From: hackbard 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}