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**Anna university - gtec**- 9 Topics
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- Electrical Properties of Materials - ( 1 - 6 )
- Quantum free electron theory - ( 7 - 17 )
- Super conductivity - ( 18 - 18 )
- BCS theory - ( 19 - 26 )
- Contents - ( 27 - 28 )
- Semiconductor Physics - ( 29 - 74 )
- Magnetic Properties of Materials - ( 75 - 81 )
- Optical Materials - ( 82 - 120 )
- Nanotecnology - ( 121 - 131 )

Topic:

(d) The electric current in a metal due to an applied field is due to drift of electrons in a direction opposite to the direction of the field. Q3. Explain/Define the terms: a) Thermal velocity b) Drift velocity c) Relaxation time c) Mean free path d) Mean collision time (a) Thermal velocity: A conductor consists of large number of free electrons of about 1029electrons/m3. Due to thermal energy these electrons are moving in between the ions with a speed of 106m/s and collide with ion cores of the conductor. After each collision velocity of the electron becomes zero. There after start moving in random direction. Thus in the absence of applied electric field, there is a kind of randomness in the motion of electrons. Though the free electrons are in motion, the net flow of current is zero or does not give rise to any current. “The average velocity with which the free electrons move inside the conductor due to thermal energy is called thermal velocity”. b) Drift velocity (vd): When an electric field is applied, an electric field is developed inside the conductor. As a result potential difference is developed between the ends of a conductor. The electrons start moving opposite to the field direction and collide with ion cores. After each collision velocity of electrons become zero; and again they gain velocity in a fresh direction but always opposite to the direction of applied electric field. Even though randomness exists; distance travelled as well as time of collision between the successive collisions is different. “The average velocity with which electrons move in a conductor under the influence of applied electric field is called drift velocity”. The expression for drift velocity. Consider a conductor of length „L‟ is subjected to an electric field E. In the steady state, conduction electrons are drifted opposite to the direction of applied electric field. If „m‟ is the mass of an electron, „vd‟ drift velocity, „τ‟ is the mean collision time, and then resistance force „Fr‟ offered to its motion is given by Fr =ma= mv d (1) If „e‟ is the charge on the electron, „E‟ is the electric field, then force experienced by electron due to applied electric field is F = eE (2) In the steady state F = Fr eE= mv d The drift velocity is given by 2 vd eE m

(c) Relaxation time (τr) : In the absence of electric field, the conduction electrons move in random direction, and hence the probability of finding an electron moving in any given direction is zero. i.e. Vav = 0 When an external field is applied, net positive value V1av for the average velocity of the conduction electrons in a direction opposite to the direction of field; which is equal to the drift velocity i.e Vav=V1av If the field is turned off, the average velocity Vav starts reducing exponentially as shown in the figure and is according to the equation. (1) Time counted from the instant the field is turned off, If = Relaxation time equation (1) becomes Hence relaxation time is defined as “The time interval during which drift velocity of electrons reduces to 1 e times the maximum value attained by them when applied field is turned off”. (d) Mean free path (λ): The average distance travelled by the conduction electrons between two successive collisions of conduction electrons under the influence of applied electric field is called mean free path. d) Mean collision time (τ): The average time interval between two consecutive collisions of an electron with the lattice cores in a conductor under the influence of applied electric field is called mean collision time. τ = λ/vth Where „λ‟ is the mean free path, v≈vth is velocity same as combined effect of thermal & drift velocities. Note: But ½ mvth2=3/2 KT Question (4): Discuss the failures of classical free electron theory. Classical free electron theory has failed to explain specific heat, dependence of conductivity with temperature and dependence of conductivity on electron concentration as follows (a) Specific heat: 3

As per classical free electron theory, free electrons in a metal behave as gas molecules and 3 hence the molar specific heat of electrons at constant volume is given by Cv= R 2 Where R is universal gas constant But experimentally molar specific heat of free electrons in a metal is given by CV=10-4RT This is not only less than the experimental value but also depends on temperature. Hence classical free electron theory failed to explain specific heat. (b) Dependence of electrical conductivity on Temperature: According to the assumptions of classical free electron theory √ √ ---------------------- (1) The mean collision time „τ‟ is inversely proportional to the thermal velocity. (vth= ) i.e. √ From the expression ------ ------------------------- (2) From (1) σ= ne 2 -------------- (3) m Substituting for from equation (3) gives 1 σα -----------------------(4) T Therefore according to classical theory, electrical conductivity is inversely proportional to the square root of absolute temperature. But experimentally σ is inversely proportional to the temperature T. i.e. cxpt 1 T ----------- (5) Therefore classical theory failed to explain conductivity with temperature. (c) Dependence of electrical conductivity on electron concentration: According to classical theory, electrical conductivity is directly proportional to electron concentration i.e σ = ne 2 m σαn Thus as n increases conductivity should increase. But it is contrary to the observation. Consider the data from the following table. 4

Metals Electron Con.(n) in/m3 Conductivity in Ω-1m-1 Cu 8.45×1028 5.88×107 Ag 5.85×1028 6.3×107 Al 18×1028 3.67×107 Electrical conductivity of aluminium is lesser than Copper (Cu) and Silver (Ag); even though electron concentration in Al is higher than that of Cu and Ag. Similar observations are made with Cu and Al. Therefore σ α n do not holds good. Hence the classical free electron theory failed to explain the dependence of σ on electron concentration. Problems on classical free electron theory 1. Calculate drift velocity and thermal velocity of electrons in a metal of thickness 1mm across which a potential difference of 1volt is applied at temperature 300K.Compare this value with thermal velocity of electrons. Given that mobility of electrons is 40cm2/VS. Solution: Vd=?, Vth= ?, L=1mm=1X10-3m; V=1volt; T=300K 40cm 2 /VS 40X 104 m2 /VS E V 1 1000V L 1X 103 Vd E =40x10-4x1000=4 ms-1 is drift velocity------(1) Vth 3KT m 3 X 1.38X 10 23 X 300 1.17X 105 mS 1 ----(2) 9.1X 1031 Vd 4 3.14 x10 5 ms 1 Vth 1.17 x10 5 2. A uniform silver wire has a resistivity of 1.54X10-8 ohm-m at room temperature. across which an electric field of 1 volt/cm is applied. Calculate (i) relaxation time (ii) drift velocity and (iii) mobility of electrons; assuming that there are 5.8X10-28 electrons per m-3of the metal. (VTU Jun 2010) Solution: 1.54X10-8 ohm-m; E=1 volt/cm=1volt/1x10-2m=100V/m; n=5.8X10-28 electrons per m-3; Vd=? ?; =? (i) Relaxation time Consider m ne2 m ne2 ne2 m 9.1x1031 3.985x1014 S 8 28 19 2 1.54x10 x5.8 x10 (1.6 x10 ) (ii) Drift velocity 19 eE 1.6 x10 X 100 Vd x3.98 x10 14 ms 1 0.699.ms 1 31 9.1x10 m 5

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