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120
S3/ExPhyIII/VL/ExIIIVL5.typ
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120
S3/ExPhyIII/VL/ExIIIVL5.typ
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// Main VL template
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#import "../preamble.typ": *
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// Fix theorems to be shown the right way in this document
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#import "@preview/ctheorems:1.1.3": *
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#show: thmrules
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// Main settings call
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#show: conf.with(
|
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// May add more flags here in the future
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num: 5,
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type: 0, // 0 normal, 1 exercise
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date: datetime.today().display(),
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//date: datetime(
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// year: 2025,
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// month: 5,
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// day: 1,
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//).display(),
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)
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= Uebersicht
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Dispersion
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$
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omega &= c k \
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omega &= c (k ) k.
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$
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Die Geschwindigkeit aus der WEllengleichung wird Phasengeschwindigkeit genannt.
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Femtosekundenlaser durch Inteferenz von monochromatischem Licht. Im Vakuum gibt es keine Dispersion.
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Verallgeminerung: Ausbreitung eines Pules
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$
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psi (x, t) &= integral_(- oo)^(oo) underbrace(A (k) e ^(i (k x - omega (k)t)), "Darstellung im Fourierraum") dif k \
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omega (k) &approx omega (k_0 ) + (k - k_0 ) (diff omega) / (diff k) (k_0 ) \
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psi (x, t) &= integral_(- oo)^(oo) dif k A (k) e ^( i (k_0 x - omega (k_0 )t) - (k - k_0 )(diff omega) / (diff t) t) \
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&= underbrace(e ^( i (k_0 x - omega ( k_0) t )), "monochromatische Welle bewegt sich mit" c_(p) ) integral_(- oo)^(oo) dif k A (k) e ^( i [(k - k_0 )(x - (diff omega) / (diff k) t)]) \
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c &= (diff omega) / (diff k) \
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c_(p) &= omega_0 /k_0.
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$
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Dispersionsrelation mit nicht konstantem $c$
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$
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omega &= c (k) k \
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c^2 &= [underbrace(g/h, "grosse Welle") + underbrace((sigma k) / (rho), "kleine Welle")] tan (k h) \
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c &= sqrt(T/rho),
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$
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wobei $h$ die Wassertiefe und $sigma$ die Oberflaechenspannung ist.
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Wo sind die Wellenzahlen gleich gross
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$
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g/h &= (Delta k_(c)) / (rho) \
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==> lambda_(c) &= 2 pi sqrt(sigma/(rho g)) \, space O ("cm").
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$
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In der zum Beispiel Nordsee
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$
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"fuer" lambda >> lambda_(c) \, space k << k_(c) \
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"fuer" h k << 1 \
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==> c^2 approx g h \, space k := "const".
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$
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In der Tiefsee
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$
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h k >> 1 \
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c = sqrt(g/k) \, space omega = c k = sqrt(g k) \
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c_(g) = (diff omega) / (diff k) = 1/2 sqrt(g/k)= 1/2 c_(p).
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$
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#theorem[
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Navier-Stokes Gleichung
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$
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rho (partial _(t) arrow(v) + arrow(v) * arrow(nabla) arrow(v)) = rho arrow(g) - arrow(nabla) p.
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$
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]
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= Von der Maxwellgleichung zur Wellengleichung
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Faradaysche und Amperesche Gesetz
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$
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arrow(nabla) times arrow(E) = - (diff arrow(B)) / (diff t) \, space arrow(nabla) times arrow(B) = epsilon_0 mu_0 (diff arrow(E)) / (diff t) \
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underbrace(arrow(nabla) times arrow(nabla) times arrow(E), = arrow(nabla) (underbrace(arrow(nabla) * arrow(E), = 0\, "da" rho = 0))- arrow(nabla) ^2 arrow(E)) = arrow(nabla) times (- (diff arrow(B)) / (diff t) ) = - diff / (diff t) ( arrow(nabla) times arrow(B))= - epsilon_0 mu_0 (diff ^2 arrow(E)) / (diff t^2 ) \
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==> arrow(nabla) ^2 arrow(E) - epsilon_0 mu_0 (diff ^2 arrow(E)) / (diff t^2 ) = 0 \
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arrow(B) (arrow(r), t) "analog" arrow(nabla) ^2 arrow(B) - epsilon_0 mu_0 (diff ^2 arrow(B)) / (diff t^2 ) = 0 \
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==> c_0 = 1/sqrt(epsilon_0 mu_0 ) approx 2.9 * 10 ^(8) "m"/"s" \
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arrow(nabla) ^2 phi - 1/c^2 (diff ^2 phi) / (diff t^2 ) = 0 .. forall phi in {E_(x) , E_(y) , E_(z) , B_(x) , B_(y) , B_(z) }.
