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32 {\LARGE {\bf Materials Physics I}\\}
37 {\Large\bf Tutorial 3}
40 \section{Drude theory of metallic conduction}
41 {\bf Motivation:} In the following excercise we will reconsider once more the
42 Drude theory of metals.
43 We will end up with an expression for the electrical conductivity of a metal.
44 In addition we will deduce the expression of power loss
45 for current flowing in a wire.
47 {\bf Our understanding of condensed matter} is based on the notion of a solid
48 being composed of heavy, positively charged ions
49 and light, negatively charged valence electrons.
50 The ions consist of the nuclei and core electrons tightly bound to the nuclei
51 which thus do not contribute to the metallic conductivity.
52 The mobile valence electrons on the other hand are responsible for the
53 electrical and thermal conductivity of the metal.
55 {\bf The basic assumptions of the Drude model} are presented in the following.
56 Basically the theory is constructed by applying the kinetic theory of gases
57 to a metal, considered as a gas of free non-interacting valence electrons.
58 Briefly outlined, the models assumptions are mentioned:
60 \item Between collisions:\\
61 Independent electron approximation
62 $\rightarrow$ no electron-electron interaction\\
63 Free electron approximation
64 $\rightarrow$ no electron-ion interaction
65 \item Electrons collide with the large heavy ions.
66 Collisions are instantaneous events abruptly altering the velocity of
67 an electron and randomly changing its direction.
68 \item On average, electrons travel for a time $\tau$
69 before its next collision.\\
70 $\Rightarrow$ Probability of a collision for an electron in an
71 infinitesimal time interval $dt$ is $dt/\tau$.
72 \item Thermal equilibrium achieved by collisions only.\\
73 $\Rightarrow$ Electron's speed after collision determined
74 according to local temperature.
77 Consider a wire of length $L$ and cross-sectional area $A$.
78 The wire has a resistance $R$.
81 \item According to Ohm's law ($U=IR$) the current $I$ flowing in that wire
82 is proportional to the potential drop $U$.
83 The resistance depends on the shape of the wire ($R=\rho\frac{L}{A}$).
84 Rewrite Ohm's law eliminating this dependence using
85 the resitivity $\rho$ which is only characterized by the metal.
86 {\bf Hint:} $U=EL$ is the potential drop along the wire
88 and $j=I/A$ is the current density.
89 \item Find an expression for the current density if $n$ electrons
90 per unit volume move with velocity $v$.
91 {\bf Hint:} What distance the electrons travel in a time $dt$?
92 How many electrons will cross an area $A$ perpendicular to the
93 direction of flow in a time $dt$?
94 Remember that the current $I$ is the derivative of charge $Q$
96 \item What is the average velocity of the electrons in the absence
98 What does this mean for the contribution of the
99 thermal electronic velocity after a collsion
100 to the average electronic velocity?
101 Find an expression for the electric field dependent
102 average electronic velocity.
103 \item Rewrite the current density using the average electronic velocity
104 and find an expression for the conductivity $\sigma=1/\rho$.
105 \item The resistance is caused by collisions of the electrons
107 Energy is not conserved in the collisions.
108 Find an expression for the power loss in the considered wire.