1. Superposition of Forces
We find the total force by adding the vector sum of the individual forces.

2. Work Problem 21-18 21-18. (III) Two charges, and are a distance apart
Work Problem (III) Two charges, and are a distance apart. These two charges are free to move but do not because there is a third charge nearby. What must be the magnitude of the third charge and its placement in order for the first two to be in equilibrium?

3. Electric Field of a Point Charge

4. Relation between F and E
Don’t confuse this charge q1 with the test charge q0 or the original charges q that produced E. The test charge q0 was used to find the electric field. This is a real charge q1 placed in the electric field.

5. Electric Field Lines for a Point Charge
See next slide for +q and – q, the dipole.

6. Electric Field Lines for Systems of Charges
We call this a dipole. It is a dipole field.

7. The Electric Field of a Charged Plate
I have transparency for the document camera for this.

8. A Parallel-Plate Capacitor
Use transparency to do this on document camera.

9. The electric field near a conducting surface must be perpendicular to the surface when in equilibrium.

10. Conductor placed around a charge +Q

11. An electron moves in a circle of radius r around a very long uniformly charged wire in a vacuum chamber, as shown in the figure. The charge density on the wire is λ = 0.14 μC/m. (a) What is the electric field at the electron (magnitude and direction in terms of r and λ? (b) What is the speed of the electron?

12. We can work all kinds of problems with charged particles moving in electric fields.
Electron entering charged parallel plates

13. Electric flux:
Electric flux through an area is proportional to the total number of field lines crossing the area.
Figure (a) A uniform electric field E passing through a flat area A. (b) E┴ = E cos θ is the component of E perpendicular to the plane of area A. (c) A┴ = A cos θ is the projection (dashed) of the area A perpendicular to the field E.

14. Flux through a closed surface:
positive
negative

15. The net number of field lines through the surface is proportional to the charge enclosed, and also to the flux, giving Gauss’s law:
This can be used to find the electric field in situations with a high degree of symmetry.

16. Electric field of charged sheet

17. Electric Potential V definition!!
Electric potential, or potential, is one of the most useful concepts in electromagnetism. This is a biggie!!
The volt is named after Alessandro Volta, an Italian physicist, who lived in the 17th and 18th centuries. Among other things, he invented the battery.

18. Electrostatic Potential Energy and Potential Difference
The electrostatic force is conservative – potential energy can be defined.
Change in electric potential energy is negative of work done by electric force:
Figure Work is done by the electric field in moving the positive charge q from position a to position b.

19. The Potentials of Charge Distributions
If the electric field is known:
For one point charge:
For many point charges:

20. The Potentials of Charge Distributions
If the electric field is known:
For differential charge:
For many point charges:
For a continuous charge distribution:

21. Equipotential Surfaces
An equipotential is a line or surface over which the potential is constant.
Electric field lines are perpendicular to equipotentials.
The surface of a conductor is an equipotential.
Figure Equipotential lines (the green dashed lines) between two oppositely charged parallel plates. Note that they are perpendicular to the electric field lines (solid red lines).

22. Equipotential Surfaces
Another case showing electric field lines are perpendicular to equipotentials.
The surface of a conductor is an equipotential.
We can also see that equipotentials are perpendicular to electric fields from the equation
Figure Equipotential lines (the green dashed lines) between two oppositely charged parallel plates. Note that they are perpendicular to the electric field lines (solid red lines).

23. Equipotential Surfaces
Equipotential surfaces are always perpendicular to field lines; they are always closed surfaces (unlike field lines, which begin and end on charges).
Electric field and equipotentials for electric dipole.
Figure Equipotential lines (green, dashed) are always perpendicular to the electric field lines (solid red) shown here for two equal but oppositely charged particles.

24. Four point charges are located at the corners of a square that is 8.0 cm on a side. The charges, going in rotation around the square, are Q, 2Q, -3Q and 2Q, where Q = 3.1 μC. What is the total electric potential energy stored in the system, relative to U = 0 at infinite separation?

25. When a capacitor is connected to a battery, the charge on its plates is proportional to the voltage:
The quantity C is called the capacitance.

26. Parallel plate capacitor
The capacitance value depends only on geometry!

27. Capacitors in Parallel
Capacitors in parallel have the same voltage across each one. The equivalent capacitor is one that stores the same charge when connected to the same battery:
Figure Capacitors in parallel: Ceq = C1 + C2 + C3 .

28. Capacitors in Series
Capacitors in series have the same charge. In this case, the equivalent capacitor has the same charge across the total voltage drop. Note that the formula is for the inverse of the capacitance and not the capacitance itself!
Figure Capacitors in series: 1/Ceq = 1/C1 + 1/C2 + 1/C3.

