×
Start learning today, and be successful in your academic & professional career. Start Today!  Dan Fullerton

Conductors

Slide Duration:

Section 1: Electricity
Electric Charge & Coulomb's Law

30m 48s

Intro
0:00
Objective
0:15
Electric Charges
0:50
Matter is Made Up of Atoms
0:52
Most Atoms are Neutral
1:02
Ions
1:11
Coulomb
1:18
Elementary Charge
1:34
Law of Conservation of Charge
2:03
Example 1
2:39
Example 2
3:42
Conductors and Insulators
4:41
Conductors Allow Electric Charges to Move Freely
4:43
Insulators Do Not Allow Electric Charges to Move Freely
4:50
Resistivity
4:58
Charging by Conduction
5:32
Conduction
5:37
Balloon Example
5:40
Charged Conductor
6:14
Example 3
6:28
The Electroscope
7:16
Charging by Induction
7:57
Bring Positive Rod Near Electroscope
8:08
Ground the Electroscope
8:27
Sever Ground Path and Remove Positive Rod
9:07
Example 4
9:39
Polarization and Electric Dipole Moment
11:46
Polarization
11:54
Electric Dipole Moment
12:05
Coulomb's Law
12:38
Electrostatic Force, Also Known as Coulombic Force
12:48
How Force of Attraction or Repulsion Determined
12:55
Formula
13:08
Coulomb's Law: Vector Form
14:18
Example 5
16:05
Example 6
18:25
Example 7
19:14
Example 8
23:21
Electric Fields

1h 19m 22s

Intro
0:00
Objectives
0:09
Electric Fields
1:33
Property of Space That Allows a Charged Object to Feel a Force
1:40
Detect the Presence of an Electric Field
1:51
Electric Field Strength Vector
2:03
Direction of the Electric Field Vector
2:21
Example 1
3:00
Visualizing the Electric Field
4:13
Electric Field Lines
4:56
E Field Due to a Point Charge
7:19
Derived from the Definition of the Electric Field and Coulomb's Law
7:24
Finding the Electric Field Due to Multiple Point Charges
8:37
Comparing Electricity to Gravity
8:51
Force
8:54
Field Strength
9:09
Constant
9:19
Charge Units vs. Mass Units
9:35
Attracts vs. Repel
9:44
Example 2
10:06
Example 3
17:25
Example 4
24:29
Example 5
25:23
Charge Densities
26:09
Linear Charge Density
26:26
Surface Charge Density
26:30
Volume Charge Density
26:47
Example 6
27:26
Example 7
37:07
Example 8
50:13
Example 9
54:01
Example 10
1:03:10
Example 11
1:13:58
Gauss's Law

52m 53s

Intro
0:00
Objectives
0:07
Electric Flux
1:16
Amount of Electric Field Penetrating a Surface
1:19
Symbol
1:23
Point Charge Inside a Hollow Sphere
4:31
Place a Point Charge Inside a Hollow Sphere of Radius R
4:39
Determine the Flux Through the Sphere
5:09
Gauss's Law
8:39
Total Flux
8:59
Gauss's Law
9:10
Example 1
9:53
Example 2
17:28
Example 3
22:37
Example 4
25:40
Example 5
30:49
Example 6
45:06
Electric Potential & Electric Potential Energy

1h 14m 3s

Intro
0:00
Objectives
0:08
Electric Potential Energy
0:58
Gravitational Potential Energy
1:02
Electric Potential Energy
1:11
Electric Potential
1:19
Example 1
1:59
Example 2
3:08
The Electron-Volt
4:02
Electronvolt
4:16
1 eV is the Amount of Work Done in Moving an Elementary Charge Through a Potential Difference of 1 Volt
4:26
Conversion Ratio
4:41
Example 3
4:52
Equipotential Lines
5:35
Topographic Maps
5:36
Lines Connecting Points of Equal Electrical Potential
5:47
Always Cross Electrical Field Lines at Right Angles
5:57
Gradient of Potential Increases As Equipotential Lines Get Closer
6:02
Electric Field Points from High to Low Potential
6:27
Drawing Equipotential Lines
6:49
E Potential Energy Due to a Point Charge
8:20
Electric Force from Electric Potential Energy
11:59
E Potential Due to a Point Charge
13:07
Example 4
14:42
Example 5
15:59
Finding Electric Field From Electric Potential
19:06
Example 6
23:41
Example 7
25:08
Example 8
26:33
Example 9
29:01
Example 10
31:26
Example 11
43:23
Example 12
51:51
Example 13
58:12
Electric Potential Due to Continuous Charge Distributions