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$
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Spezialfall ebene Welle zum Beispiel Wellenvektor $k$ ist $k hat(z)$. Welle ist konstant in $x$ und $y$ $forall z, t$ $==>$ Ebene senkrecht auf $z$ haben konstante Phase und Amplitude
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$
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arrow(E) = arrow(E) (z, t) .. "aus" arrow(nabla) * arrow(E) = 0 "folgt" partial _(z) E_(z) = 0 \
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==> E_(z) "const". \
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arrow(E)_(0) = vec(E_(x), E_(y) , 0) \, space arrow(E) (z, t) = arrow(E)_(0) e ^(i (k z - omega t)) \
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"mit" partial _(x) ^2 arrow(E) = partial _(y)^2 arrow(E)= 0 .. "wird die WG" \
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partial _(z) ^2 arrow(E) - 1/c^2 partial _(t) ^2 E= 0.
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$
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Allgemeiner
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$
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arrow(E) = arrow(E)_(0) e ^(i (arrow(k)* arrow(r) - omega t)) \, space arrow(k) = vec(k_(x) , k_(y) , k_(z) ) = (2 pi)/lambda hat(k) \
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arrow(nabla) ^2 arrow(E) - 1/c^2 partial _(t) ^2 arrow(E) = 0 \, space "3 WG" \
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"fuer jede Komponente" psi = psi_0 e ^(i (arrow(k)* arrow(r) - omega t)) \
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==> partial _(t) ^2 psi = - omega ^2 psi_0 e ^(i (arrow(k)* arrow(r) - omega)) \
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arrow(nabla) ^2 psi = psi_0 (i arrow(k))^2 e^(i (arrow(k) arrow(r) - omega t)) \
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==> - k^2 + omega^2 /c^2 = 0 ==> c = omega/k.
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$
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Das Magnetfeld der ebenen Welle. Sei
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$
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arrow(E) = underbrace(arrow(E)_(0) hat(x), "Polarisation") e ^(i (k z - omega t)) \
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==> arrow(nabla) times arrow(E) = (diff arrow(B)) / (diff t) \, space arrow(B) = arrow(B) (arrow(r), t) \
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- abs(mat( hat(x), hat(y), hat(z); partial _(x) , partial _(y) , partial _(z) ; E_(x) , 0, 0; )) = - (partial _(z) E_(x) ) hat(y) = (diff B_(y) ) / (diff t) \
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(diff B_(y) ) / (diff t) = - (diff E_(x) ) / (diff z) = - i k E_0 e ^(i (k z - omega t)) \
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==> B_(y) = k/omega E_0 e ^(i (k z - omega t)) = 1/c E_0 e ^(i (k z - omega t))
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$
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#figure(
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image("typst-assets/drawing-2025-11-14-09-19-41.rnote.svg"),
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)
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Es folgt
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$
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arrow(B) = 1/c^2 (arrow(k) times arrow(E)).
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$
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Warum dreht sich die Reflektion um in einem Koaxialkabel?
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94
S3/ExPhyIII/VL/ExIIIVL6.typ
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94
S3/ExPhyIII/VL/ExIIIVL6.typ
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@@ -0,0 +1,94 @@
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// 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(),
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)
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= Uebersicht
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== Wiederholung der Maxwellschen Gleichungen
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Anschreiben der 5 Gleichungen.
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Unterscheidung zwischen freien makroskopischen und atmoaren/molekuralen mikroskopischen Ladungen & Stroemen. Es gilt
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$
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D = epsilon_0 E + P = epsilon_(r) epsilon_0 E.
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$
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Es gilt
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$
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arrow(nabla) times (arrow(nabla) times arrow(E)) = arrow(nabla) (arrow(nabla) * E) - arrow(nabla) ^2 E.
