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Lecture PowerPoints Physics for Scientists and Engineers, 3rd edition Fishbane Gasiorowicz Thornton © 2005 Pearson Prentice Hall This work is protected by United States copyright laws and is provided solely for the use of instructors in teaching their courses and assessing student learning. Dissemination or sale of any part of this work (including on the World Wide Web) will destroy the integrity of the work and is not permitted. The work and materials from it should never be made available to students except by instructors using the accompanying text in their classes. All recipients of this work are expected to abide by these restrictions and to honor the intended pedagogical purposes and the needs of other instructors who rely on these materials. Chapter 26 Currents in Materials 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 26-1 Electric Current Definition: the total charge that passes through a given cross-sectional area per unit time. (26-2) Units: amperes 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 26-1 Electric Current • Current density: current per unit area, defined over an infinitesimal area • Current is surface integral of current density (26-4) 26-1 Electric Current Current density of moving charges: (26-7) • nq is number of charges per unit volume • q is the magnitude of each charge • v is the velocity of each charge 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 26-2 Currents in Materials • Electric field introduces overall “drift” • Drift velocity is very small compared to thermal velocity 26-2 Currents in Materials Writing the current in terms of the motion of individual charge carriers: (26-8) And solving for the drift velocity: (26-9) The current density is then: (26-10) 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 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: Units of resistance: ohms (Ω) (26-11) 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) 26-3 Resistance Resistors • Ohmic material • Specified resistance • For use in circuits 26-3 Resistance Resistivity and Conductivity • Property of a material • Independent of geometry and size • Definition: (26-13) 26-3 Resistance Calculating resistance using resistivity: (26-14) for a material of length L and crosssectional area A. The conductivity is the inverse of the resistivity: (26-15) 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) 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) 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) 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!) 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. 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 26-6 Materials and Conductivity Illustration: insulator has a gap between filled and empty bands, conductor does not 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 26-6 Materials and Conductivity Superconductor • At some critical temperature Tc, resistance goes abruptly to zero. • Zero resistance means currents can last indefinitely 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. 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 Summary of Chapter 26, cont. • Resistors in series: (26-21) • Resistors in parallel: (26-22) • Power P = V/I