1. 20. 7. 20031 II–4 Microscopic View of Electric Currents

2. 20. 7. 20032 Main Topics The Resistivity and Conductivity. Conductors, Semiconductors and Insulators. The Speed of Moving Charges. The Ohm’s Law in Differential Form. The Classical Theory of Conductivity. The Temperature Dependence of Resistivity The Thermocouple

3. 20. 7. 20033 The Resistivity and Conductivity I Let’s have an ohmic conductor i.e. the one which obeys the Ohm’s law: V = RI The resistance R depends both on the geometry and the physical properties of the conductors. If we have a homogeneous conductor of the length l and the cross-section A we can define the resistivity  and its reciprocal the conductivity  by:

4. 20. 7. 20034 The Resistivity and Conductivity II The resistivity is the ability of materials to defy the electric current. With the same geometry a stronger field is necessary if the resitivity is high to reach a certain current. The SI unit of resistivity is 1  m. The conductivity is the ability to conduct the electric current. The SI unit of conductivity is 1  -1 m -1. A special unit siemens exists 1 Si =  --1.

5. 20. 7. 20035 Mobile Charge Carriers I Generally, they are charged particles or pseudo-particles which can move freely in conductors. They can be electrons, holes or various ions. The conductive properties of materials depend on how freely their charge carriers can move and this depends on deep structure properties of the particular materials.

6. 20. 7. 20036 Mobile Charge Carriers II E.g. in solid conductors each atom shares some of its electrons, those least strongly bounded, with the other atoms. In zero electric field these electrons normally move chaotically at very high speeds and undergo frequent collisions with the array of atoms of the solid. It resembles thermal movement of gas molecules  electron gas.

7. 20. 7. 20037 Mobile Charge Carriers III In non-zero field the electrons also have some relatively very low drift speed in the opposite direction then has the field. The collisions are the predominant mechanism for the resistivity (of metals at normal temperatures) and they are also responsible for the power loses in conductors.

8. 20. 7. 20038 Differential Ohm’s Law I Let us again have a conductor of the length l and the cross-section A and consider only one type of charged carriers and a uniform current, which depends on their: density n i.e. number in unit volume charge q drift speed v d

9. 20. 7. 20039 Differential Ohm’s Law II Within some length  x of the conductor there is a charge:  Q = n q  x A The volume which passes some plane in 1 second is A  x/  t = v d A so the current is: I =  Q/  t = n q v d A = j A Where j is so called current density. Using Ohm’s law and the definition of the conductivity: I = j A = V/R = El  A/l  j =  E

10. 20. 7. 200310 Differential Ohm’s Law III j =  E This is Ohm’s law in differential form. It has a similar form as the integral law but it contains only microscopic and non- geometrical parameters. So it is a the starting point of theories which try to explain conductivity. Generally, it is valid in vector form:

11. 20. 7. 200311 Differential Ohm’s Law IV Its meaning is that the magnitude of the current density is directly proportional to the field and that the charge carriers move along the field lines. For deeper insight it is necessary to have at least rough ideas about the magnitudes of the parameters involved in the Ohms law.

12. 20. 7. 200312 An Example I Let us have a current of 10 A running through a copper conductor with the cross- section of 3 10 -6 m 2. What is the charge density and drift velocity if every atom contributes by one free electron? The atomic weight of Cu is 63.5 g/mol. The density  = 8.95 g/cm 3.

13. 20. 7. 200313 An Example II 1 m 3 contains 8.95 10 6 /63.5 = 1.4 10 5 mol. If each atom contributes by one free electron, this corresponds to n = 8.48 10 28 electrons/m 3. 10/(8.48 10 28 1.6 10 -19 3 10 -6 ) = 2.46 10 -4 m/s

14. 20. 7. 200314 The Internal Picture The drift speed is extremely low. It would take the electron 68 minutes to travel 1 meter! In comparison, the average speed of the chaotic movement is of the order of 10 6 m/s. So we have currents of the order of 10 12 A running in random directions and so compensating themselves and relatively a very little currents caused by the field. It is similar as in the case of charging something a very little un-equilibrium.

15. 20. 7. 200315 A Quiz The drift speed of the charge carriers is of the order of 10 -4 m/s. Why it doesn’t take hours before a bulb lights when we switch on the light?

16. 20. 7. 200316 The Answer By switching on the light we actually connect the voltage across the wires and the bulb and thereby create the electric field which moves the charge carriers. But the electric field spreads with the speed of light c = 3 10 8 m/s, so all the charges start to move (almost) simultaneously.