1h 1m 28s

Intro
0:00
Objectives
0:10
Potential Due to a Charged Ring
0:27
Potential Due to a Uniformly Charged Desk
3:38
Potential Due to a Spherical Shell of Charge
11:21
Potential Due to a Uniform Solid Sphere
14:50
Example 1
23:08
Example 2
30:43
Example 3
41:58
Example 4
51:41
Conductors

20m 35s

Intro
0:00
Objectives
0:08
Charges in a Conductor
0:32
Charge is Free to Move Until the
0:36
All Charge Resides at Surface
2:18
Field Lines are Perpendicular to Surface
2:34
Electric Field at the Surface of a Conductor
3:04
Looking at Just the Outer Surface
3:08
Large Electric Field Where You Have the Largest Charge Density
3:59
Hollow Conductors
4:22
Draw Hollow Conductor and Gaussian Surface
4:36
Applying Gaussian Law
4:53
Any Hollow Conductor Has Zero Electric Field in Its Interior
5:24
5:35
Electric Field and Potential Due to a Conducting Sphere
6:03
Example 1
7:31
Example 2
12:39
Capacitors

41m 23s

Intro
0:00
Objectives
0:08
What is a Capacitor?
0:42
Electric Device Used to Store Electrical Energy
0:44
Place Opposite Charges on Each Plate
1:10
Develop a Potential Difference Across the Plates
1:14
Energy is Stored in the Electric Field Between the Plates
1:17
Capacitance
1:22
Ratio of the Charge Separated on the Plates of a Capacitor to the Potential Difference Between the Plates
1:25
Units of Capacitance
1:32
1:37
Formula
1:52
Calculating Capacitance
1:59
Assume Charge on Each Conductor
2:05
Find the Electric Field
2:11
Calculate V by Integrating the Electric Field
2:21
Utilize C=Q/V to Solve for Capitance
2:33
Example 1
2:44
Example 2
5:30
Example 3
10:46
Energy Stored in a Capacitor
15:25
Work is Done Charging a Capacitor
15:28
Solve For That
15:55
Field Energy Density
18:09
Amount of Energy Stored Between the Plates of a Capacitor
18:11
Example
18:25
Dielectrics
20:44
Insulating Materials Place Between Plates of Capacitor to Increase The Devices' Capacitance
20:47
Electric Field is Weakened
21:00
The Greater the Amount of Polarization The Greater the Reduction in Electric Field Strength
21:58
Dielectric Constant (K)
22:30
Formula
23:00
Net Electric Field
23:35
Key Take Away Point
23:50
Example 4
24:00
Example 5
25:50
Example 6
26:50
Example 7
28:53
Example 8
30:57
Example 9
32:55
Example 10
34:59
Example 11
37:35
Example 12
39:57
Section 2: Current Electricity
Current & Resistance

17m 59s

Intro
0:00
Objectives
0:08
Electric Current
0:44
Flow Rate of Electric Charge
0:45
Amperes
0:49
Positive Current Flow
1:01
Current Formula
1:19
Drift Velocity
1:35
Constant Thermal Motion
1:39
Net Electron Flow
1:43
When Electric Field is Applied
1:49
Electron Drift Velocity
1:55
Derivation of Current Flow
2:12
Apply Electric Field E
2:20
Define N as the Volume Density of Charge Carriers
2:27
Current Density
4:33
Current Per Area
4:36
Formula
4:44
Resistance
5:14
Ratio of the Potential Drop Across an Object to the Current Flowing Through the Object
5:19
5:23
Resistance of a Wire
6:05
Depends on Resistivity
6:09
Resistivity Relates to the Ability of a Material to Resist the Flow of Electrons
6:25
Refining Ohm's Law
7:22
Conversion of Electric Energy to Thermal Energy
8:23
Example 1
9:54
Example 2
10:54
Example 3
11:26
Example 4
14:41
Example 5
15:24
Circuits I: Series Circuits