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$
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Es ergeben sich die Wellengleichungen
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$
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arrow(nabla) ^2 arrow(E) = mu_0 epsilon_0 (diff ^2 arrow(E)) / (diff t^2 ) \, space arrow(nabla) ^2 arrow(E) = mu_0 epsilon_0 (diff arrow(B)) / (diff t^2 ) \, space c = 1/sqrt(mu_0 epsilon_0 ) = 2.9979 * 10^(8) "m"/"s".
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$
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Flouresenz von Kastaniensaft.
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= Energie und Impuls elektromagnetischer Wellen
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Energiedichte in der Elektrostatik
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$
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mu = epsilon/v = (epsilon_0 E^2 ) / (2) + 1/ (2 mu_0 ) B^2
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$
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fuer eine ebene Welle wissen wir, dass
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$
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B = E/c ==> B^2 = epsilon_0 m_0 E^2 ==> mu = epsilon_0 E^2.
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$
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Zeitliche Anederung
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$
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partial _(t) mu (arrow(r), t) &= epsilon_0 arrow(E) * dot(arrow(E) )+ 1/mu_0 arrow(B)*dot(arrow(B)) \
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epsilon_0 dot(arrow(E)) &= 1/mu_0 arrow(nabla) times arrow(B) - arrow(j) \
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==> ^("MG (III)") dot(arrow(B)) &= - arrow(nabla) times arrow(E) \
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==> partial _(t) mu &= 1/mu_0 arrow(E)* (arrow(nabla) times arrow(B)) - arrow(E) * arrow(j) - 1/mu_0 arrow(B) * (arrow(nabla) times arrow(E)) \
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partial _(t) mu &= - 1/mu_0 arrow(nabla) * (arrow(E)times arrow(B)) - arrow(E) * arrow(j).
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$
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Betrachte mechanische Energie der Ladungsverschiebung
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$
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dif W = arrow(F) * dif arrow(s) = q arrow(E) * arrow(v) dif t \, space q = integral _(V) rho dif V \, space rho arrow(v) = arrow(j) \
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==> (dif W) / (dif t) = integral _(V) arrow(E) * arrow(j) dif^3 arrow(r) \, space partial _(t) mu_("mech") = dot(arrow(E)) * arrow(j) \
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mu_("total") = mu + mu_("mech") \, space partial _(t) mu_("total") = - 1/mu_0 arrow(nabla) * (arrow(E) times arrow(B)) .. "Poynting theorem".
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$
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Dies ist analog zur Kontinuitaetsgleichung mit der Ladungserhaltung
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$
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partial _(t) rho = - arrow(nabla) times arrow(j) \
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arrow(s):= 1/mu_0 arrow(E)times arrow(B) .. "Energiedichte als Poyntingvektor".
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$
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Fuer die ebene Welle gilt dann
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$
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arrow(s) = 1/mu_0 E B hat(k) = 1/(mu_0 c)E^2 hat(k) \
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arrow(s)/c = epsilon_0 E^2 hat(k) .. "hat die Einheit der Energiedichte" ==> arrow(s) = underbrace(mu, "Energiedichte") c hat(k) \, space [s] = "J"/("m"^2 "s" ) \
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I = lr(angle.l arrow(s) angle.r)_(t) .. "ueber eine Periode gemittelt" \
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"Impulsstromdichte" arrow(s)/c.
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$
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In der Photonenoptik als Funfact (Einschub) fuer Lichtteilchen und Photonen
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$
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E^2 = m^2 c^(4) + p^2 c^2 ==> ^(m = 0) p = E/c.
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$
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Eine ebene Welle wird an einer Flaeche reflektiert. Nun interessieren wir und fuer den Strahlungsdruck $P_("rad") $ und fuer die Strahlungskraft $F_("rad") $
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$
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arrow(F)= (dif arrow(p)) / (dif t) \, space arrow(F)_("rad") = - abs(arrow(s))/c A (hat(e)_(1) - hat(e)_(0) ) \, space P_("rad") = lr(angle.l abs(arrow(s))/c angle.r) = I/c.
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$
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Kraft eines Laserstrahls mit Leistung $W$
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$
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F ["N"] = (P ["W"]) / (3 * 10 ^(8) ) = 3.3 * 10^(- 12) P ["mW"].
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||||
$
|
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Ein Laserpointer hat $W approx 1"mW"$. Es geht also eine Kraft von 3 Piconewton raus.
|
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Experiment der Lichtmuehle.
