From e999dee2b993e91b24b003574f8ee1e7e4083cf3 Mon Sep 17 00:00:00 2001 From: trenta3 Date: Wed, 2 Nov 2016 15:17:11 +0100 Subject: [PATCH] Aggiunto Formulario di Fisica di Giada e Glaudo --- .gitignore | 2 + Fisica/Formulari/CommandsAndStyle.sty | 205 ++++++ Fisica/Formulari/FormularioElettrostatica.pdf | Bin 0 -> 201282 bytes Fisica/Formulari/FormularioElettrostatica.tex | 597 ++++++++++++++++++ 4 files changed, 804 insertions(+) create mode 100644 Fisica/Formulari/CommandsAndStyle.sty create mode 100644 Fisica/Formulari/FormularioElettrostatica.pdf create mode 100644 Fisica/Formulari/FormularioElettrostatica.tex diff --git a/.gitignore b/.gitignore index 124f581..8543381 100644 --- a/.gitignore +++ b/.gitignore @@ -6,3 +6,5 @@ *.log *.out *.pdf +*.fdb_latexmk +*.fls \ No newline at end of file diff --git a/Fisica/Formulari/CommandsAndStyle.sty b/Fisica/Formulari/CommandsAndStyle.sty new file mode 100644 index 0000000..e78c53e --- /dev/null +++ b/Fisica/Formulari/CommandsAndStyle.sty @@ -0,0 +1,205 @@ +\ProvidesPackage{CommandsAndStyle} + +\pagenumbering{gobble} +\usepackage[utf8]{inputenc} +\usepackage[italian]{babel} +\usepackage[a4paper,twoside,top=0.5in, bottom=0.5in, left=0.5in, right=0.5in]{geometry} +\usepackage{graphicx} +\usepackage{titlesec} +\usepackage{multicol} +\usepackage{hyperref} +\hypersetup{ + colorlinks = true, + linkcolor = {blue}, + urlcolor = {red} +} + +\hypersetup{ + pageanchor=false +} +\usepackage{amsmath} +\usepackage{amsthm} +\usepackage{amssymb} +\usepackage{esint} % For the average integral +\usepackage{color} +\usepackage{array} +\usepackage{hhline} +\usepackage{enumitem} +\usepackage{mathrsfs} %Just for cool sigma-algebra +\usepackage{imakeidx} +\usepackage{xparse} %In order to use ExplSyntax +\usepackage{mathtools} +\usepackage[nameinlink]{cleveref} +\newcommand{\crefpairconjunction}{ e } +\newcommand{\crefrangepreconjunction}{da } +\newcommand{\crefrangeconjunction}{ a } +\newcommand{\creflastconjunction}{ e } +\let\oldvec\vec +\renewcommand{\vec}[1]{\oldvec{#1}\mkern 2mu\vphantom{#1}} + +\let\oldhat\hat +\renewcommand{\hat}[1]{\oldhat{#1}\mkern 2mu\vphantom{#1}} +% \newcommand{\creflastgroupconjunction}{ e } +% \newcommand{\crefrangepostconjunction}{ e } +% \newcommand{\crefmiddleconjunction}{ e } +% \newcommand{\crefpairgroupconjunction}{ e } +\usepackage{fancyhdr} %Header and footer of pages +% \setlength{\headheight}{16pt} +% \pagestyle{fancy} +% \fancyhead{} +% \fancyhead[LE,RO]{\slshape\nouppercase{\leftmark}} + + +\newcounter{Results}[section] +\renewcommand{\theResults}{\thesection.\arabic{Results}} + +\newtheorem{theorem}[Results]{Teorema} +\newtheorem{lemma}[Results]{Lemma} +\newtheorem{proposition}[Results]{Proposizione} +\newtheorem{corollary}[Results]{Corollario} +\newtheorem{exercise}[Results]{Esercizio} + +\theoremstyle{remark} +\newtheorem{remark}[Results]{Nota} + +\theoremstyle{definition} +\newtheorem{definition}[Results]{Definizione} +\newtheorem*{example}{Esempio} + +\crefname{equation}{Equazione}{Equazioni} +\crefname{theorem}{Teorema}{Teoremi} +\crefname{lemma}{Lemma}{Lemmi} +\crefname{proposition}{Proposizione}{Proposizioni} +\crefname{corollary}{Corollario}{Corollari} +\crefname{exercise}{Esercizio}{Esercizi} +\crefname{remark}{Nota}{Note} +\crefname{definition}{Definizione}{Definizioni} +\crefname{section}{Sezione}{Sezioni} + +\newcommand{\ImplicationProof}[2]{$\text{\ref{#1}}\implies\text{\ref{#2}}$}% ref