29. Effect of a Dielectric on the Electric Field of a Capacitor

30. A dielectric is an insulator, and is characterized by a dielectric constant .
Capacitance of a parallel-plate capacitor filled with dielectric:
Using the dielectric constant, we define the permittivity:

31. Energy in electric field
The energy U in a capacitor is
The volume is Ad, and the energy density u is

32. Defibrillator

33. Potential energy of a charged capacitor:
All three expressions are equivalent!

34. A complete circuit is one where current can flow all the way around
A complete circuit is one where current can flow all the way around. Note that the schematic drawing doesn’t look much like the physical circuit!
Open circuit
Figure (a) A simple electric circuit. (b) Schematic drawing of the same circuit, consisting of a battery, connecting wires (thick gray lines), and a lightbulb or other device.

35. Direction of Current and Electron Flow
Bulbs are not really very ohmic.

36. Resistors are color coded to indicate the value of their resistance.

37. Resistivity
The resistance of a wire is directly proportional to its length and inversely proportional to its cross-sectional area:
The constant ρ, the resistivity, is characteristic of the material.

38. Energy and Power The unit of electrical power is watt W (J/s).
If we use Ohm’s law with this equation, we have

39. AC Voltage and Current for a Resistor Circuit
Note that I and V are in phase!!

40. 25-36. (II) A 120-V hair dryer has two settings: 850 W and 1250 W
(II) A 120-V hair dryer has two settings: 850 W and 1250 W. (a) At which setting do you expect the resistance to be higher? After making a guess, determine the resistance at (b) the lower setting; and (c) the higher setting.

41. 25-43. (II) How many 75-W lightbulbs, connected to 120 V as in Fig
(II) How many 75-W lightbulbs, connected to 120 V as in Fig. 25–20, can be used without blowing a 15-A fuse?

42. Resistors in Series
Current is not used up in each resistor. Same current I passes through each resistor in series.

43. A parallel connection splits the current; the voltage across each resistor is the same:
Figure (a) Resistances connected in parallel. (b) The resistances could be lightbulbs. (c) The equivalent circuit with Req obtained from Eq. 26–4: 1/Req = 1/R1 + 1/R2 + 1/R3

44. Analyzing a Complex Circuit of Resistors

45. Kirchhoff’s Junction Rule
In +
Out -
The sum of currents meeting at a junction must be zero. I1 – I2 – I3 = 0 or I1 = I2 + I3

46. Kirchhoff’s Loop Rule The sum of potential differences around any
closed circuit
loop is zero.
Our rules:
When going from – to + across an emf the
V is +. (+ to -, it is -).
2) When going across resistor in direction of
assumed I, the V is -. (Opposite, it is +).

47. Measuring the Current in a Circuit
We want ammeter to have very low resistance so it will not affect circuit. Ammeters go in series.

48. Measuring the Voltage in a Circuit
We want voltmeter to have very large resistance so it will not affect circuit. Voltmeters go in parallel across what is being measured.

49. An ohmmeter measures resistance; it requires a battery to provide a current. These circuits are much more complicated. Rsh is a shunt resistor to change scales. Rser is a resistor to adjust galvanometer scale zero.
Figure An ohmmeter.

50. Magnetic Field Lines for a Bar Magnet
Imagine using a test pole N; place it at any point and see where the force is. Just like we do for electric fields. We actually use small compasses to do this.

51. The Magnetic Force Right-Hand Rule

52. Units of magnetic field: teslas
The Lorentz force is the sum of the electric and magnetic forces acting on the same object:

53. 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.
Figure The Earth acts like a huge magnet; but its magnetic poles are not at the geographic poles, which are on the Earth’s rotation axis.

54. Operating Principle of a Mass Spectrometer
Several applications

55. Magnetic Force on a Current-Carrying Wire

56. The Magnetic-Field Right-Hand Rule
Put thumb along direction of current, and fingers curl in direction of B.
We did this demo last time. This result is found experimentally, not derived here. Discuss the dependence on I and r.

57. Magnetic Forces on a Current Loop
Forces cause a torque I
Ftotal = 0

58. An electric motor uses the torque on a current loop in a magnetic field to turn magnetic energy into kinetic energy.
Figure Diagram of a simple dc motor.
Figure The commutator-brush arrangement in a dc motor ensures alternation of the current in the armature to keep rotation continuous. The commutators are attached to the motor shaft and turn with it, whereas the brushes remain stationary.

59. 27-23. (II) A 6. 0-MeV (kinetic energy) proton enters a 0
(II) A 6.0-MeV (kinetic energy) proton enters a 0.20-T field, in a plane perpendicular to the field. What is the radius of its path? See Section 23–8.

