Chapter 21

1. Chapter 21 Current and Direct Current Circuits

2. 21.1 Electric Current • Electric current is the rate of flow of charge through a surface • The SI unit of current is the Ampere (A) • 1 A = 1 C / s • The symbol for electric current is I

3. Average Electric Current • Assume charges are moving perpendicular to a surface of area A • If DQ is the amount of charge that passes through A in time Dt, the average current is

4. Instantaneous Electric Current • If the rate at which the charge flows varies with time, the instantaneous current, I, can be found

5. Direction of Current • The charges passing through the area could be positive or negative or both • It is conventional to assign to the current the same direction as the flow of positive charges • The direction of current flow is opposite the direction of the flow of electrons • It is common to refer to any moving charge as a charge carrier

6. Model of current in a Conductor • The zig-zag black line represents the motion of charge carrier in a conductor • The sharp changes in direction are due to collisions • When an external electric field is applied on the conductor, the electric field exerts a force on the electrons • The force accelerates the electrons and produces a current • The net motion of electrons is opposite the direction of the electric field • Vd is the drift velocity, which is small • If there is no electric field, Vd is zero

7. Drift Velocity, Example • Assume a copper wire, with one free electron per atom contributed to the current • The drift velocity for a 12 gauge copper wire carrying a current of 10 A is 2.22 x 10-4 m/s • This is a typical order of magnitude for drift velocities

8. Current and Drift Speed • Charged particles move through a conductor of cross-sectional area A • n is the number of charge carriers per unit volume • n A Δx is the total number of charge carriers

9. Current and Drift Speed • The total charge is the number of carriers times the charge per carrier, q • ΔQ = (n A Δ x) q • The drift speed, vd, is the speed at which the carriers move • vd = Δ x/ Δt • Rewritten: ΔQ = (n A vd Δt) q • Finally, current, I = ΔQ/Δt = nqvdA

12. Current Density • J is the current density of a conductor • It is defined as the current per unit area • J = I / A = n q vd • This expression is valid only if the current density is uniform and A is perpendicular to the direction of the current • J has SI units of A / m2 • The current density is in the direction of the positive charge carriers

13. Conductivity • A current density J and an electric field E are established in a conductor whenever a potential difference is maintained across the conductor • J = sE • The constant of proportionality, s, is called the conductivity of the conductor

14. 21.2 Resistance • In a conductor, the voltage applied across the ends of the conductor is proportional to the current through the conductor • The constant of proportionality is the resistance of the conductor SI units of resistance are ohms (Ω) 1 Ω = 1 V / A

15. Ohm’s Law • Ohm’s Law states that for many materials, the resistance is constant over a wide range of applied voltages • Most metals obey Ohm’s Law • Materials that obey Ohm’s Law are said to be ohmic • Not all materials follow Ohm’s Law • Materials that do not obey Ohm’s Law are said to be nonohmic • Ohm’s Law is not a fundamental law of nature • Ohm’s Law is an empirical relationship valid only for certain materials

16. Ohmic Material, Graph • An ohmic device • The resistance is constant over a wide range of voltages • The relationship between current and voltage is linear • The slope is related to the resistance

17. Nonohmic Material, Graph • Non-ohmic materials are those whose resistance changes with voltage or current • The current-voltage relationship is nonlinear • A diode is a common example of a non-ohmic device

18. Resistivity • Resistance is related to the geometry of the device: • r is called the resistivity of the material • The inverse of the resistivity is the conductivity: • s = 1 / r and R = l / sA • Resistivity has SI units of ohm-meters (W. m)

20. Resistance and Resistivity, Summary • Resistivity is a property of a material • Resistance is a property of an object • The resistance of a objector depends on its geometry and its resistivity • An ideal (perfect) conductor would have zero resistivity • An ideal insulator would have infinite resistivity