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102
S3/ExPhyIII/VL/ExIIIVL7.typ
Normal file
102
S3/ExPhyIII/VL/ExIIIVL7.typ
Normal file
@@ -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
|
||||
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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)) \
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||||
arrow(nabla) * (epsilon_0 arrow(E) + arrow(P)) = rho_(f) \
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||||
arrow(nabla) * arrow(D) = phi_(f) \
|
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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.
|
||||
$
|
||||
|
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175
S3/ExPhyIII/VL/ExIIIVL8.typ
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S3/ExPhyIII/VL/ExIIIVL8.typ
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// Main VL template
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#import "../preamble.typ": *
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// Fix theorems to be shown the right way in this document
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#import "@preview/ctheorems:1.1.3": *
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#show: thmrules
|
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// Main settings call
|
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#show: conf.with(
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// May add more flags here in the future
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num: 5,
|
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type: 0, // 0 normal, 1 exercise
|
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date: datetime.today().display(),
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//date: datetime(
|
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// year: 2025,
|
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// month: 5,
|
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// day: 1,
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//).display(),
|
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)
|
||||
|
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= Uebersicht
|
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== Reflexion und Brechung
|
||||
|
||||
Wechselstroeme und das Zeigerdiagramm
|
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$
|
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I (t) = I_0 cos [omega t + phi_(I) ] \
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U (t) = U_0 cos [ omega t + phi _(U) ] \
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Z = a + i b = abs(z) exp(i phi).
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$
|
||||
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Wiederholung von Kombinantion von Wechselstromwiederstaenden und deren Impedanz
|
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$
|
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abs(Z) = sqrt(R^2 + (omega L - 1/ (omega C))^2 ) \
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tan phi = (omega L - 1/(omega C)) / (R) = (Im Z) / (Re Z).
|
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$
|
||||
|
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Es gilt
|
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$
|
||||
(diff V) / (diff x) = - ((diff L) / (diff x) )(diff I) / (diff t) \, space (diff I) / (diff x) = - ((diff C) / (diff x) )(diff V) / (diff t) \
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(diff ^2 V) / (diff t^2 ) = - 1/C_0 diff / (diff x) (diff I) / (diff t) = - 1/C_0 diff / (diff x) (- 1/L_0 (diff V) / (diff x) )= 1 / (L_0 C_0 )(diff ^2 V) / (diff x^2 ) \, space c = 1/sqrt(L_0 C_0 ) \
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partial _(x) I = - C_0 partial _(t) V ==> I_(+) = I_(0 +) e^(i (k x - omega t)) \, space V_(+) = V_(0 +) e^(i (k x - omega t)) \
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I_(0 +) i k = - C_0 (i omega) V_(0 +) \
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==> (V_(0 +) ) / (I _(0 +) ) = k/omega 1/C_0 = 1/c 1/C_0 = sqrt(L_0 C_0 )1/C_0 = sqrt((L_0 ) / (C_0 ) ) = Z_0.
|
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$
|
||||
|
||||
= Koaxialkabel
|
||||
Es gilt
|
||||
$
|
||||
E = 1/(2 pi epsilon_0 ) (lambda) / (r) = 1/(2 pi epsilon_0 ) (Q) / (L r) \
|
||||
Delta V = V_(b) - V_(a) \
|
||||
dif C = (2 pi epsilon_0 ) / (ln (R/r)) dif l \
|
||||
dif L = (mu mu_0 ) / (2 pi) ln (R/r) dif l.
|
||||
$
|
||||
|
||||
Wir haben Wellen der Form
|
||||
$
|
||||
I, U, arrow(E), arrow(B).
|
||||
$
|
||||
Fuer die Impedanz gilt dann
|
||||
$
|
||||
Z := sqrt(L/C) \, space [Z]= Omega.
|
||||
$
|
||||
Die Grenzflaeche ist dann zwischen $Z_(1) $ und $Z_2 $. Es gilt fuer die drei Stroeme
|
||||
$
|
||||
I = I_(1 + ) + I _(1 - ) = I _(2 + ) \
|
||||
U = U_(1 + ) + U_(1 - ) = Z_(1) I _(1 + ) - Z_(1) I_(1 - ) = U _(2 + ) = Z_(2) I_(2 + ) \
|
||||
==> I_(2 + ) = I _(1 + ) + I _(1 - ) \, space Z_(2) I_(2 + ) = Z_(1) I _(1 + ) + Z_(1) I _(1 - ) \
|
||||
==> 1 + (I_(1 - ) ) / (I _(1 + ) ) = (I _(2 + ) ) / (I _(1 + ) ) := underbrace(t, "Transmission") = 1 + underbrace(r, "Reflexion") \
|
||||
Z_(1) - Z_(2) (I_(1 - ) ) / (I _(1 + ) ) = Z_(2) (I_(2 + ) ) / (I _(1 + ) ) \
|
||||
==> r = (Z_(1) - Z_(2) ) / (Z_(1) + Z_(2) ) \, space t = (2 Z_(1) ) / (Z_(1) + Z_(2) ).