29m 8s

Intro
0:00
Objectives
0:08
Ohm's Law Revisited
0:39
Relates Resistance, Potential Difference, and Current Flow
0:39
Formula
0:44
Example 1
1:09
Example 2
1:44
Example 3
2:15
Example 4
2:56
Electrical Power
3:26
Transfer of Energy Into Different Types
3:28
Light Bulb
3:37
Television
3:41
Example 5
3:49
Example 6
4:27
Example 7
5:12
Electrical Circuits
5:42
Closed-Loop Path Which Current Can Flow
5:43
Typically Comprised of Electrical Devices
5:52
Conventional Current Flows from High Potential to Low Potential
6:04
Circuit Schematics
6:26
Three-dimensional Electrical Circuits
6:37
Source of Potential Difference Required for Current to Flow
7:29
Complete Conducting Paths
7:42
Current Only Flows in Complete Paths
7:43
Left Image
7:46
Right Image
7:56
Voltmeters
8:25
Measure the Potential Difference Between Two Points in a Circuit
8:29
Can Remove Voltmeter from Circuit Without Breaking the Circuit
8:47
Very High Resistance
8:53
Ammeters
9:31
Measure the Current Flowing Through an Element of a Circuit
9:32
Very Low Resistance
9:46
Put Ammeter in Correctly
10:00
Example 8
10:24
Example 9
11:39
Example 10
12:59
Example 11
13:16
Series Circuits
13:46
Single Current Path
13:49
Removal of Any Circuit Element Causes an Open Circuit
13:54
Kirchhoff's Laws
15:48
Utilized in Analyzing Circuits
15:54
Kirchhoff's Current Law
15:58
Junction Rule
16:02
Kirchhoff's Voltage Law
16:30
Loop Rule
16:49
Example 12
16:58
Example 13
17:32
Basic Series Circuit Analysis
18:36
Example 14
22:06
Example 15
22:29
Example 16
24:02
Example 17
26:47
Circuits II: Parallel Circuits

39m 9s

Intro
0:00
Objectives
0:16
Parallel Circuits
0:38
Multiple Current Paths
0:40
Removal of a Circuit Element May Allow Other Branches of the Circuit to Continue Operating
0:44
Draw a Simple Parallel Circuit
1:02
Basic Parallel Circuit Analysis
3:06
Example 1
5:58
Example 2
8:14
Example 3
9:05
Example 4
11:56
Combination Series-Parallel Circuits
14:08
Circuit Doesn't Have to be Completely Serial or Parallel
14:10
Look for Portions of the Circuit With Parallel Elements
14:15
Lead to Systems of Equations to Solve
14:42
Analysis of a Combination Circuit
14:51
Example 5
20:23
Batteries
28:49
Electromotive Force
28:50
Pump for Charge
29:04
Ideal Batteries Have No Resistance
29:10
Real Batteries and Internal Resistance
29:20
Terminal Voltage in Real Batteries
29:33
Ideal Battery
29:50
Real Battery
30:25
Example 6
31:10
Example 7
33:23
Example 8
35:49
Example 9
38:43