speciale per implicazioni X => Y + +%Qui di seguito ridefiniamo il comando cref per piazzare pure i nomi dei teoremi quando presenti. +\makeatletter +\ExplSyntaxOn +\usepackage{ifthen} +\usepackage{xstring} +\let\@tempcref\cref +\newcommand{\myref}[1]{ + \IfSubStr{#1}{,}{ + \@tempcref{#1}% + } + { + \ifcsname r@#1\endcsname%La reference è definita? + \edef\@RefInfo{\csname r@#1\endcsname}%Contiene tutte le informazioni sulla reference + \edef\@CompleteCounter{\expandafter\@fourthoffive\@RefInfo}%Contiene il nome espanso del contatore + + \expandafter\IfBeginWith{\@CompleteCounter}{equation}{%La reference è un'equazione? + \@tempcref{#1}% + } + { + \edef\@RefName{\expandafter\@thirdoffive\@RefInfo}%Contiene, se definito, il nome della reference + \StrLen{\@RefName}[\@RefNameLen] + + \ifthenelse{\@RefNameLen<2}{%Il nome è vuoto o di un solo carattere? (tocca lasciare la possibilità che sia di un solo carattere sennò al carducci non va) + \@tempcref{#1}% + } + { + \@tempcref{#1}~(\nameref*{#1})% + } + } +% \begin{itemize} +% \item \expandafter\@firstoffive\@mytxt +% \item \expandafter\@secondoffive\@mytxt +% \item \expandafter\@thirdoffive\@mytxt +% \item \expandafter\@fourthoffive\@mytxt +% \item \expandafter\@fifthoffive\@mytxt +% \end{itemize} + \else + \@tempcref{#1} + \fi + } +} +% \renewcommand{\cref}{\myref} +\ExplSyntaxOff +\makeatother + +\numberwithin{equation}{section} + +\newcommand{\sigalg}[1][a]{$\sigma$-algebr#1} +\DeclareRobustCommand{\semiring}[1][o]{semianell#1} +\DeclareRobustCommand{\Semiring}[1][o]{Semianell#1} +\DeclareRobustCommand{\sigadd}[1][a]{$\sigma$-additiv#1} +\DeclareRobustCommand{\sigfin}[1][a]{$\sigma$-finit#1} +\newcommand{\sigsubadd}[1][a]{$\sigma$-subadditiv#1} +\DeclareRobustCommand{\carat}{Carathéodory} +\newcommand{\Rpiu}{\ensuremath{[0,+\infty]}} +\newcommand{\M}{\ensuremath{\mathcal M}}%Misurabili +\renewcommand{\L}{\ensuremath{\mathcal L^1}}%Integrabili +\newcommand{\A}{\ensuremath{\mathscr A}}%Sigma-algebra +\newcommand{\B}{\ensuremath{\mathscr B}}%Altro nome da sigma-algebra +\newcommand{\N}{\ensuremath{\mathbb N}}%Naturali +\newcommand{\Z}{\ensuremath{\mathbb Z}}%Interi +\newcommand{\Q}{\ensuremath{\mathbb Q}}%Razionali +\newcommand{\R}{\ensuremath{\mathbb R}}%Reali +\newcommand{\Rbar}{\overline{\R}}%Reali U +- infinito +\renewcommand{\S}{\ensuremath{\mathcal S}}%Semianello +\newcommand{\simp}{\ensuremath{\varphi}}%Funzione semplice +\newcommand{\Borel}{\ensuremath{\mathcal B}}%Boreliani + + +\newcommand{\de}{\ensuremath{\,\mathrm d}}%Il de degli integrali +\newcommand{\Diff}{\ensuremath{\mathrm D}}%Simbolo del differenziale +\newcommand{\LNorm}[1]{\ensuremath{\Vert #1\Vert_1}}%Norma L^1 +\newcommand{\aint}{\ensuremath{\fint}}%Integrale mediato +% \newcommand{\div}{\mathop{\mathrm{div}}\nolimits} +\let\divsymb\div +\let\div\undefined +% \DeclareMathOperator{\div}{div} +\makeatletter +\newcommand*\dotp{\mathpalette\dotp@{.4}} +\newcommand*\dotp@[2]{\mathbin{\vcenter{\hbox{\scalebox{#2}{$\m@th#1\bullet$}}}}} +\makeatother +\newcommand{\div}{\ensuremath{\vec{\nabla} \dotp}} +\DeclareMathOperator{\tr}{tr} +% \DeclareMathOperator{\grad}{\nabla} +\newcommand{\grad}{\ensuremath{\nabla}} +\DeclareMathOperator{\graf}{graf} +\DeclareMathOperator{\supp}{supp} + +\newcommand{\bigO}{\ensuremath{\mathcal{O}}} +\newcommand{\smallO}{\ensuremath{\mathrm{o}}} + +%Definiamo gli intervalli semiaperti, aperti, chiusi. +\newcommand{\co}[2]{\ensuremath{\left[\,#1,\,#2\,\right)}} +\newcommand{\oo}[2]{\ensuremath{\left(\,#1,\,#2\,\right)}} +\newcommand{\cc}[2]{\ensuremath{\left[\,#1,\,#2\,\right]}} +\newcommand{\oc}[2]{\ensuremath{\left(\,#1,\,#2\,\right]}} + +% Definiamo le parentesi left( e right) per metterci dentro cose +\newcommand{\lrb}[1]{\ensuremath{\left({#1}\right)}} + +%Punteggiatura nelle formule +\ifdefined\virgola + 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+\setlength{\columnsep}{1cm} +\usepackage{CommandsAndStyle} + +\author{Idea originaria di Federico Glaudo e Giada Franz \\ Piccole aggiunte di Dario Balboni} +\title{Formulario di Fisica II} + +\begin{document} +\maketitle + +\begin{multicols}{2} +\section{Costanti fisiche} +\formula{Costante dielettrica nel vuoto}{ + \begin{equation*} + \ez=8.854\cdot 10^{-12}\frac Fm + \end{equation*} + \begin{equation*} + k=\frac 1{4\pi\ez}=8.987 \cdot 10^9 + \end{equation*} +} +\formula{Permeabilità magnetica nel vuoto}{ + \begin{equation*} + \muz=4\pi\cdot 10^{-7}=1.256 \cdot 10^{-6} \frac Hm + \end{equation*} +} + +\formula{Velocità della luce nel vuoto}{ + \begin{equation*} + c=2.997\cdot 10^8\frac ms=\frac 1{\sqrt{\muz\ez}} + \end{equation*} +} + +\section{Leggi fondamentali del campo elettrico} +\formula{Legge di Coulomb}{ + Chiamando $\epsilon=\ez$ se siamo nel vuoto e $\epsilon=\ez\epsilon_r$ se siamo in un mezzo: + \begin{equation*} + \vec F=\frac 1{4\pi\epsilon}\frac{q_1q_2}{r^2}\hat r + \end{equation*} +} + +\formula{Campo elettrico}{ + \begin{equation*} + \vec E(\vec r)=\frac 1{4\pi\ez}\int \rho(\vec r')\frac{\vec r-\vec r'}{\abs{\vec r-\vec r'}^3}\de \vec r' + \end{equation*} +} + +\formula{Potenziale}{ + \begin{equation*} + V(\vec r)=\frac 1{4\pi\ez}\int \frac{\rho(\vec r')}{\abs{\vec r-\vec r'}}\de \vec r' + \end{equation*} + + \begin{equation*} + \vec E=-\grad V + \end{equation*} +} + +\formula{Legge di Gauss}{ + \begin{equation*} + \frac{Q_{int}}{\ez}=\frac 1\ez\int_\Omega \rho =\int_{\partial \Omega}\vec E\cdot \hat n + \end{equation*} +} + +\formula{I equazione di Maxwell}{ + \begin{equation*} + \div \vec E=\frac\rho\ez + \end{equation*} +} + +\formula{II equazione di Maxwell (Elettrostatica)}{ + \begin{equation*} + \rot \vec E=0 + \end{equation*} +} + +\formula{Equazione di Poisson}{ + \begin{equation*} + \lapl V=-\frac \rho \ez + \end{equation*} +} + +\formula{Energia elettrostatica}{ + \begin{equation*} + U_E=\frac 12\int \int \frac 1{4\pi\ez}\cdot\frac{\rho(\vec r)\rho(\vec r')}{\abs{\vec r-\vec r'}}=\frac 12\int\rho V=\frac \ez 2\int \vec E^2 + \end{equation*} +} + +\formula{Forza agente su una carica}{ + La forza di Lorentz agente