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32 {\LARGE {\bf Materials Physics I}\\}
37 {\Large\bf Tutorial 3 - proposed solutions}
40 \section{Drude theory of metallic conduction}
42 \item $U=IR \Rightarrow EL=jA\rho\frac{L}{A}
45 \item distance: $v\,dt$
46 \item number of electrons crossing $A$: $n(v\,dt)A$
48 $\Rightarrow$ $j=\frac{I}{A}=\frac{dQ/dt}{A}=\frac{-e\,n(v\,dt)A/dt}{A}
51 \item In the absence of an electric field, electrons are as likely
52 to be moving in any one direction as in any other.
53 The velocity averages to zero.
54 As expected, according to the above equation, there is no
55 net electric current density.
56 \item Since electrons emerge in a random direction
57 there will be no contribution from the thermal velocity
58 to the average electronic velocity.
59 \item $v_{average}=at=\frac{F}{m}\tau=-\frac{eE}{m}\tau$
62 \item $j=\left(\frac{ne^2\tau}{m}\right)E$\\
63 \item $j=\sigma E \Rightarrow \sigma=\frac{ne^2\tau}{m}$
66 \item Energy transfer: $\frac{m}{2}v_{drift}^2$,
68 final drift velocity of the accelerated electron
69 \item $v_{drift}=-\frac{eE}{m}t_0$, $\quad t_0$:
70 free flight time (no collision) of the electron
71 \item $v_{average}=\frac{1}{t_0}\int_{0}^{t_0} v(t) dt
72 =-\frac{eE}{m}\frac{1}{t_0}[\frac{t^2}{2}]_{0}^{t_0}
73 =-\frac{eE}{m}\frac{t_0}{2}=:-\frac{eE}{m}\tau$,
74 $\qquad t_0=2\tau$, $v_{drift}=2v_{average}$
75 \includegraphics[width=12cm]{drude_v.eps}
76 \item Each of the $n$ electrons per unit volume
77 transfer the kinetic energy $\frac{1}{2}mv^2_{drift}$
78 once per $t_0$ to the lattice
81 \Rightarrow \frac{P}{V}=\frac{E_{kin}}{Vt_0}
82 =\frac{n\frac{1}{2}m\frac{e^2E^2}{m^2}t_0^2}{t_0}
83 =n\frac{1}{2}\frac{e^2E^2}{m}2\tau
84 =\sigma E^2=jE=j^2\rho=\frac{I^2}{A^2}\frac{A}{L}R
88 \Rightarrow P=I^2R \textrm{ (Joule heating)}