diff --git a/v0.1/Formelsammlung.tex b/v0.1/Formelsammlung.tex index d15751e2ef9e89ff0b00adcbdf666f037932304e..ee893d858cea2fbee00d738c9fb30eb1f12fc984 100644 --- a/v0.1/Formelsammlung.tex +++ b/v0.1/Formelsammlung.tex @@ -1,6 +1,6 @@ \chapter*{Formelsammlung} \begin{enumerate}[a)] -\item Ladung/Ströme +\item \textbf{Ladung/Ströme} \begin{enumerate}[i)] \item Ladung: $q,Q$, Einheit $[q]$=Coulomb (C) \item Ladungsdichte: $\rho$, Einheit Ladung/Volumen $[\rho]=$C~m$^{-3}$ @@ -10,14 +10,14 @@ \item Elektrische Feldkonstante $\epsilon_0=\frac{1}{\mu_0~c^2}) \approx 8,85\times 10^{-12}~\mathrm{C}^2/(\mathrm{Nm}^2)$. \end{enumerate} -\item Felder +\item \textbf{Felder} \begin{enumerate}[i)] \item Elektrisches Feld $\bm{E}$, $[\bm{E}]=$V/m \item Magnetisches Feld $\bm{B}$, $[\bm{B}]=$Tesla (T) \item Elektrische Feldlinien: Zeigen von $+$ nach $-$ \item Magnetische Feldlinien: Laufen von Nordpol zu Südpol, geschlossen! \end{enumerate} -\item Ladungsverteilungen +\item \textbf{Ladungsverteilungen} \begin{enumerate}[i)] \item Superpositionsprinzip diskreter Ladungen \[ \bm{E}(\bm{r}) = \frac{1}{4\pi\epsilon_0} \sum\limits_{i} @@ -26,7 +26,7 @@ \[ \bm{E}(\bm{r}) = \frac{1}{4\pi\epsilon_0} \int dV' \rho(\bm{r}') \frac{\bm{r}-\bm{r}'}{|\bm{r}-\bm{r}'|^3} \] \end{enumerate} - \item Stromverteilungen + \item \textbf{Stromverteilungen} \begin{enumerate}[i)] \item bewegte diskrete Ladungen: \[ \bm{B}(\bm{r}) = \frac{\mu_0}{4\pi} \sum\limits_{i} @@ -35,12 +35,12 @@ \[ \bm{B}(\bm{r}) = \frac{\mu_0}{4\pi} \int dV' \bm{j}(\bm{r}') \times \frac{ \bm{r}-\bm{r}'}{|\bm{r}-\bm{r}'|^3}\] \end{enumerate} -\item Kräfte +\item \textbf{Kräfte} \begin{enumerate}[i)] \item Lorentzkraft: \[ \bm{F} = q( \bm{E} + \bm{v}\times \bm{B})\] \item Ströme (Kraft/Längeneinheit): $\bm{F}/L=I \bm{e}_L \times \bm{B}$. \end{enumerate} -\item Potenzial, potenzielle Energie: +\item \textbf{Potenzial, potenzielle Energie:} \begin{enumerate}[i)] \item skalares Potenzial $\phi(\bm{r})$, so dass $\bm{E} = -\bm{\nabla}\phi$. \item Vektorpotenzial $\bm{A}(\bm{r})$, so dass $\bm{B}=\bm{\nabla}\times \bm{A}$. @@ -56,7 +56,7 @@ bei gegegebenem $\bm{j}(\bm{r})$: \[ \bm{A}(\bm{r}) = \frac{\mu_0}{4\pi}\int dV' \frac{\bm{j}(\bm{r}')}{|\bm{r}-\bm{r}'|} \] \end{enumerate} -\item Dipolmoment +\item \textbf{Dipolmoment} \begin{enumerate}[i)] \item Elektrisches Dipolmoment $\bm{p}_e = \int dV' \bm{r}' \rho(\bm{r}')$ \item magnetisches Dipolmoment $\bm{p}_m = \frac{1}{2}\int dV' \bm{r}'\times \bm{j}(\bm{r}')$ @@ -66,9 +66,12 @@ \item Potenzielle Energie: $W_\mathrm{pot}=-\bm{p}_e\cdot \bm{E}$ bzw. $W_\mathrm{pot}=-\bm{p}_m\cdot \bm{B}$. \end{enumerate} -\item Felder in Materie +\item \textbf{Felder in Materie} \begin{enumerate}[i)] - \item Polarisation + \item Polarisation im äußeren Feld $\bm{E}_\mathrm{vak}$ + reduziert das Feld $\bm{E}_\mathrm{net}=\bm{E}_\mathrm{vak}/\epsilon_r$ + mit der relativen Permittivität $\epsilon_r$ (einheitenlos). + \item elektrische Verschiebung $\epsilon_0 \epsilon_r \bm{D} = \bm{E}$ \end{enumerate} \end{enumerate} diff --git a/v0.1/Graphiken/drawing_capacity_dielectric.svg b/v0.1/Graphiken/drawing_capacity_dielectric.svg new file mode 100644 index 0000000000000000000000000000000000000000..7de3e958f3c283db7591888390a30872133d89ce --- /dev/null +++ b/v0.1/Graphiken/drawing_capacity_dielectric.svg @@ -0,0 +1,527 @@ +<?xml version="1.0" encoding="UTF-8" standalone="no"?