su una carica {\bf in assenza di campo magnetico} è + \begin{equation*} + \vec F = q \vec E + \end{equation*} +} + +\section{Primi termini dello sviluppo del potenziale} +Nel caso in cui la distribuzione di carica sia sufficientemente localizzata, si può sviluppare il potenziale ai primi ordini con le formule: +\formula{Termine di monopolo}{ + \begin{equation*} + V(\vec r)\approx\frac 1{4\pi\ez r}\left(\int \rho\right) + \end{equation*} +} +\formula{Termine di dipolo}{ + \begin{equation*} + V(\vec r)\approx\frac {\vec r}{4\pi\ez r^3}\cdot\left(\int \rho\left(\vec r'\right)\vec r'\de \vec r'\right) + \end{equation*} +} + + +\section{Conduttori} + +\formula{Pressione elettrostatica}{ + \begin{equation*} + p=\frac{\de F^\perp}{\de S}=\frac{\de q}{\de S} \cdot\frac{E^\perp_{in}+E^\perp_{out}}2=\sigma \cdot\frac{E^\perp_{in}+E^\perp_{out}}2 + \end{equation*} +} + +\formula{Campo e pressione sulla superficie di un conduttore}{ + \begin{equation*} + E^\perp=\frac \sigma\ez + \end{equation*} + \begin{equation*} + p=\sigma\frac {E^\perp}2=\frac{\sigma^2}{2\ez} + \end{equation*} + +} + +\formula{Sistema di conduttori}{ + Chiamiamo $\vec Q=(Q_1,\dots,Q_n)$ il vettore delle cariche e $\vec V=(V_1,\dots,V_n)$ il vettore dei potenziali degli $n$ conduttori, allora esiste una matrice $C$ tale che $\vec Q=C\vec V$. Inoltre: + \begin{equation*} + U=\frac 12\sum Q_iV_i + \end{equation*} +} + +\section{Configurazioni speciali di carica} +\formula{Campi e potenziali particolari}{ + \begin{description} + \item[Carica puntiforme]\begin{equation*} + \vec E(r)=\frac1{4\pi\ez}\cdot \frac Q{r^2} + \end{equation*} + \begin{equation*} + V(r)=\frac1{4\pi\ez}\cdot \frac Qr + \end{equation*} + + + \item[Filo]\begin{equation*} + \vec E(r)=\frac 1{2\pi\ez}\cdot\frac \lambda r + \end{equation*} + \begin{equation*} + V(r)=-\frac\lambda{2\pi\ez}\ln(r) + \end{equation*} + + \item[Piano]\begin{equation*} + \vec E=\frac \sigma{2\ez} + \end{equation*} + \begin{equation*} + V(r)=-\frac{\sigma r}{2\ez} + \end{equation*} + \end{description} +} + +\formula{Dipolo}{ + \begin{equation*} + V(\vec r)=\frac 1{4\pi\ez}\cdot \frac{\vec p\dotp \vec r}{r^3} + \end{equation*} + \begin{equation*} + \vec E(\vec r)=\frac 1{4\pi\ez}\left(\frac{3(\vec p\dotp \hat r)\hat r-\vec p}{r^3} \right) + \end{equation*} + \begin{equation*} + \tau=\vec p\times \vec E + \end{equation*} + \begin{equation*} + \vec F=\Diff \vec E\dotp \vec p + \end{equation*} +} + +\formula{Carica immagine}{ + \begin{description} + \item [Piano] Nel simmetrico rispetto al piano una carica $-q$. + \item [Sfera a terra] Nell'inverso rispetto alla sfera, cioè in $x'=\frac{R^2}{x}$, una carica $q'=-\frac{qR}x$. + \item [Sfera isolata] Se c'è una carica $q$ all'esterno della sfera, le cariche immagine (per risolvere il problema all'esterno della sfera) sono due: una nell'inverso rispetto alla sfera di carica $q'=-\frac{qR}x$ e una nel centro della sfera di carica $-q'$ (in modo che la carica totale nella sfera sia 0). + \end{description} +} + +\section{Condensatori} +\formula{Relazione tra carica e capacità}{ + \begin{equation*} + Q = CV + \end{equation*} +} + +\formula{Relazione tra corrente e