1. Copyright © 2009 Pearson Education, Inc. Chapter 26 DC Circuits

2. Copyright © 2009 Pearson Education, Inc. Midterm 1 - Recap Statistics: Average = 78.9 σ = 16.3 Problem questions: Three identical capacitors are connected in series to a battery. If a total charge of Q flows from the battery, how much charge does each capacitor carry? A solid non-conducting sphere of radius R carries a uniform charge density. At a radial distance r 1 = R/4 the electric field has a magnitude E 0. What is the magnitude of the electric field at a radial distance r 2 = 2R?

3. Copyright © 2009 Pearson Education, Inc. When the switch is closed, the capacitor will begin to charge. As it does, the voltage across it increases, and the current through the resistor decreases. 26-5 Circuits Containing Resistor and Capacitor (RC Circuits)

4. Copyright © 2009 Pearson Education, Inc. 26-5 RC Circuits To find the voltage as a function of time, we write the equation for the voltage changes around the loop: Since I = dQ/dt, we can integrate to find the charge as a function of time:

5. Copyright © 2009 Pearson Education, Inc. 26-5 RC Circuits The voltage across the capacitor is V C = Q/C : The quantity RC that appears in the exponent is called the time constant of the circuit:

6. Copyright © 2009 Pearson Education, Inc. 26-5 RC Circuits Example 26-11: RC circuit, with emf. The capacitance in the circuit shown is C = 0.30 μ F, the total resistance is 20 kΩ, and the battery emf is 12 V. Determine (a) the time constant, (b) the maximum charge the capacitor could acquire, (c) the time it takes for the charge to reach 99% of this value, (d) the current I when the charge Q is half its maximum value, (e) the maximum current, and (f) the charge Q when the current I is 0.20 its maximum value.

7. Copyright © 2009 Pearson Education, Inc. If an isolated charged capacitor is connected across a resistor, it discharges: 26-5 RC Circuits

8. Copyright © 2009 Pearson Education, Inc. 26-5 RC Circuits Once again, the voltage and current as a function of time can be found from the charge: and

9. Copyright © 2009 Pearson Education, Inc. 26-5 RC Circuits Example 26-12: Discharging RC circuit. In the RC circuit shown, the battery has fully charged the capacitor, so Q 0 = C E. Then at t = 0 the switch is thrown from position a to b. The battery emf is 20.0 V, and the capacitance C = 1.02 μF. The current I is observed to decrease to 0.50 of its initial value in 40 μs. (a) What is the value of Q, the charge on the capacitor, at t = 0? (b) What is the value of R ? (c) What is Q at t = 60 μs?

10. Copyright © 2009 Pearson Education, Inc. 26-5 RC Circuits Conceptual Example 26-13: Bulb in RC circuit. In the circuit shown, the capacitor is originally uncharged. Describe the behavior of the lightbulb from the instant switch S is closed until a long time later.

11. Copyright © 2009 Pearson Education, Inc. A source of emf transforms energy from some other form to electrical energy. A battery is a source of emf in parallel with an internal resistance. Resistors in series: Summary of Chapter 26

12. Copyright © 2009 Pearson Education, Inc. Resistors in parallel: Kirchhoff’s rules: 1.Sum of currents entering a junction equals sum of currents leaving it. 2.Total potential difference around closed loop is zero. Summary of Chapter 26

13. Copyright © 2009 Pearson Education, Inc. RC circuit has a characteristic time constant: To avoid shocks, don’t allow your body to become part of a complete circuit. Summary of Chapter 26

14. Copyright © 2009 Pearson Education, Inc. Chapter 27 Magnetism

15. Copyright © 2009 Pearson Education, Inc. Magnets and Magnetic Fields Electric Currents Produce Magnetic Fields Force on an Electric Current in a Magnetic Field; Definition of B Force on an Electric Charge Moving in a Magnetic Field Torque on a Current Loop; Magnetic Dipole Moment Units of Chapter 27

16. Copyright © 2009 Pearson Education, Inc. Applications: Motors, Loudspeakers, Galvanometers Discovery and Properties of the Electron The Hall Effect Mass Spectrometer Units of Chapter 27

17. Copyright © 2009 Pearson Education, Inc. Magnets have two ends – poles – called north and south. Like poles repel; unlike poles attract. 27-1 Magnets and Magnetic Fields

18. Copyright © 2009 Pearson Education, Inc. However, if you cut a magnet in half, you don’t get a north pole and a south pole – you get two smaller magnets. 27-1 Magnets and Magnetic Fields Demo

19. Copyright © 2009 Pearson Education, Inc. Magnetic fields can be visualized using magnetic field lines, which are always closed loops. 27-1 Magnets and Magnetic Fields Demo (Except for one April Fool’s Day in California … but that’s another story.)

