1. Electric Forces and Electric Fields
Chapter 19
Electric Forces
and
Electric Fields

2. Electricity and Magnetism, Some History
Many applications
Macroscopic and microscopic
Chinese
Documents suggests that magnetism was observed as early as 2000 BC
Greeks
Electrical and magnetic phenomena as early as 700 BC
Experiments with amber and magnetite

3. Electricity and Magnetism, Some History, 2
1600
William Gilbert showed electrification effects were not confined to just amber
The electrification effects were a general phenomena
1785
Charles Coulomb confirmed inverse square law form for electric forces

4. Electricity and Magnetism, Some History, 3
1820
Hans Oersted found a compass needle deflected when near a wire carrying an electric current
1831
Michael Faraday and Joseph Henry showed that when a wire is moved near a magnet, an electric current is produced in the wire

5. Electricity and Magnetism, Some History, 4
1873
James Clerk Maxwell used observations and other experimental facts as a basis for formulating the laws of electromagnetism
Unified electricity and magnetism
1888
Heinrich Hertz verified Maxwell’s predictions
He produced electromagnetic waves

6. Electric Charges There are two kinds of electric charges
Called positive and negative
Negative charges are the type possessed by electrons
Positive charges are the type possessed by protons
Charges of the same sign repel one another and charges with opposite signs attract one another
正負電極性

7. Electric Charges, 2 The rubber rod is negatively charged
The glass rod is positively charged
The two rods will attract

8. Electric Charges, 3 The rubber rod is negatively charged
The second rubber rod is also negatively charged
The two rods will repel

9. More About Electric Charges
The net charge in an isolated system is always conserved
For example, charge is not created in the process of rubbing two objects together
The electrification is due to a transfer of electrons from one object to another

10. Quantization of Electric Charges
The electric charge, q, is said to be quantized
q is the standard symbol used for charge as a variable
Electric charge exists as discrete packets
q = N e
N is an integer
e is the fundamental unit of charge
|e| = 1.6 x C
Electron: q = -e
Proton: q = +e

11. Conservation and Quantization of Electric Charges, Example
A glass rod is rubbed with silk
Electrons are transferred from the glass to the silk
Each electron adds a negative charge to the silk
An equal positive charge is left on the rod
The charges on the two objects are ±e, or ±2e, …

12. Conductors
Electrical conductors are materials in which some of the electrons move relatively freely
Free electrons are not bound to the atoms
These electrons can move relatively freely through the material
Examples of good conductors include copper, aluminum and silver
When a good conductor is charged in a small region, the charge readily distributes itself over the entire surface of the material

13. Insulators
Electrical insulators are materials in which electric charges do not move freely
Examples of good insulators include glass, rubber and wood
When a good insulator is charged in a small region, the charge is unable to move to other regions of the material

14. Semiconductors
The electrical properties of semiconductors are somewhere between those of insulators and conductors
Examples of semiconductor materials include silicon and germanium
The electrical properties of semiconductors can be changed over many orders of magnitude by adding controlled amounts of foreign atoms to the materials

15. Charging by Induction
Charging by induction requires no contact with the object inducing the charge
Assume we start with a neutral metallic sphere
The sphere has the same number of positive and negative charges

16. Charging by Induction, 2
A negatively charged rubber rod is placed near the sphere
It does not touch the sphere
The electrons in the neutral sphere are redistributed
The migration of electrons leaves the side near the rod with an effective positive charge

17. Charging by Induction, 3 The sphere is grounded
Grounded means the conductor is connected to an infinite reservoir for electrons, such as the Earth
Some electrons can leave the sphere through the ground wire

18. Charging by Induction, 4 The ground wire is removed
There will now be more positive charges in the sphere
The positive charge has been induced in the sphere

19. Charging by Induction, 5 The rod is removed
The rod has lost none of its charge during this process
The electrons remaining on the sphere redistribute themselves
There is still a net positive charge on the sphere

20. Charge Rearrangement in Insulators
A process similar to induction can take place in insulators
The charges within the molecules of the material are rearranged
The effect is called polarization