tensione}{ + \begin{equation*} + I = C \ded{V}{t} + \end{equation*} +} + +\formula{Energia di un condensatore}{ + \begin{equation*} + U=\frac 12QV=\frac 12CV^2=\frac12\frac {Q^2}C + \end{equation*} +} + +\formula{Capacità di vari condensatori}{ + Nota: la presenza di un materiale fra le armature di un condensatore ne aumenta la capacità di un fattore $\epsilon_r$ (numero puro) detto costante + dielettrica relativa che dipende dalle caratteristiche del materiale. (Tutte le formule della capacità vengono moltiplicate per $\epsilon_r$) + \begin{description} + \item[Condensatore piano] \begin{equation*} + C=\frac{\ez S}d + \end{equation*} + dove $S$ è la superficie del condesatore e $d$ è la distanza fra le due piastre. + \item[Condensatore cilindrico] \begin{equation*} + C=\frac{2\pi\ez h}{\ln(\frac Rr)} + \end{equation*} + dove $h$ è l'altezza del cilindro e $R,r$ sono rispettivamente il raggio della piastra esterna e quello della piastra interna. + \item[Condensatore sferico] \begin{equation*} + C=4\pi\ez\frac{Rr}{R-r} + \end{equation*} + dove $R$ ed $r$ sono rispettivamente il raggio della piastra esterna e quello della piastra interna. + \item[Conduttore sferico isolato] \begin{equation*} + C=4\pi\ez r + \end{equation*} + + \item[Condensatori in parallelo] \begin{equation*} + C_{eq}=C_1+C_2 + \end{equation*} + \item[Condensatori in serie]\begin{equation*} + C_{eq}=\frac 1{\frac 1{C_1}+\frac 1{C_2}} + \end{equation*} + \end{description} +} + +\section{Correnti} + +\formula{Equazione di continuità}{ + Indichiamo con $\vec J$ la densità di corrente. + \begin{equation*} + \frac {\partial \rho}{\partial t}+\div \vec J=0 + \end{equation*} + \begin{equation*} + \frac{\de Q}{\de t}=-\int_S \vec J\dotp \hat n + \end{equation*} +} + +\formula {Legge di Ohm (conducibilità e resistività)} { + La resistività $\rho$ è tale che + \begin{equation*} + \vec J=\frac{\vec E}\rho + \end{equation*} + e la conducibilità è $\sigma=\frac 1\rho$. +} + +\formula{Effetto Joule}{ + La potenza dissipata da un certo materiale in un certo istante è + \begin{equation*} + P=\int \vec E\dotp \vec J=\int \frac{\vec E^2}\rho + \end{equation*} + dove per l'ultima uguaglianza abbiamo utilizzato la legge di Ohm. +} + +\formula{Tempo di scarica}{ + Se la carica rispetta un'equazione del tipo + \begin{equation*} + \frac{\de Q}{\de t}=-\frac{Q}{\tau} + \end{equation*} + allora chiamiamo $\tau$ il tempo di scarica. + La soluzione di una tale equazione è + \begin{equation*} + Q(t) = Q(0) e^{-\frac{t}{\tau}} + \end{equation*} + + Per circuiti che si scaricano si ha + \begin{equation*} + I=- \ded{Q}{t} + \end{equation*} +} + +\section{Resistenze} +\begin{description} + \item[Legge di Ohm] + \begin{equation*} + V=RI + \end{equation*} + \item[Relazione con Lunghezza e Sezione] + \begin{equation*} + R=\frac dA \rho + \end{equation*} + \item[Potenza dissipata] + \begin{equation*} + P=VI=RI^2 + \end{equation*} + \item[Resistenze in serie] + \begin{equation*} + R_{eq} = R_1 + R_2 + \end{equation*} + \item[Resistenze in parallelo] + \begin{equation*} + R_{eq} = \frac{1}{\frac{1}{R_1} + \frac{1}{R_2}} + \end{equation*} +\end{description} + +\section{Leggi