From 4a8ecfc6e10661b8755ff9f1aa9bc32568e6bd4c Mon Sep 17 00:00:00 2001
From: BAI2795 <dieter.horns@uni-hamburg.de>
Date: Tue, 12 Jan 2021 21:00:21 +0100
Subject: [PATCH] update ,  bug fix on Poynting vector definition

---
 v0.1/Formelsammlung.tex                       |   21 +-
 .../Graphiken/drawing_capacity_dielectric.svg |  527 +++++
 ...y_dielectric.svg.2020_11_20_12_42_26.0.svg |  616 +++++
 v0.1/Graphiken/drawing_capacity_liquid.svg    |  527 +++++
 v0.1/Graphiken/drawing_exercise.svg           |  161 ++
 v0.1/Graphiken/drawing_filter.svg             |  290 +++
 v0.1/Graphiken/drawing_massspectro.png        |  Bin 0 -> 66465 bytes
 v0.1/Graphiken/drawing_massspectro.svg        | 1028 ++++++++
 v0.1/Graphiken/plattenkondensator.svg         | 2092 +++++++++++++++++
 v0.1/elektrostatik.tex                        |    2 +-
 v0.1/em_waves.tex                             |    8 +-
 v0.1/scripts/plot_dipole_fieldlines_pot.py    |    6 +-
 v0.1/scripts/plot_example.py                  |  135 ++
 v0.1/skript.pdf                               |  Bin 26400178 -> 26390593 bytes
 14 files changed, 5396 insertions(+), 17 deletions(-)
 create mode 100644 v0.1/Graphiken/drawing_capacity_dielectric.svg
 create mode 100644 v0.1/Graphiken/drawing_capacity_dielectric.svg.2020_11_20_12_42_26.0.svg
 create mode 100644 v0.1/Graphiken/drawing_capacity_liquid.svg
 create mode 100644 v0.1/Graphiken/drawing_exercise.svg
 create mode 100644 v0.1/Graphiken/drawing_filter.svg
 create mode 100644 v0.1/Graphiken/drawing_massspectro.png
 create mode 100644 v0.1/Graphiken/drawing_massspectro.svg
 create mode 100644 v0.1/Graphiken/plattenkondensator.svg
 create mode 100644 v0.1/scripts/plot_example.py

diff --git a/v0.1/Formelsammlung.tex b/v0.1/Formelsammlung.tex
index d15751e..ee893d8 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 0000000..7de3e95
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diff --git a/v0.1/elektrostatik.tex b/v0.1/elektrostatik.tex
index d7e22f9..49bf7ea 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 c6db1b2..f5704b0 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 21b45e7..b78e19e 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 0000000..2ab58ca
--- /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
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