60. We have to look closely at fields and forces to see how the forces occur.

61. Ampère’s Law
We have seen that

62. To find the field inside, we use Ampère’s law along the path indicated in the figure.
Figure 28-16: Cross-sectional view into a solenoid. The magnetic field inside is straight except at the ends. Red dashed lines indicate the path chosen for use in Ampère’s law.

63. Induced Current Produced by a Moving Magnet

64. We conclude that it is the change in magnetic flux that causes induced current.

65. This is called Faraday’s Law of Induction after Michael Faraday.

66. Lenz’s Law
The induced current will always be in the direction to oppose the change that produced it.

67. Applying Lenz’s Law to a Magnet Moving Toward and Away From a Current Loop
Induced current

68. An Electrical Generator
Current is induced
Magnetic flux changes!
Produces AC power

69. A Simple Electric Motor/Generator

70. Inductance
The inductance L is a proportionality constant that depends on the geometry of the circuit

71. There will be a magnetic flux in Loop 1 due to current I1 flowing in Loop 1 and due to current I2 flowing in Loop 2.
Now it is clearer why we call L self inductance and M mutual inductance.

72. Solenoid Self-Induction

73. General energy density

74. (II) A 425-pF capacitor is charged to 135 V and then quickly connected to a 175-mH inductor. Determine (a) the frequency of oscillation, (b) the peak value of the current, and (c) the maximum energy stored in the magnetic field of the inductor.

75. RL Circuit I
Think about what will happen when the switch is closed.

76. Current as a Function of Time in an RL Circuit

77. Oscillations in LC Circuits
Start with charged capacitor. Close switch.
It will discharge through inductor, and then recharge in opposite sense.
If no resistance, will continue indefinitely.

78. The charge and current are both sinusoidal, but with different phases.
Figure Charge Q and current I in an LC circuit. The period T = 1/f = 2π/ω = 2π(L/C)1/2.

79. LC Oscillations with Resistance (LRC Circuit)
Any real (nonsuperconducting) circuit will have resistance.
Figure An LRC circuit.

80. Damped Oscillations in RLC Circuits
Charge equation:
Solution:
where
and

81. Mention terminology shown on this slide.

82. This is a step-up transformer – the emf in the secondary coil is larger than the emf in the primary:
Figure Step-up transformer (NP = 4, NS = 12).

83. Lots of applications for transformers, the bug zapper.

84. Power distribution
Transformers work only if the current is changing; this is one reason why electricity is transmitted as ac.

85. Single elements with AC Source
Resistors, capacitors, and inductors have different phase relationships between current and voltage when placed in an ac circuit.
Resistive element
The current through a resistor is in phase with the voltage.
Figure (a) Resistor connected to an ac source. (b) Current (blue curve) is in phase with the voltage (red) across a resistor.

86. Inductive Element The voltage across the inductor is determined by or
Figure (a) Inductor connected to an ac source. (b) Current (blue curve) lags voltage (red curve) by a quarter cycle or 90°. The voltage curve reaches its peak at time t1 before the current does at time t2.
Therefore, the current through an inductor lags the voltage by 90°.

87. Inductive Circuit
The voltage across the inductor is related to the current through it:
The quantity XL is called the inductive reactance, and has units of ohms:
For very low frequencies the inductive reactance is small. That is because for direct currents (zero frequency), an inductor has little or no effect. Direct current passes right through an inductor.

88. Capacitive Circuit The voltage across the capacitor is given by
Therefore, in a capacitor, the current leads the voltage by 90°.
Figure (a) Capacitor connected to an ac source. (b) Current leads voltage by a quarter cycle, or 90°.

89. Capacitive Circuit
The voltage across the capacitor is related to the current through it:
The quantity XC is called the capacitive reactance, and (just like the inductive reactance) has units of ohms:

90. Effects of frequency on capacitive reactance
Note that when the frequency increases to large values that XC becomes very small. The current then becomes very large.
Why?
The frequency is so high that the capacitor doesn’t have time to fully charge. It almost acts as a short circuit. At low frequencies, it acts as an open circuit.

91. Either capacitors or inductors can be used to make either AC or DC filters:
AC & DC input

92. LRC Series AC Circuit
Analyzing the LRC series AC circuit is complicated, as the voltages are not in phase – this means we cannot simply add them. Furthermore, the reactances depend on the frequency.
Figure An LRC circuit.

93. Some Applications Half-wave Rectifier Diodes and Rectifiers
A diode conducts electricity in one direction only
Can use diodes and combinations of diodes to make half- and full-wave rectifiers

94. Some Applications
Full-wave
Rectifier