34m 3s

Intro
0:00
Objectives
0:17
Capacitors in Parallel
0:51
Store Charge on Plates
0:52
Can Be Replaced with an Equivalent Capacitor
0:56
Capacitors in Series
1:12
Must Be the Same
1:13
Can Be Replaced with an Equivalent Capacitor
1:15
RC Circuits
1:30
Comprised of a Source of Potential Difference, a Resistor Network, and Capacitor
1:31
RC Circuits from the Steady-State Perspective
1:37
Key to Understanding RC Circuit Performance
1:48
Charging an RC Circuit
2:08
Discharging an RC Circuit
6:18
The Time Constant
8:49
Time Constant
8:58
By 5 Time Constant
9:19
Example 1
9:45
Example 2
13:27
Example 3
16:35
Example 4
18:03
Example 5
19:39
Example 6
26:14
RC Circuits: Transient Analysis

1h 1m 7s

Intro
0:00
Objectives
0:13
Charging an RC Circuit
1:11
Basic RC Circuit
1:15
Graph of Current Circuit
1:29
Graph of Charge
2:17
Graph of Voltage
2:34
Mathematically Describe the Charts
2:56
Discharging an RC Circuit
13:29
Graph of Current
13:47
Graph of Charge
14:08
Graph of Voltage
14:15
Mathematically Describe the Charts
14:30
The Time Constant
20:03
Time Constant
20:04
By 5 Time Constant
20:14
Example 1
20:39
Example 2
28:53
Example 3
27:02
Example 4
44:29
Example 5
55:24
Section 3: Magnetism
Magnets

8m 38s

Intro
0:00
Objectives
0:08
Magnetism
0:35
Force Caused by Moving Charges
0:36
Dipoles
0:40
Like Poles Repel, Opposite Poles Attract
0:53
Magnetic Domains
0:58
Random Domains
1:04
Net Magnetic Field
1:26
Example 1
1:40
Magnetic Fields
2:03
Magnetic Field Strength
2:04
Magnets are Polarized
2:16
Magnetic Field Lines
2:53
Show the Direction the North Pole of a Magnet Would Tend to Point if Placed on The Field
2:54
Direction
3:25
Magnetic Flux
3:41
The Compass
4:05
Earth is a Giant Magnet
4:07
Earth's Magnetic North Pole
4:10
Compass Lines Up with the Net Magnetic Field
4:48
Magnetic Permeability
5:00
Ratio of the magnetic Field Strength Induced in a Material to the Magnetic Field Strength of the Inducing Field
5:01
Free Space
5:13
Permeability of Matter
5:41
Highly Magnetic Materials
5:47
Magnetic Dipole Moment
5:54
The Force That a Magnet Can Exert on Moving Charges
5:59
Relative Strength of a Magnet
6:04
Example 2
6:26
Example 3
6:52
Example 4
7:32
Example 5
7:57
Moving Charges In Magnetic Fields

29m 7s

Intro
0:00
Objectives
0:08
Magnetic Fields
0:57
Vector Quantity
0:59
Tesla
1:08
Gauss
1:14
Forces on Moving Charges
1:30
Magnetic Force is Always Perpendicular to the Charged Objects Velocity
1:31
Magnetic Force Formula
2:04
Magnitude of That
2:20
Image
2:29
Direction of the Magnetic Force
3:54
Right-Hand Rule
3:57
Electron of Negative Charge
4:04
Example 1
4:51
Example 2
6:58
Path of Charged Particles in B Fields
8:07
Magnetic Force Cannot Perform Work on a Moving Charge
8:08
Magnetic Force Can Change Its Direction
8:11
Total Force on a Moving Charged Particle
9:40
E Field
9:50
B Field
9:54
Lorentz Force
9:57
Velocity Selector
10:33
Charged Particle in Crosses E and B Fields Can Undergo Constant Velocity Motion
10:37
Particle Can Travel Through the Selector Without Any Deflection
10:49
Mass Spectrometer
12:21
Magnetic Fields Accelerate Moving Charges So That They Travel in a Circle
12:26
Used to Determine the Mass of An Unknown Particle
12:32
Example 3
13:11
Example 4
15:01
Example 5
16:44
Example 6
17:33
Example 7
19:12
Example 8
19:50
Example 9
24:02
Example 10
25:21
Forces on Current-Carrying Wires

