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2. Chapter 26 Currents in Materials

3. Main Points of Chapter 26 Electric current Current density Currents in materials: conductors and insulators Conservation of charge Resistors, resistance, and conductivity Series and parallel combinations of resistors Materials and conductivity: semiconductors and superconductors Electric power

4. 26-1 Electric Current Definition: the total charge that passes through a given cross-sectional area per unit time. (26-2) Units: amperes

5. 26-1 Electric Current Direction: the direction a positive charge would take, even if the current consists of negative charges moving in the opposite direction Note: current is a signed scalar, not a vector

6. 26-1 Electric Current Current density: current per unit area, defined over an infinitesimal area Current is surface integral of current density (26-4)

7. 26-1 Electric Current Current density of moving charges: (26-7) n q is number of charges per unit volume q is the magnitude of each charge v is the velocity of each charge

8. 26-2 Currents in Materials Electrons in conductors are always colliding with molecules Average (rms) speed increases with temperature If no electric field, no net speed in any direction

9. 26-2 Currents in Materials Electric field introduces overall “drift” Drift velocity is very small compared to thermal velocity

10. 26-2 Currents in Materials Writing the current in terms of the motion of individual charge carriers: And solving for the drift velocity: (26-8) (26-9) The current density is then: (26-10)

11. 26-2 Currents in Materials Current and the conservation of charge: If the diameter of the conductor changes, the current density and drift velocity change too

12. 26-3 Resistance Resistance is a measure of how easily current flows in a material For the same voltage, more resistance means less current, and vice versa Definition of resistance: (26-11) Units of resistance: ohms (Ω)

13. 26-3 Resistance Ohm’s Law: An ohmic material is one where the resistance is nearly constant over a wide range of voltages. In that case: (26-12)

14. Resistors 26-3 Resistance Ohmic material Specified resistance For use in circuits

15. 26-3 Resistance Resistivity and Conductivity Property of a material Independent of geometry and size Definition: (26-13)

16. 26-3 Resistance Calculating resistance using resistivity: (26-14) for a material of length L and cross- sectional area A. The conductivity is the inverse of the resistivity: (26-15)

17. 26-3 Resistance The Temperature Dependence of Resistivity Resistivity has its origin in collisions between electrons and atoms or molecules Higher temperature = faster thermal velocities = more collisions = higher resistivity (26-18)

18. 26-4 Resistances in Series and in Parallel Resistors in series: have the same current, but voltage depends on resistance. Equivalent resistance (same current and A-B voltage drop): (26-21)

19. 26-4 Resistances in Series and in Parallel Resistors in parallel: have the same voltage, but current depends on resistance. Equivalent resistance (same current and A-B voltage drop): (26-22)

20. 26-5 Free-Electron Model of Resistivity Assumptions: Conductors contain free electrons, not attached to particular atoms Electrons form a “gas” at temperature T Electric field creates drift; collisions create drag, yielding constant drift velocity Yields resistivity that depends only on mean free path of electrons in that material (a success!)

21. 26-5 Free-Electron Model of Resistivity Failures: Electrons move much faster than predicted Model predicts electrons move faster with temperature – they don’t Mean free path should be independent of temperature – it isn’t, and is much larger than predicted Successful model uses quantum mechanics.

22. 26-6 Materials and Conductivity Quantum mechanics tells us that energy levels of electrons in materials are quantized – only certain values are possible Each level can contain only two electrons Results in filled and empty levels Insulators: have gap between filled and empty energy levels

23. 26-6 Materials and Conductivity Illustration: insulator has a gap between filled and empty bands, conductor does not

24. 26-6 Materials and Conductivity Semiconductor: also has gap between filled and empty bands, but gap is small and can be breached using external field

25. 26-6 Materials and Conductivity Superconductor At some critical temperature T c, resistance goes abruptly to zero. Zero resistance means currents can last indefinitely

26. 26-7 Electric Power Using the definition of power and of voltage: (26-26) And now the definition of current: (26-27) This is the power lost in resistive circuit elements, and is valid for all materials.

27. Summary of Chapter 26 Electric current: I = dQ/dt Current density (a vector) is current passing through a unit area Electrical resistance is the ratio of voltage to current: R = V/I In ohmic materials, this ratio is nearly constant Resistivity is a property of a material; to find resistance: R = ρL/A Resistivity depends on temperature

28. Summary of Chapter 26, cont. Resistors in series: (26-21) Resistors in parallel: (26-22) Power P = V/I