> +<!-- Created with Inkscape (http://www.inkscape.org/) --> + +<svg + xmlns:dc="http://purl.org/dc/elements/1.1/" + xmlns:cc="http://creativecommons.org/ns#" + xmlns:rdf="http://www.w3.org/1999/02/22-rdf-syntax-ns#" + xmlns:svg="http://www.w3.org/2000/svg" + xmlns="http://www.w3.org/2000/svg" + xmlns:sodipodi="http://sodipodi.sourceforge.net/DTD/sodipodi-0.dtd" + xmlns:inkscape="http://www.inkscape.org/namespaces/inkscape" + width="210mm" + height="297mm" + viewBox="0 0 210 297" + version="1.1" + id="svg8" + inkscape:version="0.92.3 (2405546, 2018-03-11)" + 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-0.57902,-43.88677 -0.60884,-45.28441 l -0.0542,-2.54116 h 3.62844 3.62843 l 0.0516,1.61308 c 0.0284,0.8872 0.17409,14.44045 0.32377,30.11833 0.37887,39.68332 0.51395,51.50076 0.60976,53.34237 0.0455,0.87505 0.11038,2.63508 0.14413,3.91119 l 0.0614,2.32019 -0.25429,-0.0105 c -0.13986,-0.006 -1.76658,-0.17146 -3.61493,-0.36817 z" + id="path1063" + inkscape:connector-curvature="0" + transform="scale(0.26458333)" /> + </g> +</svg> diff --git a/v0.1/elektrostatik.tex b/v0.1/elektrostatik.tex index d7e22f95be6ec2fae17406a7738361a18ba8e346..49bf7ea467d4c272e85bf5bb1a69e0c9aabab7e6 100644 --- a/v0.1/elektrostatik.tex +++ b/v0.1/elektrostatik.tex @@ -629,7 +629,7 @@ mit zunehmendem Abstand ab. $\varphi\ll 1$ um die Positionen mit $\theta=\pi$ und $\theta=0$! } \subsection{Elektrischer Dipol in einem inhomogenen äußeren Feld} - Wir betrachten den einfachen Fall eines inhomogenen Magnetfeldes mit zunehmender + Wir betrachten den einfachen Fall eines inhomogenen elektrischen Feldes mit zunehmender Feldstärke in $x$-Richtung, so dass $d\bm{E}/dx\ne0$ gilt. Dann ist die Kraft auf dem Dipol \[ \bm{F} = \bm{F}_+ + \bm{F}_- = Q(\bm{E}(x)-\bm{E}(x+\Delta x)) \approx diff --git a/v0.1/em_waves.tex b/v0.1/em_waves.tex index c6db1b2aeffdde0f4724128f5104b5bb268c1ec0..f5704b0d87f6fdd948040019e1cf460e3a1b6efc 100644 --- a/v0.1/em_waves.tex +++ b/v0.1/em_waves.tex @@ -215,8 +215,8 @@ Anmerkungen: \section{Energie- und Impulstransport} Die Energiedichte des elektrischen und magnetischen Feldes kennen wir bereits f\"ur den statischen Fall: -$$w= \frac{1}{2}\epsilon_r\,\epsilon_0\,\bm{E}^2 + \frac{1}{2}\mu_r\,\mu_0\,\bm{H}^2 - = \frac{1}{2}\epsilon_r\,\epsilon_0\,\bm{E}^2 + \frac{1}{2}\frac{\bm{B}^2}{\mu_r\,\mu_0}.$$ +\[ w= \frac{1}{2}\epsilon_r\,\epsilon_0\,\bm{E}^2 + \frac{1}{2}\mu_r\,\mu_0\,\bm{H}^2 + = \frac{1}{2}\epsilon_r\,\epsilon_0\,\bm{E}^2 + \frac{1}{2}\frac{\bm{B}^2}{\mu_r\,\mu_0}.\] F\"ur eine elektromagnetische Welle ist $|\bm{B}|=n|\bm{E}|/c=\sqrt{\epsilon_r\,\mu_r\,\epsilon_0\,\mu_0}|\bm{E}|$, also l\"asst sich die Energiedichte in Abh\"angigkeit der elektrischen Feldst\"arke schreiben als $$w=\frac{1}{2}\epsilon_r\,\epsilon_0\,\bm{E}^2 + \frac{1}{2}\frac{n^2\bm{E}^2}{c^2\mu_r\mu_0} =\epsilon_r\,\epsilon_0\,\bm{E}^2.