|
||||
$
|
||||
|
||||
= Dopplereffekt
|
||||
|
||||
Fuer Schall
|
||||
$
|
||||
f' = (1 +- v_(e) /c) / (1 minus.plus v_(s) /c) f_0,
|
||||
$
|
||||
wobei $+ $ aufeinander zu und $- $ voneinander weg ist. Auch ist $v_(e) $ die Geschwindigkeit des Empfaengers und $v_(s) $ die des Senders.
|
||||
Lorentztransformation impliziert
|
||||
$
|
||||
f = sqrt((c +- v) / (c minus.plus v) ) f_0.
|
||||
$
|
||||
Fuer die Wellenlaenge $lambda$
|
||||
$
|
||||
lambda = c T \, space f_0 = c /T \
|
||||
lambda' = (c + v_(s) ) T \, space lambda'' = (c - v_(s) ) T \
|
||||
f' = c/lambda' = (c) / ((c + v_(s) )T) = c / (v + v_(s) )f_0 ==> f' = 1/(1 +- v_(s) /c) f_0.
|
||||
$
|
||||
|
||||
Fuer den bewegten Empfaenger
|
||||
$
|
||||
T' = (lambda) / (c + v_(e) ) = T_0 /(1 + v_(e) /c) \
|
||||
==> f'= (1 + v_(e) /c) f_0 \
|
||||
f' = (1 +- v_(e) /c) / (1 minus.plus v_(s) /c) f_0.
|
||||
$
|
||||
Fuer Relativitaet
|
||||
$
|
||||
x' = gamma (x - v t) \, space y' = y \, space z' = z \, space t' = gamma (t - (x v) / (c^2 ) ) \, space gamma = (1) / (sqrt(1 - v^2 /c^2 )) \
|
||||
( c Delta t - v Delta t)/N \, space f = c/lambda = (c N) / (c Delta t - v Delta t) \
|
||||
N = f_0 Delta t' \, space Delta t' = Delta t = (Delta t)/gamma \
|
||||
f = (c) / (c Delta t - v Delta t) f_0 (Delta t) / (gamma) = 1/ (1 - v/c) f_0 /gamma = sqrt((1 + v/c)/(1 - v/c)) f_0
|
||||
$
|
||||
|
||||
= Brechung und Reflexion
|
||||
Betrachte Wellen
|
||||
$
|
||||
psi_(1) = a_1 e ^(i (arrow(k)_(1) arrow(r) - omega t)) \
|
||||
psi_(r) = a_(r) e^(i (arrow(k)_(r) arrow(r) - omega _(r) t))
|
||||
psi_(2) = a_(2) e^(i (arrow(k)_(2) arrow(r) - omega _(2) t)).
|
||||
$
|
||||
Das Ziel ist die Bestimmung der Geometrie. Also $Theta_(2) , Theta_(1) "und" Theta_(r) $ und $R = abs((a_(r) ) / (a_(1) ) )^2 \, space R = R (n_1, n_2, Theta_(1) )$ sowie $T = abs(a_2 /a_1 )^2 \, space T = T (n_1, n_2, Theta_(1) ) $. Es gilt dabei $R = abs(r)^2 \, space T = abs(t)^2 $.
|
||||
Alles folgt aus den Randbedingungen. So ist $psi$ hier stetig und
|
||||
$
|
||||
arrow(nabla) psi "stetig an der Grenzflaeche" z = 0.
|
||||
$
|
||||
|
||||
Stetigkeit von $psi$ impliziert
|
||||
$
|
||||
a_1 e^(i (arrow(k)_(1) arrow(r) - omega t)) + a_(r) e^(i (arrow(k)_(r) arrow(r) - omega_(r) t)) = a_(2) e^(i (arrow(k)_(2) arrow(r) - omega_(2) t)) space forall arrow(r) = vec(x, y, 0) space forall t \
|
||||
==> omega := omega_(1) = omega_(r) = omega_(2).