21. Charles Coulomb
1736 – 1806
Major contributions in the fields of electrostatics and magnetism
Also investigated
Strengths of materials
Structural mechanics
Ergonomics
How people and animals can best do work

22. Coulomb’s Law
Charles Coulomb measured the magnitudes of electric forces between two small charged spheres
He found the force depended on the charges and the distance between them

23. Coulomb’s Law, 2
The electrical force between two stationary charged particles is given by Coulomb’s Law
The force is inversely proportional to the square of the separation r between the particles and directed along the line joining them
The force is proportional to the product of the charges, q1 and q2, on the two particles

24. Point Charge
The term point charge refers to a particle of zero size that carries an electric charge
The electrical behavior of electrons and protons is well described by modeling them as point charges

25. Coulomb’s Law, Equation
Mathematically,
The SI unit of charge is the Coulomb, C
ke is called the Coulomb Constant
ke = x 109 N.m2/C2 = 1/(4peo)
eo is the permittivity free space
eo = x C2 / N.m2

26. Coulomb's Law, Notes Remember the charges need to be in Coulombs
e is the smallest unit of charge
Except quarks
e = 1.6 x C
So 1 C needs 6.24 x 1018 electrons or protons
Typical charges can be in the µC range
Remember that force is a vector quantity

27. Vector Nature of Electric Forces
In vector form,
is a unit vector directed from q1 to q2
The like charges produce a repulsive force between them

28. Vector Nature of Electrical Forces, 2
Electrical forces obey Newton’s Third Law
The force on q1 is equal in magnitude and opposite in direction to the force on q2
With like signs for the charges, the product q1q2 is positive and the force is repulsive
With opposite signs for the charges, the product q1q2 is negative and the force is attractive

29. Vector Nature of Electrical Forces, 3
Two point charges are separated by a distance r
The unlike charges produce an attractive force between them
With unlike signs for the charges, the product q1q2 is negative and the force is attractive

30. A Final Note About Directions
The sign of the product of q1q2 gives the relative direction of the force between q1 and q2
The absolute direction is determined by the actual location of the charges

31. Hydrogen Atom Example
The electrical force between the electron and proton is found from Coulomb’s Law
Fe = keq1q2 / r2 = 8.2 x 10-8 N
This can be compared to the gravitational force between the electron and the proton
Fg = Gmemp / r2 = 3.6 x N

32. The Superposition Principle
The resultant force on any one particle equals the vector sum of the individual forces due to all the other individual particles
Remember to add the forces as vectors
The resultant force on q1 is the vector sum of all the forces exerted on it by other charges:

33. Zero Resultant Force, Superposition Example
Where is the resultant force equal to zero?
The magnitudes of the individual forces will be equal
Directions will be opposite
Will result in a quadratic
Choose the root that gives the forces in opposite directions

34. Electric Field – Test Particle
The electric field is defined in terms of a test particle, qo
By convention, the test particle is always a positive electric charge
The test particle is used to detect the existence of the field
It is also used to evaluate the strength of the field
The test charge is assumed to be small enough not to disturb the charge distribution responsible for the field

35. Electric Field – Definition
An electric field is said to exist in the region of space around a charged object
This charged object is the source particle
When another charged object, the test charge, enters this electric field, an electric force acts on it

36. Electric Field – Definition, cont
The electric field is defined as the electric force on the test charge per unit charge
The electric field vector, , at a point in space is defined as the electric force, , acting on a positive test charge, qo placed at that point divided by the test charge:

37. Electric Field, Notes
is the field produced by some charge or charge distribution, separate from the test charge
The existence of an electric field is a property of the source charge
The presence of the test charge is not necessary for the field to exist
The test charge serves as a detector of the field

38. Relationship Between F and E
This is valid for a point charge only
One of zero size
For larger objects, the field may vary over the size of the object
If q is positive, and are in the same direction
If q is negative, and are in opposite directions

39. Electric Field Notes, Final
The direction of is that of the force on a positive test charge
The SI units of are N/C
We can also say that an electric field exists at a point if a test charge at that point experiences an electric force

40. Electric Field, Vector Form
Remember Coulomb’s Law, between the source and test charges, can be expressed as
Then, the electric field will be

