From: hackbard <hackbard@sage.physik.uni-augsburg.de>
Date: Thu, 22 Nov 2007 16:19:42 +0000 (+0100)
Subject: solution tutorial 3
X-Git-Url: https://hackdaworld.org/cgi-bin/gitweb.cgi?a=commitdiff_plain;h=24a37a3b2c9313eabf336293e51dee6e452af5b0;p=lectures%2Flatex.git

solution tutorial 3
---

diff --git a/solid_state_physics/tutorial/1_03s.tex b/solid_state_physics/tutorial/1_03s.tex
index 1174731..dd97aa2 100644
--- a/solid_state_physics/tutorial/1_03s.tex
+++ b/solid_state_physics/tutorial/1_03s.tex
@@ -41,8 +41,10 @@
 \begin{enumerate}
  \item $U=IR \Rightarrow EL=jA\rho\frac{L}{A}
              \Rightarrow E=j\rho$
- \item distance: $v\,dt$\\
-       number of electrons crossing $A$: $n(v\,dt)A$\\
+ \item \begin{itemize}
+        \item distance: $v\,dt$
+        \item number of electrons crossing $A$: $n(v\,dt)A$
+       \end{itemize}
        $\Rightarrow$ $j=\frac{I}{A}=\frac{dQ/dt}{A}=\frac{-e\,n(v\,dt)A/dt}{A}
                        =-nev$
  \item \begin{itemize}
@@ -60,12 +62,30 @@
 	\item $j=\left(\frac{ne^2\tau}{m}\right)E$\\
 	\item $j=\sigma E \Rightarrow \sigma=\frac{ne^2\tau}{m}$
        \end{itemize}
- \item Energy transfer: $\frac{m}{2}v_{drift}^2$,
-                        $\qquad v_{drift}$:
-			end drift velocity of the accelerated electron\\
-       $v_{drift} \ne v_{average}$
-
-       
+ \item \begin{itemize}
+        \item Energy transfer: $\frac{m}{2}v_{drift}^2$,
+                               $\quad v_{drift}$:
+		               final drift velocity of the accelerated electron
+	\item $v_{drift}=-\frac{eE}{m}t_0$, $\quad t_0$:
+	      free flight time (no collision) of the electron
+        \item $v_{average}=\frac{1}{t_0}\int_{0}^{t_0} v(t) dt
+	                  =-\frac{eE}{m}\frac{1}{t_0}[\frac{t^2}{2}]_{0}^{t_0}
+			  =-\frac{eE}{m}\frac{t_0}{2}=:-\frac{eE}{m}\tau$,
+              $\qquad t_0=2\tau$
+	\item Each of the $n$ electrons per unit volume
+	      transfer the kinetic energy $\frac{1}{2}mv^2_{drift}$
+	      once per $t_0$ to the lattice
+       \end{itemize}
+       \[
+       \Rightarrow \frac{P}{V}=\frac{E_{kin}}{Vt_0}
+                              =\frac{n\frac{1}{2}m\frac{e^2E^2}{m^2}t_0^2}{t_0}
+			      =n\frac{1}{2}\frac{e^2E^2}{m}2\tau
+			      =\sigma E^2=jE=j^2\rho=\frac{I^2}{A^2}\frac{A}{L}R
+			      =\frac{I^2R}{V}
+       \]
+       \[
+       \Rightarrow P=I^2R \textrm{ (Joule heating)}
+       \]
 \end{enumerate}
 
 \end{document}