subsection{Vector derivatives}+ \begin{definition}[Gradient vector] The \emph{gradient} of a function $$f(\vec{x}), \vec{x}\in\mathbb{R}^m$$ is a column vector containing the partial derivatives@@ -228,11 +232,13 @@ If there is a relation $$x(y)$$ or $$y(x)$$ the above does not hold. \begin{theorem}[Stokes' theorem] $- \int_\mathcal{S} \curl \vec{F} \dotp d\vec{s}- = \oint_{\partial\mathcal{S}} \vec{F} \dotp d\vec{r}+ \int_S \curl \vec{F} \dotp d\vec{s}+ = \oint_{\partial S} \vec{F} \dotp d\vec{r}$ \end{theorem} +\subsection{Second vector derivatives}+ \begin{definition}[Laplacian operator] A second vector derivative is so important that it has a special name. For a scalar function $$f: \mathbb{R}^m \to \mathbb{R}$$ the divergence of the@@ -330,15 +336,15 @@ Maxwell's equations in matter in their integral form are Where $$\vec{J}$$ and $$\rho$$ are the \emph{free current density} and \emph{free charge density} respectively. -\subsection{Isotropic linear materials and boundary conditions}+\subsection{Linear materials and boundary conditions} -Inside of so called isotropic linear materials the fields and flux (or current)-densities are proportional, i.e.+Inside of so called isotropic linear materials fluxes and current+densities are proportional and parallel to the fields, i.e. \begin{align*}- \vec{D} &= \varepsilon \vec{E}, & \vec{J} &= \sigma \vec{E}, & \vec{B} &= \mu \vec{H}.+ \vec{D} &= \epsilon \vec{E}, & \vec{J} &= \sigma \vec{E}, & \vec{B} &= \mu \vec{H}. \end{align*} -Between two materials (1) and (2) the following boundary conditions must be satisfied:+Where two materials meet the following boundary conditions must be satisfied: \begin{align*} &\uvec{n} \dotp \vec{D}_1 = \uvec{n} \dotp \vec{D}_2 + \rho_s & &\uvec{n} \crossp \vec{E}_1 = \uvec{n} \crossp \vec{E}_2 \\@@ -348,9 +354,57 @@ Between two materials (1) and (2) the following boundary conditions must be sati &\uvec{n} \crossp \vec{M}_1 = \uvec{n} \crossp \vec{M}_2 + \vec{J}_{s,m} \end{align*} -\subsection{Magnetic vector potential}+\subsection{Potentials}++Because $$\vec{E}$$ is often conservative ($$\curl \vec{E} = \vec{0}$$), and+$$\div \vec{B}$$ is always zero, it is often useful to use \emph{potentials} to+describe these quantities instead. The electric scalar potential and magnetic+vector potentials are in their integral form:+\begin{align*}+ \varphi &= \int_\mathsf{A}^\mathsf{B} \vec{E} \dotp d\vec{l}, &+ \vec{A} &= \frac{\mu_0}{4\pi} \int_V \frac{\vec{J} dv}{R}+\end{align*}+With differential operators:+\begin{align*}+ \vec{E} &= - \grad \varphi, &+ \mu_0 \vec{J} &= - \vlaplacian \vec{A}.+\end{align*}+By taking the divergence on both sides of the equation with the electric field+we get $$\rho/\epsilon = - \laplacian \varphi$$, which also contains the+Laplacian operator. We will study equations with of form in \S \ref{sec:poisson}.++% \subsection{Energy density}++\section{Laplace and Poisson's equations} \label{sec:poisson}++The so called \emph{Poisson's equation} has the form+$+ \laplacian \varphi = - \frac{\rho}{\epsilon}.+$+When the right side of the equation is zero, it is also known as \emph{Laplace's+equation}.++\subsection{Easy solutions of Laplace and Poisson's equations} -\section{Laplace and Poisson's equation}+\subsubsection{Geometry with zenithal and azimuthal symmetries (\"Ubung 2)} +Suppose we have a geometry where, using spherical coordinates, there is a+symmetry such that the solution does not depend on $$\phi$$ or $$\theta$$.+Then Laplace's equation reduces down to+$+ \laplacian \varphi = \frac{1}{r^2} \partial_r ( r^2 \partial_r \varphi) = 0,+$+which has solutions of the form+$+ \varphi(r) = \frac{C_1}{r} + C_2.+$++\subsection{Geometry with azimuthal and translational symmetry (\"Ubung 3)}++Suppose that when using cylindrical coordinates, the solution does not depend+on $$\phi$$ or $$z$$. Then Laplace's equation becomes+$+ \laplacian A_z = \frac{1}{r} \partial_r (r \partial_r A_z) = 0.+$ \end{document}