41. More About Electric Field Direction
a) q is positive, the force is directed away from q
b) The direction of the field is also away from the positive source charge
c) q is negative, the force is directed toward q
d) The field is also toward the negative source charge

42. Superposition with Electric Fields
At any point P, the total electric field due to a group of source charges equals the vector sum of electric fields at that point due to all the particles

43. Superposition Example
Find the electric field due to q1,
Find the electric field due to q2,
Remember, the fields add as vectors
The direction of the individual fields is the direction of the force on a positive test charge

44. Electric Field – Continuous Charge Distribution
The distances between charges in a group of charges may be much smaller than the distance between the group and a point of interest
In this situation, the system of charges can be modeled as continuous
The system of closely spaced charges is equivalent to a total charge that is continuously distributed along some line, over some surface, or throughout some volume

45. Electric Field – Continuous Charge Distribution, cont
Procedure:
Divide the charge distribution into small elements, each of which contains Dq
Calculate the electric field due to one of these elements at point P
Evaluate the total field by summing the contributions of all the charge elements

46. Electric Field – Continuous Charge Distribution, equations
For the individual charge elements
Because the charge distribution is continuous

47. Charge Densities
Volume charge density – when a charge is distributed evenly throughout a volume
r = Q / V
Surface charge density – when a charge is distributed evenly over a surface area
s = Q / A
Linear charge density – when a charge is distributed along a line
l = Q / l

48. Problem Solving Strategy
Conceptualize
Imagine the type of electric field that would be created by the charges or charge distribution
Categorize
Analyzing a group of individual charges or a continuous charge distribution?
Think about symmetry

49. Problem Solving Hints, cont
Analyze
Group of individual charges: use the superposition principle
The resultant field is the vector sum of the individual fields
Continuous charge distributions: the vector sums for evaluating the total electric field at some point must be replaced with vector integrals
Divide the charge distribution into infinitesimal pieces, calculate the vector sum by integrating over the entire charge distribution

50. Problem Solving Hints, final
Analyze, cont
Symmetry: take advantage of any symmetry in the system
Finalize
Check to see if your field is consistent with the mental representation and reflects any symmetry
Imagine varying parameters to see if the result changes in a reasonable way

51. Electric Field Lines
Field lines give us a means of representing the electric field pictorially
The electric field vector is tangent to the electric field line at each point
The line has a direction that is the same as that of the electric field vector
The number of lines per unit area through a surface perpendicular to the lines is proportional to the magnitude of the electric field in that region

52. Electric Field Lines, General
The density of lines through surface A is greater than through surface B
The magnitude of the electric field is greater on surface A than B
The lines at different locations point in different directions
This indicates the field is non-uniform

53. Electric Field Lines, Positive Point Charge
The field lines radiate outward in all directions
In three dimensions, the distribution is spherical
The lines are directed away from the source charge
A positive test charge would be repelled away from the positive source charge

54. Electric Field Lines, Negative Point Charge
The field lines radiate inward in all directions
The lines are directed toward the source charge
A positive test charge would be attracted toward the negative source charge

55. Electric Field Lines – Dipole
The charges are equal and opposite
The number of field lines leaving the positive charge equals the number of lines terminating on the negative charge

56. Electric Field Lines – Like Charges
The charges are equal and positive
The same number of lines leave each charge since they are equal in magnitude
At a great distance, the field is approximately equal to that of a single charge of 2q

57. Electric Field Lines, Unequal Charges
The positive charge is twice the magnitude of the negative charge
Two lines leave the positive charge for each line that terminate on the negative charge
At a great distance, the field would be approximately the same as that due to a single charge of +q

58. Electric Field Lines – Rules for Drawing
The lines must begin on a positive charge and terminate on a negative charge
In the case of an excess of one type of charge, some lines will begin or end infinitely far away
The number of lines drawn leaving a positive charge or approaching a negative charge is proportional to the magnitude of the charge
Field lines cannot intersect

59. Motion of Charged Particles
When a charged particle is placed in an electric field, it experiences an electrical force
If this is the only force on the particle, it must be the net force
The net force will cause the particle to accelerate according to Newton’s Second Law