17. 20. 7. 200317 The Classical Model I Let’s try to explain the drift speed using more elementary parameters. We suppose that during some average time between the collision  the charge carriers are accelerated by the field. And non-elastic collision stops them. Using what we know from electrostatics: v d = qE  /m

18. 20. 7. 200318 The Classical Model II We substitute the magnitude of the drift velocity into the formula for the current density: j = n q v d = n q 2  E/m So we obtain conductivity and resistivity:  = n q 2  /m  = 1/  = m/nq 2 

19. 20. 7. 200319 The Classical Model III It may seem that we have just replaced one set of parameters by another. But here only the average time is unknown and it can be related to mean free path and the average thermal speed using well established theories similar to those studying ideal gas properties. This model predicts dependence of the resistivity on the temperature but not on the electric field.

20. 20. 7. 200320 Temperature Dependence of Resistivity I In most cases the behavior is close to linear. We define a change in resistivity in relation to some reference temperature t 0 (0 or 20° C):   =  (t) –  (t 0 ) The relative change of resistivity is directly proportional to the change of the temperature:

21. 20. 7. 200321 Temperature Dependence of Resistivity II  [K -1 ] is the linear temperature coefficient. It is given by the temperature dependence of n and v d. It can be negative e.g. in the case of semiconductors (but exponential behavior). In larger temperature span we have to add a quadratic term etc.  /  (t 0 ) =  (t – t 0 ) =   t +  (  t) 2 + …   (t) =  (t 0 )(1 +   t +  (  t) 2 + …)

22. 20. 7. 200322 The Thermocouple I The thermocouple is an example of a transducer, a device which transfers some physical quality (here temperature) to an electrical one. Unlike other temperature sensors e.g. the platinum thermometer or thermistor which use the thermal conductivity change of metals or semiconductors, the thermocouple is a power-source.

23. 20. 7. 200323 The Thermocouple II It is based on thermoelectric or Seebeck (Thomas 1821) effect : If we keep a difference of temperature on two ends of a conductive wire also potential difference appears between these ends. This voltage is proportional to the temperature difference and some a material parameter Seebeck’s coefficient.

24. 20. 7. 200324 The Thermocouple III Let’s connect two conductors A and B in one point, which we keep at temperature t 1. The other ends, which are at room temperature t 0 will have voltages with respect to their contact point : V A =k A (t 1 -t 0 ) and V B =k B (t 1 -t 0 ) A voltmeter connected between these ends shows : V AB = V B - V A = (k B - k A )(t 1 - t 0 )

25. 20. 7. 200325 The Thermocouple IV As a thermocouple two wires with sufficiently different Seebeck’s coefficient can be used. Usually around ten selected pairs of materials are frequently used. They are named J, K … and their calibration parameters are known. They differ e.g. in temperature span where they are used. When using one thermocouple its voltage depends on room temperature which is not a very convenient property.

26. 20. 7. 200326 The Thermocouple V A simple possibility to get rid of this dependence is to use a pair of thermocouples. Let use make a second connection of conductors A and B and place it into known temperature t 2. The we cut one of the conductors (e.g. B) in a place on room temperature t 0. The voltages of the points of disconnection X and Y with respect to the first common point is :V X = k B (t 1 - t 0 ) V Y = k A (t 1 - t 2 ) + k B (t 2 - t 0 )

27. 20. 7. 200327 The Thermocouple VI And the voltage between these points is : V XY = V Y - V X = k A (t 1 - t 2 ) + k B (t 2 - t 0 ) - k B (t 1 - t 0 ) so finally :V XY = (k A - k B )(t 1 - t 2 ) The dependence on the room temperature has really vanished. The price is the necessity to use a bath with the reference temperature t 2. Usually some well defined phase transitions e.g. (melting of ice in water) are used. But care has to be taken e.g. for pressure dependence.

28. 20. 7. 200328 The Thermocouple VII Modern instruments (equipped with microprocessors) usually measure the room temperature, so they can simulate the “cold junction” (reference junction) and using only one thermocouple is sufficient. They can be, however, only used with the types of thermocouples for which they are preprogrammed and instructions how to precisely connect the thermocouple have to be obeyed.

29. 20. 7. 200329 Peltier’s Effect Thermoelectric effect works also the other way. If current flows through a junction of two different materials, heat can be transferred into or from this junction. This is so called Peltier effect (Jean 1834). Peltier cells are commercially available. They can be used to control conveniently temperature of some volume of interest in a temperature span of circa – 50 to 200 °C. They can both heat and cool! In special cases e.g. in space ships they can even be used as power sources.

30. 20. 7. 200330 Homework 26 – 3, 4, 10, 11, 40 Study guides

31. 20. 7. 200331 Things to read This lecture covers : Chapter 25 – 4, 8, 9 and 26 – 6 Advance reading Chapters 21 – 26 except 25 – 7, 26 – 4 See demonstrations: http://buphy.bu.edu/~duffy/semester2/semester2.html