$$ @@ -226,10 +226,10 @@ der Energie pro Zeitintervall $dt$ und Fl\"achenelement $dA$ $$\frac{dU}{dA\,dt}=\frac{dU}{dA\,dx}\frac{dx}{dt}=\frac{dU}{dV}\dot{x}=w\,v,$$ mit $v=\dot{x}$. Die Geschwindigkeit der Wellenausbreitung im Dielektrikum mit Brechungsindex $n$ ist $v=c/n$, so dass wir f\"ur die Leistung/Fl\"ache erhalten: -$$\frac{dU}{dA\,dt}=w\,\frac{c}{n}=\epsilon_r\,\epsilon_0\,|\bm{E}|\,\frac{|\bm{B}|\,c}{n}\,\frac{c}{n}=\frac{|\bm{E}|\,|\bm{B}|}{\mu_r\,\epsilon_r}.$$ +$$\frac{dU}{dA\,dt}=w\,\frac{c}{n}=\epsilon_r\,\epsilon_0\,|\bm{E}|\,\frac{|\bm{B}|\,c}{n}\,\frac{c}{n}=\frac{|\bm{E}|\,|\bm{H}|}{\mu_r\,\epsilon_r}.$$ Wir k\"onnen eine neue abgeleitete Gr\"oße einf\"uhren, den \textit{Poynting}-Vektor, der parallel zu $\bm{k}$ definiert ist: \begin{wichtig}[Poynting-Vektor] -$$\bm{S}:= \frac{1}{\mu_r\,\epsilon_r}\bm{E}\times\bm{B}.$$ +$$\bm{S}:= \frac{1}{\mu_r\,\epsilon_r}\bm{E}\times\bm{H}.$$ \end{wichtig} Der Richtung des Poynting-Vektors entspricht der Ausbreitungsrichtung der elektromagnetischen Welle, der Betrag des Poynting-Vektors ist die Leistung, die pro Fl\"acheneinheit (Fl\"achennormale parallel zur Richtung des Poynting-Vektors) transportiert wird.\\ Der so definierte Poynting-Vektor ist im allgemeinen zeit- und ortsabh\"angig. Wir k\"onnen f\"ur den Spezialfall einer harmonischen Welle einen zeitlichen Mittelwert angeben: diff --git a/v0.1/scripts/plot_dipole_fieldlines_pot.py b/v0.1/scripts/plot_dipole_fieldlines_pot.py index 21b45e71968dfa747186eb2417a67ca8505d4b55..b78e19efbd33a268050f81ab07d743588726cb71 100644 --- a/v0.1/scripts/plot_dipole_fieldlines_pot.py +++ b/v0.1/scripts/plot_dipole_fieldlines_pot.py @@ -45,8 +45,8 @@ def V_total(x, y, charges): charges=[ charge(+1, [-1, 0]), charge(-1, [1, 0])] # calculate field lines -x0, x1=-3, 3 -y0, y1=-3, 3 +x0, x1=-6, 6 +y0, y1=-6, 6 R=0.001 # loop over all charges xs,ys = [],[] @@ -122,7 +122,7 @@ vvs[np.where(vvs<clim0)] = clim0*0.999999 # to avoid error vvs[np.where(vvs>clim1)] = clim1*0.999999 # to avoid error plt.tricontour(xxs,yys,vvs,30,colors="0.3") plt.tricontourf(xxs,yys,vvs,100,cmap=cm.jet) -#cbar = plt.colorbar() +cbar = plt.colorbar() #plt.xlabel('$x$') #plt.ylabel('$y$') plt.xlim(x0, x1) diff --git a/v0.1/scripts/plot_example.py b/v0.1/scripts/plot_example.py new file mode 100644 index 0000000000000000000000000000000000000000..2ab58ca58ace07f577da0ba6f52fa6df3fd7721d --- /dev/null +++ b/v0.1/scripts/plot_example.py @@ -0,0 +1,135 @@ +import matplotlib.pyplot as plt +import numpy as np +from scipy.integrate import ode as ode +from matplotlib import cm +from itertools import product + + +class charge: + def __init__(self, q, pos): + self.q=q + self.pos=pos + +def E_point_charge(q, a, x, y): + return q*(x-a[0])/((x-a[0])**2+(y-a[1])**2)**(1.5), \ + q*(y-a[1])/((x-a[0])**2+(y-a[1])**2)**(1.5) + +def E_total(x, y, charges): + Ex, Ey=0, 0 + for C in charges: + E=E_point_charge(C.q, C.pos, x, y) + Ex=Ex+E[0] + Ey=Ey+E[1] + return [ Ex, Ey ] + +def E_dir(t, y, charges): + Ex, Ey=E_total(y[0], y[1], charges) + n=np.sqrt(Ex**2+Ey**2) + return [Ex/n, Ey/n] # normalized + +def V_point_charge(q, a, x, y): + return q/((x-a[0])**2+(y-a[1])**2)**(0.5) + +def V_total(x, y, charges): + V=0 + for C in charges: + Vp=V_point_charge(C.q, C.pos, x, y) + V = V+Vp + return V + + + + +# charges and positions +#charges=[ charge(-1, [-1, 0]), charge(-1, [1, 0]), charge(1, [0, 1]), charge(1, [0, -1]) ] +charges=[ charge(+2, [-1, 0]), charge(-1, [1, 0])] + +# calculate field lines +# in this range +x0, x1=-6, 6 +y0, y1=-6, 6 +R=0.005 +xs,ys = [],[] +# loop over all charges +for C in charges: + # plot field lines starting in current charge + dt=0.8*R +# if C.q<0: +# continue + if C.q<0: + dt=-dt + # loop over field lines starting in different directions + # around current charge + # this should be scaled with the charge! + # unit charge is equal to 10 lines etc. + nfield = np.int(np.sqrt(C.q*C.q))*20 + nfield = np.int(np.sqrt(C.q*C.q))*10 + for alpha in np.linspace(0., 2*np.pi*(nfield-1)/nfield, nfield): + if (alpha==0 and C.pos[0]<0) or (alpha==np.pi and C.pos[0]>0): + continue + r=ode(E_dir) + r.set_integrator('vode') + r.set_f_params(charges) + x=[ C.pos[0] + np.cos(alpha)*R ] + y=[ C.pos[1] + np.sin(alpha)*R ] + r.set_initial_value([x[0], y[0]], 0) + while r.successful(): + r.integrate(r.t+dt) + x.append(r.y[0]) + y.append(r.y[1]) + hit_charge=False + # check if field line left drwaing area or ends in some charge + for C2 in charges: + if np.sqrt((r.y[0]-C2.pos[0])**2+(r.y[1]-C2.pos[1])**2)<R: + hit_charge=True + if hit_charge or (not (x0<r.y[0] and r.y[0]<x1)) or \ + (not (y0<r.y[1] and r.y[1]<y1)): + break + xs.append(x) + ys.append(y) + + + +# calculate electric potential +vvs = [] +xxs = [] +yys = [] +numcalcv = 300 +for xx,yy in product(np.linspace(x0,x1,numcalcv),np.linspace(y0,y1,numcalcv)): + xxs.append(xx) + yys.append(yy) + vvs.append(V_total(xx,yy,charges)) +xxs = np.array(xxs) +yys = np.array(yys) +vvs = np.array(vvs) + + +plt.figure(figsize=(4.5, 4.5),facecolor="w") + +# plot field line +for x, y in zip(xs,ys): + plt.plot(x, y, color="b",linewidth=0.5) + +#plt.plot([-3,3],[0,0],'k',lw=1.5) +#plt.plot([0,0],[-3,3],'k') +# plot point charges +for C in charges: + if C.q>0: + plt.plot(C.pos[0], C.pos[1], 'ro', ms=8*np.sqrt(C.q)) + if C.q<0: + plt.plot(C.pos[0], C.pos[1], 'go', ms=8*np.sqrt(-C.q)) + +clim0,clim1 = -3,3 +vvs[np.where(vvs<clim0)] = clim0*0.999999 # to avoid error +vvs[np.where(vvs>clim1)] = clim1*0.999999 # to avoid error +plt.tricontour(xxs,yys,vvs,30,colors="0.3") +plt.tricontourf(xxs,yys,vvs,100,cmap=cm.jet) +#cbar = plt.colorbar() +#plt.xlabel('$x$') +#plt.ylabel('$y$') +plt.xlim(x0, x1) +plt.ylim(y0, y1) +plt.axes().set_aspect('equal','datalim') +#plt.savefig('electric_force_lines_4.png',dpi=250,bbox_inches="tight",pad_inches=0.02) +plt.show() + diff --git a/v0.1/skript.pdf b/v0.1/skript.pdf index f0feee8c35aa680192b047e4e46761de4922e364..ab77f9c23dfe917ec0a27c4f5bcc04cf4695073a 100644 Binary files a/v0.1/skript.pdf and b/v0.1/skript.pdf differ