60. Motion of Particles, cont
If E is uniform, then a is constant
If the particle has a positive charge, its acceleration is in the direction of the field
If the particle has a negative charge, its acceleration is in the direction opposite the electric field
Since the acceleration is constant, the kinematic equations can be used

61. Electron in a Uniform Field, Example
The electron is projected horizontally into a uniform electric field
The electron undergoes a downward acceleration
The charge is negative, so the acceleration is opposite the field
Its motion is parabolic while between the plates

62. Cathode Ray Tube (CRT)
A CRT is commonly used to obtain a visual display of electronic information in oscilloscopes, radar systems, televisions, etc
The CRT is a vacuum tube in which a beam of electrons is accelerated and deflected under the influence of electric or magnetic fields

63. CRT, cont
The electrons are deflected in various directions by two sets of plates
The placing of charge on the plates creates the electric field between the plates and allows the beam to be steered

64. Electric Flux
Electric flux is the product of the magnitude of the electric field and the surface area, A, perpendicular to the field
FE = E A

65. Electric Flux, General Area
The electric flux is proportional to the number of electric field lines penetrating some surface
The field lines may make some angle q with the perpendicular to the surface
Then FE = E A cos q

66. Electric Flux, Interpreting the Equation
The flux is a maximum when the surface is perpendicular to the field
The flux is zero when the surface is parallel to the field
If the field varies over the surface, F = E A cos q is valid for only a small element of the area

67. Electric Flux, General
In the more general case, look at a small area element
In general, this becomes

68. Electric Flux, final
The surface integral means the integral must be evaluated over the surface in question
In general, the value of the flux will depend both on the field pattern and on the surface
The units of electric flux will be N.m2/C2

69. Electric Flux, Closed Surface
Assume a closed surface
The vectors point in different directions
At each point, they are perpendicular to the surface
By convention, they point outward

70. Flux Through Closed Surface, cont
At (1), the field lines are crossing the surface from the inside to the outside; q <90o, F is positive
At (2), the field lines graze the surface; q =90o, F = 0
At (3), the field lines are crossing the surface from the outside to the inside;180o > q >90o, F is negative

71. Flux Through Closed Surface, final
The net flux through the surface is proportional to the net number of lines leaving the surface
This net number of lines is the number of lines leaving the volume surrounding the surface minus the number entering the volume
If En is the component of E perpendicular to the surface, then

72. Gauss’ Law, Introduction
Gauss’ Law is an expression of the general relationship between the net electric flux through a closed surface and the charge enclosed by the surface
The closed surface is often called a Gaussian surface
Gauss’ Law is of fundamental importance in the study of electric fields

73. Gauss’ Law – General
A positive point charge, q, is located at the center of a sphere of radius r
The magnitude of the electric field everywhere on the surface of the sphere is
E = ke q / r2

74. Gauss’ Law – General, cont.
The field lines are directed radially outward and are perpendicular to the surface at every point
This will be the net flux through the Gaussian surface, the sphere of radius r
We know E = kq/r2 and Asphere = 4pr2, so

75. Gauss’ Law – General, notes
The net flux through any closed surface surrounding a point charge, q, is given by q/eo and is independent of the shape of that surface
The net electric flux through a closed surface that surrounds no charge is zero
Since the electric field due to many charges is the vector sum of the electric fields produced by the individual charges, the flux through any closed surface can be expressed as

76. Gauss’ Law – Final Gauss’ Law states
qin is the net charge inside the surface
represents the total electric field at any point on the surface
The total electric field may have contributions from charges both inside and outside of the surface

77. Applying Gauss’ Law
Gauss’ Law is valid for the electric field of any system of charges or continuous distribution of charge.
Although Gauss’ Law can, in theory, be solved to find E for any charge configuration, in practice it is limited to symmetric situations
Particularly spherical, cylindrical, or plane symmetry
Remember, the Gaussian surface is a surface you choose, it does not have to coincide with a real surface

78. Conditions for a Gaussian Surface
Try to choose a surface that satisfies one or more of these conditions:
The value of the electric field can be argued from symmetry to be constant over the surface
The dot product can be expressed as a simple algebraic produce E dA because and are parallel
The dot product is 0 because and are perpendicular
The field can be argued to be zero everywhere over the surface