20. Copyright © 2009 Pearson Education, Inc. The Earth’s magnetic field is similar to that of a bar magnet. Note that the Earth’s “North Pole” is really a south magnetic pole, as the north ends of magnets are attracted to it. 27-1 Magnets and Magnetic Fields

21. Copyright © 2009 Pearson Education, Inc. A uniform magnetic field is constant in magnitude and direction. The field between these two wide poles is nearly uniform. 27-1 Magnets and Magnetic Fields

22. Copyright © 2009 Pearson Education, Inc. Experiment shows that an electric current produces a magnetic field. The direction of the field is given by a right-hand rule. 27-2 Electric Currents Produce Magnetic Fields Demo

23. Copyright © 2009 Pearson Education, Inc. 27-2 Electric Currents Produce Magnetic Fields Here we see the field due to a current loop; the direction is again given by a right-hand rule. Demo

24. Copyright © 2009 Pearson Education, Inc. A magnet exerts a force on a current- carrying wire. The direction of the force is given by a right- hand rule. 27-3 Force on an Electric Current in a Magnetic Field; Definition of B Demo

25. Copyright © 2009 Pearson Education, Inc. The force on the wire depends on the current, the length of the wire, the magnetic field, and its orientation: This equation defines the magnetic field B. In vector notation: 27-3 Force on an Electric Current in a Magnetic Field; Definition of B

26. Copyright © 2009 Pearson Education, Inc. Unit of B : the tesla, T: 1 T = 1 N/A·m. Another unit sometimes used: the gauss ( G ): 1 G = 10 -4 T. 27-3 Force on an Electric Current in a Magnetic Field; Definition of B

27. Copyright © 2009 Pearson Education, Inc. 27-3 Force on an Electric Current in a Magnetic Field; Definition of B Example 27-1: Magnetic Force on a current-carrying wire. A wire carrying a 30-A current has a length l = 12 cm between the pole faces of a magnet at an angle θ = 60°, as shown. The magnetic field is approximately uniform at 0.90 T. We ignore the field beyond the pole pieces. What is the magnitude of the force on the wire?

28. Copyright © 2009 Pearson Education, Inc. 27-3 Force on an Electric Current in a Magnetic Field; Definition of B Example 27-3: Magnetic Force on a semicircular wire. A rigid wire, carrying a current I, consists of a semicircle of radius R and two straight portions as shown. The wire lies in a plane perpendicular to a uniform magnetic field B 0. Note choice of x and y axis. The straight portions each have length l within the field. Determine the net force on the wire due to the magnetic field B 0.

29. Copyright © 2009 Pearson Education, Inc. The force on a moving charge is related to the force on a current since a current is just a bunch of moving charges: Once again, the direction is given by a right-hand rule. 27-4 Force on an Electric Charge Moving in a Magnetic Field

30. Copyright © 2009 Pearson Education, Inc. 27-4 Force on an Electric Charge Moving in a Magnetic Field Conceptual Example 27-4: Negative charge near a magnet. A negative charge -Q is placed at rest near a magnet. Will the charge begin to move? Will it feel a force? What if the charge were positive, +Q ?

31. Copyright © 2009 Pearson Education, Inc. 27-4 Force on an Electric Charge Moving in a Magnetic Field Example 27-5: Magnetic force on a proton. A magnetic field exerts a force of 8.0 x 10 -14 N toward the west on a proton moving vertically upward at a speed of 5.0 x 10 6 m/s (a). When moving horizontally in a northerly direction, the force on the proton is zero (b). Determine the magnitude and direction of the magnetic field in this region. (The charge on a proton is q = +e = 1.6 x 10 -19 C.)

32. Copyright © 2009 Pearson Education, Inc. If a charged particle is moving perpendicular to a uniform magnetic field, its path will be a circle. 27-4 Force on an Electric Charge Moving in a Magnetic Field

33. Copyright © 2009 Pearson Education, Inc. 27-4 Force on an Electric Charge Moving in a Magnetic Field Example 27-7: Electron’s path in a uniform magnetic field. An electron travels at 2.0 x 10 7 m/s in a plane perpendicular to a uniform 0.010-T magnetic field. Describe its path quantitatively.

34. Copyright © 2009 Pearson Education, Inc. Problem solving: Magnetic fields – things to remember: 1.The magnetic force is perpendicular to the magnetic field direction. 2.The right-hand rule is useful for determining directions. 3.Equations in this chapter give magnitudes only. The right-hand rule gives the direction. 27-4 Force on an Electric Charge Moving in a Magnetic Field

35. 1) out of the page 2) into the page 3) zero 4) to the right 5) to the left  v q ConcepTest 27.1c Magnetic Force III A positive charge enters a uniform magnetic field as shown. What is the direction of the magnetic force?

36. into the page perpendicularto BOTH the B field and the velocity Using the right-hand rule, you can see that the magnetic force is directed into the page. Remember that the magnetic force must be perpendicular to BOTH the B field and the velocity. 1) out of the page 2) into the page 3) zero 4) to the right 5) to the left  v q F  ConcepTest 27.1c Magnetic Force III A positive charge enters a uniform magnetic field as shown. What is the direction of the magnetic force?

37. ConcepTest 27.3 Magnetic Field xy A proton beam enters a magnetic field region as shown below. What is the direction of the magnetic field B? 1) + y 2) – y 3) + x 4) + z (out of page) 5) – z (into page)

38. +y direction into the page out of the plane B  vB  F The picture shows the force acting in the +y direction. Applying the right-hand rule leads to a B field that points into the page. The B field must be out of the plane because B  v and B  F. ConcepTest 27.3 Magnetic Field xy A proton beam enters a magnetic field region as shown below. What is the direction of the magnetic field B? 1) + y 2) – y 3) + x 4) + z (out of page) 5) – z (into page) Follow-up: What would happen to a beam of atoms?

39. Copyright © 2009 Pearson Education, Inc. 27-4 Force on an Electric Charge Moving in a Magnetic Field Conceptual Example 27-9: A helical path. What is the path of a charged particle in a uniform magnetic field if its velocity is not perpendicular to the magnetic field?

40. Copyright © 2009 Pearson Education, Inc. 27-4 Force on an Electric Charge Moving in a Magnetic Field The aurora borealis (northern lights) is caused by charged particles from the solar wind spiraling along the Earth’s magnetic field, and colliding with air molecules.