fondamentali del campo magnetico} +\formula{Forza di Lorentz}{ + Il campo magnetico $\vec B$ è uno pseudovettore che genera su una carica in movimento la forza + \begin{equation*} + \vec F=q\vec v \times \vec B + \end{equation*} +} + +\formula{III legge di Maxwell (assenza di monopolo magnetico)}{ + \begin{equation*} + \div \vec B=0 + \end{equation*} +} + +\formula{Legge di Ampère}{ + \begin{equation*} + \rot\vec B=\muz\vec J + \end{equation*} +} + +\formula{Legge della circuitazione}{ + \begin{equation*} + \oint_\gamma \vec B\dotp\de\vec l=\muz \int_{S_\gamma} \vec J\dotp \de\vec S=\muz I_{\text{concatenata}} + \end{equation*} +} + +\formula{Biot-Savart}{ + Nel caso di una distribuzione di carica localizzata: + \begin{equation*} + \vec B(\vec r)=\frac{\muz}{4\pi}\int \frac{\vec J(\vec r')\times(\vec r-\vec r')}{\abs{\vec r-\vec r'}^3}\de\vec r' + \end{equation*} + e nel caso di fili: + \begin{equation*} + \vec B(\vec r)=\frac{\muz}{4\pi}\int_{\gamma} \frac{I\de \vec l\times(\vec r-\vec r')}{\abs{\vec r-\vec r'}^3} + \end{equation*} +} + +\formula{Densità di energia magnetica}{ + \begin{equation*} + u_B=\frac{\vec B^2}{2\muz} + \end{equation*} +} + +\formula{Pressione magnetica}{ + \begin{equation*} + \abs p=u_B=\frac{\vec B^2}{2\muz} + \end{equation*} +} + +\formula{Energia magnetica}{ + \begin{equation*} + U_B=\int u_B=\frac 1{2\muz}\int \vec B^2 + \end{equation*} +} + +\section{Potenziale vettore} +\formula{Relazioni fondamentali con $\vec B$}{ + \begin{align*} + \div \vec A&=0 \\ + \rot\vec A &=\vec B + \end{align*} +} +\formula{Biot-Savart per $\vec A$}{ + \begin{equation*} + \vec A(\vec r)=\frac{\muz}{4\pi}\int \frac{\vec J(\vec r')}{\abs{\vec r-\vec r'}}\de\vec r' + \end{equation*} +} + +\section{Momento di dipolo magnetico} +\formula{Definizione di momento di dipolo}{ + \begin{equation*} + \vec m=\frac 12\int \vec r\times\vec J + \end{equation*} +} + +\formula{Caso di filo chiuso planare}{ + Nel caso di un filo percorso da corrente $I$ che racchiude una regione di piano con area (orientata) $\vec S$, vale + \begin{equation*} + \vec m=I\vec S + \end{equation*} +} + +\formula{Termine di dipolo nel potenziale vettore}{ + \begin{equation*} + \vec A(\vec r)=\frac{\muz}{4\pi}\frac{\vec m\times\vec r}{r^3} + \end{equation*} +} + +\formula{Termine di dipolo nel campo magnetico}{ + \begin{equation*} + \vec B(\vec r)=\frac{\muz}{4\pi}\left(\frac{3\vec r\left(\vec m\dotp\vec r\right)}{r^5}-\frac{\vec m}{r^3}\right) + \end{equation*} +} + +\formula{Costante giromagnetica}{ + Un corpo di massa $m$ e carica $q$, ammette una costante $g$ che lega il momento di dipolo al momento angolare: + \begin{equation*} + \vec m=g\frac{q}{2m}\vec L + \end{equation*} + + Nel caso di una carica puntiforme o una sfera uniformemente carica $g=1$; per una sfera carica superficialmente $g=\frac 53$; per un elettrone $g=2.002$. +} + +\section{Forza e momento generati dal campo magnetico} +\formula{Forza del campo magnetico}{ + Data una distribuzione di correnti $\vec J$, ed un campo magnetico \emph{esterno} $\vec B$, la forza agente sulla distribuzione è: + \begin{equation*} + \vec F=\int \vec J\times \vec B + \end{equation*} +} + +\formula{Forza