21. Temperature variation of Resistivity of a metal • For metals, the resistivity is nearly proportional to the temperature • A nonlinear region always exists at very low temperatures • As the temperature approaches absolute zero, the resistivity usually reaches some finite value r0, called the residual resistivity

22. Residual Resistivity • The residual resistivity near absolute zero is caused primarily by the collisions of electrons with impurities and imperfections in the metal • High temperature resistivity is predominantly characterized by collisions between electrons and the vibrations of the metal atoms • This is the linear range on the graph

23. Over a limited temperature range, the resistivity of a conductor varies approximately linearly with the temperature • ρo is the resistivity at some reference temperature To • To is usually taken to be 20° C •  is the temperature coefficient of resistivity • SI units of a are oC-1

24. Resistors • Most circuits use elements called resistors • Resistors are used to control the current level in parts of the circuit • Resistors can be composite or wire-wound

25. Resistor Values • Values of resistors are commonly marked by colored bands

26. 21.3 Superconductors • A class of metals and compounds whose resistances go to zero below a certain temperature, TC • TC is called the critical temperature • The graph is the same as a normal metal above TC, but suddenly drops to zero at TC

27. Superconductors, cont • The value of TC is sensitive to • Chemical composition • Pressure • Crystalline structure • Once a current is set up in a superconductor, it persists without any applied voltage • Since R = 0

28. 21.4 A Model of Electrical Conduction • The free electrons in a conductor move with average speeds on the order of 106 m/s • Not totally free since they are confined to the interior of the conductor • The motion of an electron is random • The electrons undergo many collisions • The average velocity of the electrons is zero • There is zero current in the conductor

29. Conduction Model, 2 • As an external electric field is applied, the field modifies the motion of the charge carriers • The electrons drift in the direction opposite of the field • The average drift speed is on the order of 10-4 m/s, much less than the average speed between two successive collisions

30. Conduction Model, 3 • Assumptions: • The excess energy acquired by the electrons from the external field loses to the atoms of the conductor during the collisions • The energy given up to the atoms increases their vibration and therefore the temperature of the conductor increases • The motion of an electron after a collision is independent of its motion before the collision

31. Conduction Model, 4 • The force experienced by an electron is • From Newton’s Second Law, the acceleration is • The equation of motion between two collisions • Since the initial velocities are random, their average value is zero

32. Conduction Model, 5 • Let t be the average time interval between two successive collisions • The average value of the final velocity is the drift velocity • This is also related to the current: I = n e vd A = (n e2 E / me) t A

33. Conduction Model, final • Using Ohm’s Law, an expression for the resistivity of a conductor can be found: • Note, the resistivity does not depend on the strength of the field • The collision time t is also related to the free mean path lavg and average speed vavg • t = lavg / vavg

37. 21.5 Electrical Power • As a charge moves from a to b, the electric potential energy of the system increases by QDV • The chemical energy in the battery must decrease by this same amount

38. Electrical Power, 2 • As the charge moves through the resistor (c to d), the system loses this electric potential energy during collisions of the electrons with the atoms of the resistor • This energy is transformed into internal energy in the resistor • Corresponding to increased vibrational motion of the atoms in the resistor

39. Electric Power, 3 • The resistor is normally in contact with the air, so its increased temperature will result in a transfer of energy by heat into the air • The resistor also emits thermal radiation • After some time interval, the resistor reaches a constant temperature • The input of energy from the battery is balanced by the output of energy by heat and radiation

40. Electric Power, 4 • The rate at which the system loses potential energy as the charge passes through the resistor is equal to the rate at which the system gains internal energy in the resistor • The power is the rate at which the energy is delivered to the resistor

41. Electric Power, final • The power is given by the equation: • Applying Ohm’s Law, alternative expressions can be found: • Units: I is in A, R is in W, V is in V, and P is in W

42. Electric Power Transmission • Real power lines have resistance • Power companies transmit electricity at high voltages and low currents to minimize power losses