79. Field Due to a Point Charge
Choose a sphere as the Gaussian surface
E is parallel to dA at each point on the surface

80. Field Due to a Spherically Symmetric Charge Distribution
Select a sphere as the Gaussian surface
For r> a

81. Spherically Symmetric, cont
Select a sphere as the Gaussian surface, r < a
qin < Q
qin = r (4/3pr3)

82. Spherically Symmetric Distribution, final
Inside the sphere, E varies linearly with r
E ® 0 as r ® 0
The field outside the sphere is equivalent to that of a point charge located at the center of the sphere

83. Field at a Distance from a Line of Charge
Select a cylindrical charge distribution
The cylinder has a radius of r and a length of l
E is constant in magnitude and perpendicular to the surface at every point on the curved part of the surface

84. Field Due to a Line of Charge, cont
The end view confirms the field is perpendicular to the curved surface
The field through the ends of the cylinder is 0 since the field is parallel to these surfaces

85. Field Due to a Line of Charge, final
Use Gauss’ Law to find the field

86. Field Due to a Plane of Charge
must be perpendicular to the plane and must have the same magnitude at all points equidistance from the plane
Choose a small cylinder whose axis is perpendicular to the plane for the Gaussian surface

87. Field Due to a Plane of Charge, cont
is parallel to the curved surface and there is no contribution to the surface area from this curved part of the cylinder
The flux through each end of the cylinder is EA and so the total flux is 2EA

88. Field Due to a Plane of Charge, final
The total charge in the surface is sA
Applying Gauss’ Law
Note, this does not depend on r
Therefore, the field is uniform everywhere

89. Electrostatic Equilibrium
When there is no net motion of charge within a conductor, the conductor is said to be in electrostatic equilibrium
Every charge in the conductor is a particle in equilibrium
They experience zero net force

90. Properties of a Conductor in Electrostatic Equilibrium
The electric field is zero everywhere inside the conductor
If an isolated conductor carries a charge, the charge resides entirely on its surface
The electric field just outside a charged conductor is perpendicular to the surface and has a magnitude of s/eo
On an irregularly shaped conductor, the surface charge density is greatest at locations where the radius of curvature is the smallest

91. Property 1: Einside = 0
Consider a conducting slab in an external field E
If the field inside the conductor was not zero, free electrons in the conductor would experience an electrical force
These electrons would accelerate
These electrons would not be in equilibrium
Therefore, there cannot be a field inside the conductor

92. Property 1: Einside = 0, cont
Before the external field was applied, free electrons are distributed throughout the conductor
When the external field is applied, the electron redistribute until the magnitude of the internal field equals the magnitude of the external field
There is a net field of zero inside the conductor

93. Property 2: Charge Resides on the Surface
Choose a Gaussian surface inside but close to the actual surface
The electric field inside is zero (prop. 1)
There is no net flux through the Gaussian surface
Because the Gaussian surface can be as close to the actual surface as desired, there can be no charge inside the surface

94. Property 2: Charge Resides on the Surface, cont
Since no net charge can be inside the surface, any net charge must reside on the surface
Gauss’ Law does not indicate the distribution of these charges, only that it must be on the surface of the conductor

95. Property 3: Field’s Magnitude and Direction
Choose a cylinder as the Gaussian surface
The field must be perpendicular to the surface
If there was a parallel component to E, charges would experience a force and accelerate along the surface and it would not be in equilibrium

96. Atmospheric Electric Field
Charge distributions are created resulting in an electric field in the atmosphere
The average negative charge distributed over the surface of the Earth is about 5 x 105 C
The average surface charge density over the surface of the Earth is ~10-9 C/m2

97. Atmospheric Electric Field, cont
The average magnitude of the electric field at the surface of the Earth is ~102 N/C
This is the typical value of the fair-weather electric field
The direction of the field is downward
The Earth’s surface is negatively charged

98. Thundercloud
The electric field under the thundercloud is significantly higher than the fair-weather field
The charge distribution in the cloud can be modeled as tripolar
The high electric fields cause lightning discharges between the cloud and the ground