17m 52s

Intro
0:00
Objectives
0:08
Forces on Current-Carrying Wires
0:42
Moving Charges in Magnetic Fields Experience Forces
0:45
Current in a Wire is Just Flow of Charges
0:49
Direction of Force Given by RHR
4:04
Example 1
4:22
Electric Motors
5:59
Example 2
8:14
Example 3
8:53
Example 4
10:09
Example 5
11:04
Example 6
12:03
Magnetic Fields Due to Current-Carrying Wires

24m 43s

Intro
0:00
Objectives
0:08
Force on a Current-Carrying Wire
0:38
Magnetic Fields Cause a Force on Moving Charges
0:40
Current Carrying Wires
0:44
How to Find the Force
0:55
Direction Given by the Right Hand Rule
1:04
Example 1
1:17
Example 2
2:26
Magnetic Field Due to a Current-Carrying Wire
4:20
Moving Charges Create Magnetic Fields
4:24
Current-Carrying Wires Carry Moving Charges
4:27
Right Hand Rule
4:32
Multiple Wires
4:51
Current-Carrying Wires Can Exert Forces Upon Each Other
4:58
First Right Hand Rule
5:15
Example 3
6:46
Force Between Parallel Current Carrying Wires
8:01
Right Hand Rules to Determine Force Between Parallel Current Carrying Wires
8:03
Find Magnetic Field Due to First Wire, Then Find Direction of Force on 2nd Wire
8:08
Example
8:20
Gauss's Law for Magnetism
9:26
Example 4
10:35
Example 5
12:57
Example 6
14:19
Example 7
16:50
Example 8
18:15
Example 9
18:43
The Biot-Savart Law

21m 50s

Intro
0:00
Objectives
0:07
Biot-Savart Law
0:24
Brute Force Method
0:49
Draw It Out
0:54
Diagram
1:35
Example 1
3:43
Example 2
7:02
Example 3
14:31
Ampere's Law

26m 31s

Intro
0:00
Objectives
0:07
Ampere's Law
0:27
Finds the Magnetic Field Due to Current Flowing in a Wire in Situations of Planar and Cylindrical Symmetry
0:30
Formula
0:40
Example
1:00
Example 1
2:19
Example 2
4:08
Example 3
6:23
Example 4
8:06
Example 5
11:43
Example 6
13:40
Example 7
17:54
Magnetic Flux

7m 24s

Intro
0:00
Objectives
0:07
Magnetic Flux
0:31
Amount of Magnetic Field Penetrating a Surface
0:32
Webers
0:42
Flux
1:07
Total Magnetic Flux
1:27
Magnetic Flux Through Closed Surfaces
1:51
Gauss's Law for Magnetism
2:20
Total Flux Magnetic Flux Through Any Closed Surface is Zero
2:23
Formula
2:45
Example 1
3:02
Example 2
4:26

1h 4m 33s

Intro
0:00
Objectives
0:08
0:44
0:46
Direction of the Induced Current is Given by Lenz's Law
1:09
Formula
1:15
Lenz's Law
1:49
Lenz's Law
2:14
Lenz's Law
2:16
Example
2:30
Applying Lenz's Law
4:09
If B is Increasing
4:13
If B is Decreasing
4:30
Maxwell's Equations
4:55
Gauss's Law
4:59
Gauss's Law for Magnetism
5:16
Ampere's Law
5:26
5:39
Example 1
6:14
Example 2
9:36
Example 3
11:12
Example 4
19:33
Example 5
26:06
Example 6
31:55
Example 7
42:32
Example 8
48:08
Example 9
55:50
Section 4: Inductance, RL Circuits, and LC Circuits
Inductance

6m 41s

Intro
0:00
Objectives
0:08
Self Inductance
0:25
Ability of a Circuit to Oppose the Magnetic Flux That is Produced by the Circuit Itself
0:27
Changing Magnetic Field Creates an Induced EMF That Fights the Change
0:37
Henrys
0:44
Function of the Circuit's Geometry
0:53
Calculating Self Inductance
1:10
Example 1
3:40
Example 2
5:23
RL Circuits