di un campo costante}{ + \begin{equation*} + \vec F=0 + \end{equation*} +} + +\formula{Coppia torcente del campo magnetico}{ + Data una distribuzione di correnti $\vec J$, ed un campo magnetico \emph{esterno} $\vec B$, la coppia torcente, rispetto all'origine, agente sulla distribuzione è: + \begin{equation*} + \vec\tau=\int \vec r\times\left(\vec J\times \vec B\right) + \end{equation*} +} + +\formula{Coppia torcente di un campo costante}{ + \begin{equation*} + \vec\tau=\vec m\times \vec B_0 + \end{equation*} +} + +\section{Distribuzioni di corrente particolari} +\formula{Filo}{ + Un filo lungo l'asse $\hat z$ percorso da corrente $I$ genera + \begin{equation*} + \vec B(\vec r)=\hat\varphi \frac{\muz I}{2\pi\rho} + \end{equation*} + dove $\hat\varphi$ è orientato in modo che se il pollice della mano destra è $\hat z$, le restanti dita ne descrivono il verso. +} + +\formula{Solenoide}{ + Un solenoide di raggio $a$, con $n$ spire per unità di lunghezza percorse da corrente $I$, disposto lungo l'asse $\hat z$ genera + \begin{equation*} + \vec B(\vec r)=\hat z\muz nI + \end{equation*} + che è diretto in modo che se le dita della mano destra seguono $I$, allora il pollice detta il verso di $\vec B$. +} + +\formula{Spira}{ + Una spira di raggio $a$ percorsa da corrente $I$ con asse lungo $\hat z$, genera sull'asse $\hat z$ + \begin{equation*} + \vec B(z)=\hat z\frac{\muz a^2I}{2\left(z^2+a^2\right)^{\frac 32}} + \end{equation*} + dove anche questa volta $\vec B$ è il pollice della mano destra e le altre dita seguono $I$. +} + +\section{Elettrodinamica} +Ora tratteremo i casi, finora non considerati, di campo elettrico non costante e come questo incida sul campo magnetico. +Questo farà sì che il campo elettrico non sia più conservativo e perciò non esista più un potenziale. + +\formula{II equazione di Maxwell}{ + \begin{equation*} + \rot\vec E=-\frac{\partial \vec B}{\partial t} + \end{equation*} +} + +\formula{Legge di Lenz}{ + Nel caso di un circuito, la forma integrale della II equazione di Maxwell diventa: + \begin{equation*} + \mathcal E=-\frac \de{\de t}\int_S \vec B\dotp\de\vec S=-\frac {\de\Phi_B}{\de t} + \end{equation*} + dove $\mathcal E$ è la forza elettromotrice (f.e.m.) indotta sul circuito e $\Phi_B$ è il flusso di campo magnetico attraverso la superficie racchiusa dal circuito. + Il segno della f.e.m. indotta è tale da opporsi sempre alla variazione di corrente. +} + +\formula{IV equazione di Maxwell}{ + Questa legge generalizza sia l'equazione di continuità sia la legge di Ampère al caso elettrodinamico: + \begin{equation*} + \frac 1{c^2}\frac{\partial \vec E}{\partial t}-\rot\vec B+\muz\vec J=0 + \end{equation*} +} + +\section{Induzione magnetica} + +\formula{Induttanza}{ + L'induttanza è una costante $L$ dipendente dalla geometria del circuito, tale che + \begin{equation*} + \Phi_B=LI + \end{equation*} + dove $\Phi_B$ è il flusso di campo magnetico generato dal circuito stesso. +} + +\formula{Induttore}{ + Un induttore è una componente di un circuito la cui induttanza non è trascurabile. L'esempio tipico è un solenoide. +} + +\formula{Mutua induzione}{ + Chiamando $\vec{\mathcal