|
||||
$
|
||||
Es muessen also auch die Tangentialkomponenten von $arrow(k)$ gleich sein, also gilt auch
|
||||
$
|
||||
k_(1) ^((x)) = k_(r) ^((x)) = k_(2) ^((x)).
|
||||
$
|
||||
Also
|
||||
$
|
||||
k_(1) ^((x)) = k n_(1) sin Theta_(1) = k n_(1) sin Theta_(r) \
|
||||
==> Theta_(1) = Theta_(r) "Reflexionsgesetz".
|
||||
$
|
||||
Aus $k_(1) ^((x)) = k_(2) ^((x)) $ folgt
|
||||
$
|
||||
k n_(1) sin Theta_(1) = k_(2) n_(2) sin Theta_(2) \
|
||||
==> n_1 / n_2 = (sin Theta_(2) ) / (sin Theta_(1) ).
|
||||
$
|
||||
Mit $k = (2 pi )/lambda$ im Vakuum. Auch
|
||||
$
|
||||
a_1 + a_(r) = a_(2).
|
||||
$
|
||||
Aus der Stetigkeit des Gradienten folgt
|
||||
$
|
||||
arrow(k)_(1) psi_(1) + arrow(k)_(r) psi_(r) = arrow(k)_(2) psi_(2) \
|
||||
==> k n_1 a_1 sin Theta_(1) + k n_1 a_1 sin Theta_(1) = k n_2 a_2 sin Theta_2 \
|
||||
==> (a_1 + a_(2) ) n_1 sin Theta_1 = a_2 n_2 sin Theta_2 "wieder der Snellius".
|
||||
$
|
||||
Die senkrechte Komponenente liefert
|
||||
$
|
||||
- k n_1 a_1 cos Theta_1 + k n_1 a_(r) cos Theta_1 = - k n_2 a_2 cos Theta_2 \
|
||||
==> n_1 cos Theta_1 (a_1 - a_(r) ) = n_2 (a_1 + a_(r) )cos Theta_2 \
|
||||
( 1 - a_(r) /a_(1) ) = n_2 /n_1 (cos Theta_2 )/(cos Theta_1 ) (1 + a_(r) /a_(1) ) "mit" r := a_(r) /a_(1) ==> (1 - sigma) = n_2 /n_1 (cos Theta_2 ) / (cos Theta_1 ) space "?"
|
||||
$
|
||||
|
||||
Also
|
||||
$
|
||||
1 - n_2 /n_1 (cos Theta_2 ) / (cos Theta_1) = r (n_2 /n_1 (cos Theta_(2) ) / (Theta_(1) ) + 1) \
|
||||
==> r = (n_2 cos Theta_1 - n_2 cos Theta_2 ) / (n_1 cos Theta_(1) + n_2 cos Theta_(2) ) \
|
||||
==> t = a_(2) /a_(1) = (a_1 + a_(r) ) / (a_1 ) = 1 + r \
|
||||
==> t = (2 n_1 cos Theta_1 ) / (n_1 cos Theta_1 + n_2 cos Theta_2 ).
|
||||
$
|
||||
Jetzt als Spezialfall den senkrechten Einfall
|
||||
$
|
||||
Theta_1 = 0 degree \, space r = (n_1 - n_2 ) / (n_1 + n_2 ) \
|
||||
"fuer" n_1 = 1 \, space n_2 = 1.5 space ("Glas") \
|
||||
==> r = (- 0.5) / (2.5) = - 0.2 \, space R := r^2 = 0.04.
|
||||
$
|
||||
|
||||
Eine Grenzflaeche mit Normalen in zwei Medien mit Brechungsindizes $n_1 "und" n_2 $
|
||||
$
|
||||
Theta_1 = Theta_(c) <==> Theta_(2) = 90 degree.
|
||||
$
|
||||
Ist der Snellius fuer $Theta > Theta_(c) $ erfuellbar? Gibt es Verhaeltnisse fuer die der Reflexionskoeffizient $R = abs(r)^2 = 1$ sein kann?
|
||||
|
||||
#highlight[TODO: Totalreflexion herleiten]
|
||||
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S3/ExPhyIII/VL/typst-assets/drawing-2025-11-14-09-19-41.rnote
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Reference in New Issue
Block a user