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S3/ExPhyIII/VL/ExIIIVL7.typ

@@ -0,0 +1,102 @@
+// Main VL template
+#import "../preamble.typ": *
+
+// Fix theorems to be shown the right way in this document
+#import "@preview/ctheorems:1.1.3": *
+#show: thmrules
+
+// Main settings call
+#show: conf.with(
+	// May add more flags here in the future
+	num: 5,
+	type: 0, // 0 normal, 1 exercise
+	date: datetime.today().display(),
+	//date: datetime(
+	//	year: 2025,
+	//	month: 5,
+	//	day: 1,
+	//).display(),
+)
+
+= Uebersicht
+
+== Maxwellgleichungen in Materie
+
+Fuer $arrow(E) "und" arrow(B)$ Felder ist die Idee die Aufteilung der Ladungin freie und in Materie gebundene Ladung. Es gilt 
+$
+  arrow(nabla)  * arrow(E) =  (rho) / (epsilon_0 ) =  (rho_(f) + rho_(g) ) / (epsilon_0 ) \ 
+arrow(nabla) * arrow(E) =  1/epsilon_0 (rho_(f) - arrow(nabla) *arrow(p)) \
+arrow(nabla)  * (epsilon_0 arrow(E) + arrow(P)) = rho_(f)  \ 
+arrow(nabla) * arrow(D) = phi_(f) \
+  arrow(P) = underbrace(n, "Dichte") .. underbrace(arrow(p), "Dipole") .. "ist die Polarisation".
+$
+Es gilt weiterhin 
+$
+  arrow(nabla)  times (arrow(B)/mu_0 - arrow(M)) = arrow(j)_(f)  \ 
+  arrow(nabla)  times arrow(H).
+$
+
+== Zeitlich veraenderliche Felder
+
+Vom Ampereschen Gesetz auf das Maxwellsche Gesetz 
+$
+  arrow(nabla)  times arrow(B) =  mu_0 (arrow(j) +  epsilon_0 (diff arrow(E)) / (diff t) ) =  mu_0 (arrow(j)_(f) +  arrow(j)_(g) + arrow(j)_(g) +  epsilon_0 (diff arrow(E)) / (diff t)    ).
+$
+Polarisatoinsstroeme sind also 
+$
+  arrow(j)_(P) = (diff arrow(P)) / (diff t).
+$
+Es folgt 
+$
+  arrow(nabla)  times arrow(B) =  mu_0 (arrow(j)_(f) +  (diff (arrow(P) +  epsilon_0 arrow(E))) / (diff t) +  arrow(nabla)  times arrow(M)) \ 
+  arrow(nabla)  times ((arrow(B)) / (mu_0 ) - arrow(M)) = arrow(j)_(f) +  (diff arrow(D)) / (diff t)  \ 
+  arrow(nabla) times arrow(H) = arrow(j)_(f) +  partial _(t) arrow(D).
+$
+Also fuer die Maxwellgleichungen 
+$
+  arrow(nabla)  * arrow(D) &=  rho_(f) \ 
+  arrow(nabla)  * arrow(B) &=  0 \ 
+  arrow(nabla) times arrow(E) &=  - partial _(t) arrow(B)\ 
+  arrow(nabla)  times arrow(H) &=  arrow(j)_(f) +  partial _(t) arrow(D) \ 
+  arrow(D) &=  epsilon_(r) epsilon_0 arrow(E) .. "(wenn" arrow(P) prop arrow(E)) .. "isotrop" \ 
+  arrow(H) &= 1/ (mu_(r) mu_0 ) arrow(B) .. "linear".
+$
+
+Ohne Stroeme ergibt sich dann 
+$
+  arrow(nabla)  * arrow(D) =  0 \ 
+  arrow(nabla)  *arrow(B) =  0 \ 
+  arrow(nabla)  times arrow(E) =  - partial _(t) arrow(B) \ 
+  arrow(nabla)  times arrow(H) =  partial _(t) arrow(D).
+$
+Fuer  $epsilon (arrow(r)) "und" mu (arrow(r)) "const."$ betrachte
+$
+  arrow(nabla)  * arrow(E) =  0 \ 
+  arrow(nabla)  *arrow(B) =  0 \ 
+  arrow(nabla)  times arrow(E) =  - partial _(t) arrow(B) \ 
+  arrow(nabla)  times arrow(B) = mu_(r) mu_(0) epsilon_(r) epsilon_0  partial _(t) arrow(E)\ 
+  ==> c = c_0 /n "mit" n = sqrt(epsilon_(r) mu_(r) ).
+$
+
+In der Optik ist $n$ als Brechungsindex wichtig auch wenn $n$ nicht konstant ist 
+$
+  n = n (arrow(r)) approx sqrt(epsilon (arrow(r))) \, space "da" mu_(r) approx 1.
+$
+Lichtausbreitung im Dielektrikum mit $rho_(f) = 0 "und" arrow(j)_(f) = 0$. Oft ist 
+$n (arrow(r)) "abschnittsweise konstant."
+$
+Dadurch ist dann die Loesung wieder trivial auf den einzelnen Abschnitten.
+
+= Herleitung der Wellengleichung in nicht homogenen Materialien 
+Es gilt
+$
+  arrow(nabla)  times (arrow(nabla) times arrow(E)) =  - mu_0 (arrow(nabla) times partial _(t) arrow(H) ) = - mu_0 partial _(t) (arrow(nabla) times arrow(H)) = ^("IV") - mu_0 epsilon_0 n^2 partial _(t) ^2 arrow(E) \ 
+  arrow(nabla)  times  (arrow(nabla)  times arrow(E)) =  arrow(nabla)  (arrow(nabla)  * arrow(E )) -  arrow(nabla) ^2 arrow(E) =  - mu_0 epsilon_0 n^2  (arrow(r)) partial _(t) ^2 arrow(E) \ 
+  arrow(nabla) * arrow(D) = 0 = epsilon_0  arrow(nabla) (n^2 *arrow(E)) = epsilon_0 [arrow(nabla) (n^2 ) * arrow(E) +  n^2 (arrow(nabla)  * arrow(E))] \ 
+  ==> arrow(nabla)  * arrow(E) =  -  1/n^2  (arrow(nabla)  n^2 * arrow(E) ).
+$
+Es folgt dann fuer die Wellengleichung 
+$
+  arrow(nabla)  ^2 arrow(E) +  arrow(nabla)  [1/(n (arrow(r))^2 ) (arrow(nabla) n^2 ) * arrow(E)] - epsilon_0 mu_0 n^2 (arrow(r)) (diff ^2 arrow(E)) / (diff t^2 ) = 0.
+$
+