E}$ il vettore delle f.e.m. indotte e $\vec I$ il vettore delle correnti di $n$ circuiti, esiste una matrice $L$ simmetrica, che sulla diagonale ha le autoinduttanze dei singoli circuiti, tale che: + \begin{equation*} + \vec {\mathcal E}=-L\frac{\de \vec I}{\de t} + \end{equation*} + e il valore $L_{ij}=L_{ji}$ si calcola con + \begin{equation*} + L_{ij}=L_{ji}=\frac {\muz}{4\pi}\oint_{\gamma_i}\oint_{\gamma_j}\frac{\de \vec l_i\dotp\de\vec l_j}{r} + \end{equation*} + dove con $r$ indichiamo la distanza tra i punti in cui stiamo integrando. +} + +\formula{Energia di un induttore}{ + \begin{equation*} + U=\frac 12LI^2 + \end{equation*} + e nel caso di un sistema di induttori: + \begin{equation*} + U=\frac 12 \vec I^{\,T}L\vec I + \end{equation*} +} + +\formula{Circuiti RLC}{ + \begin{equation*} + \mathcal E=RI+\frac QC + L\frac{\de I}{\de t}=\frac QC+R\frac{\de Q}{\de t}+L\frac{\de^2Q}{\de t^2} + \end{equation*} +} + +\section{Integrali ``Noti''} +\begin{equation*} + \int \frac{r \de r}{(r^2+1)^{\frac{3}{2}}} = - \frac{1}{\sqrt{r^2 + 1}} +\end{equation*} + +\begin{equation*} + \int e^{\alpha t} \de t = \frac{1}{\alpha} e^{\alpha t} +\end{equation*} + +\section{Operatori differenziali} + + \formula{Gradiente in cartesiane} + \begin{equation*} + \grad f = \dep{f}{x} \hat{x} + + \dep{f}{y} \hat{y} + + \dep{f}{z} \hat{z} + \end{equation*} + + \formula{Gradiente in cilindriche} + \begin{equation*} + \grad f = \dep{f}{\rho} \hat{\rho} + + \frac{1}{\rho} \dep{f}{\phi} \hat{\phi} + + \dep{f}{z} \hat{z} + \end{equation*} + + \formula{Gradiente in sferiche} + \begin{equation*} + \grad f = \dep{f}{r} \hat{r} + + \frac{1}{r} \dep{f}{\theta} \hat{\theta} + + \frac{1}{r \sin \theta} \dep{f}{\phi} \hat{\phi} + \end{equation*} + + \formula{Divergenza in cartesiane} + \begin{equation*} + \div A = \dep{A_x}{x} + \dep{A_y}{y} + \dep{A_z}{z} + \end{equation*} + + \formula{Divergenza in cilindriche} + \begin{equation*} + \div A = \frac{1}{\rho} \dep{(\rho A_\rho)}{\rho} + + \frac{1}{\rho} \dep{A_\phi}{\phi} + + \dep{A_z}{z} + \end{equation*} + + \formula{Divergenza in sferiche} + \begin{equation*} + \div A = \frac{1}{r^2} \dep{(r^2 A_r)}{r} + + \frac{1}{r \sin \theta} \dep{(A_\theta \sin\theta)}{\theta} + + \frac{1}{r \sin\theta} \dep{A_\phi}{\phi} + \end{equation*} + +\formula{Rotore in cartesiane} +\begin{align*} + \rot A = \lrb{\dep{A_z}{y} - \dep{A_y}{z}} \hat{x} + \\ + \lrb{\dep{A_x}{z} - \dep{A_z}{x}} \hat{y} + \\ + \lrb{\dep{A_y}{x} - \dep{A_x}{y}} \hat{z} +\end{align*} + +\formula{Rotore in cilindriche} +\begin{align*} + \rot A = \lrb{\frac{1}{\rho} \dep{A_z}{\phi} - \dep{A_\phi}{z}} \hat{\rho} + \\ + \lrb{\dep{A_\rho}{z} - \dep{A_z}{\rho}} \hat{\phi} + \\ + \frac{1}{\rho} \lrb{\dep{(\rho A_\phi)}{\rho} - \dep{A_\rho}{\phi}} \hat{z} +\end{align*} + +\formula{Rotore in sferiche} +\begin{align*} + \rot A = \frac{1}{r \sin\theta} \lrb{\dep{(A_\phi \sin\theta)}{\phi} - \dep{A_\theta}{\phi}} \hat{r} + \\ + \frac{1}{r} \lrb{\frac{1}{\sin\theta} \dep{A_r}{\phi} - \dep{(r A_\phi)}{r}} \hat{\theta} + \\ + \frac{1}{r} \lrb{\dep{(r A_\theta)}{r} - \dep{A_r}{\theta}} \hat{\phi} +\end{align*} +\end{multicols} +\end{document} +