42m 17s

Intro
0:00
Objectives
0:11
Inductors in Circuits
0:49
Inductor Opposes Current Flow and Acts Like an Open Circuit When Circuit is First Turned On
0:52
Inductor Keeps Current Going and Acts as a Short
1:04
If the Battery is Removed After a Long Time
1:16
Resister Dissipates Power, Current Will Decay
1:36
Current in RL Circuits
2:00
Define the Diagram
2:03
Mathematically Solve
3:07
Voltage in RL Circuits
7:51
Voltage Formula
7:52
Solve
8:17
Rate of Change of Current in RL Circuits
9:42
Current and Voltage Graphs
10:54
Current Graph
10:57
Voltage Graph
11:34
Example 1
12:25
Example 2
23:44
Example 3
34:44
LC Circuits

9m 47s

Intro
0:00
Objectives
0:08
LC Circuits
0:30
Assume Capacitor is Fully Charged When Circuit is First Turned On
0:38
Interplay of Capacitor and Inductor Creates an Oscillating System
0:42
Charge in LC Circuit
0:57
Current and Potential in LC Circuits
7:14
Graphs of LC Circuits
8:27
Section 5: Maxwell's Equations
Maxwell's Equations

3m 38s

Intro
0:00
Objectives
0:07
Maxwell's Equations
0:19
Gauss's Law
0:20
Gauss's Law for Magnetism
0:44
1:00
Ampere's Law
1:18
Revising Ampere's Law
1:49
Allows Us to Calculate the Magnetic Field Due to an Electric Current
1:50
Changing Electric Field Produces a Magnetic Field
1:58
Conduction Current
2:33
Displacement Current
2:44
Maxwell's Equations (Complete)
2:58
Section 6: Sample AP Exams
1998 AP Practice Exam: Multiple Choice Questions

32m 33s

Intro
0:00
0:11
Multiple Choice 36
0:36
Multiple Choice 37
2:07
Multiple Choice 38
2:53
Multiple Choice 39
3:32
Multiple Choice 40
4:37
Multiple Choice 41
4:43
Multiple Choice 42
5:22
Multiple Choice 43
6:00
Multiple Choice 44
8:09
Multiple Choice 45
8:27
Multiple Choice 46
9:03
Multiple Choice 47
9:30
Multiple Choice 48
10:19
Multiple Choice 49
10:47
Multiple Choice 50
12:25
Multiple Choice 51
13:10
Multiple Choice 52
15:06
Multiple Choice 53
16:01
Multiple Choice 54
16:44
Multiple Choice 55
17:10
Multiple Choice 56
19:08
Multiple Choice 57
20:39
Multiple Choice 58
22:24
Multiple Choice 59
22:52
Multiple Choice 60
23:34
Multiple Choice 61
24:09
Multiple Choice 62
24:40
Multiple Choice 63
25:06
Multiple Choice 64
26:07
Multiple Choice 65
27:26
Multiple Choice 66
28:32
Multiple Choice 67
29:14
Multiple Choice 68
29:41
Multiple Choice 69
31:23
Multiple Choice 70
31:49
1998 AP Practice Exam: Free Response Questions

29m 55s

Intro
0:00
0:14
Free Response 1
0:22
Free Response 2
10:04
Free Response 3
16:22

• ## Related Books

 1 answerLast reply by: Professor Dan FullertonFri Mar 17, 2017 6:25 AMPost by Mark Sim on March 16, 2017Why is a charge free to move until the E=0? Why can't they move when E=o?

### Conductors

• Charges are free to move in conductors.
• At electrostatic equilibrium, there are no moving charges in a conductor, therefore there is no net force, and the electric field inside the conductor must be zero.
• All excess charge on a conductor lies on the surface of the conductor.
• The electric field on the surface of a conductor must be perpendicular to the surface.
• The electric field inside any hollow conductor is zero (assuming there are no charges in the hollow region). This allows hollow conductors to be utilized to isolate regions completely from electric fields. In this configuration, the hollow conductor is known as a Faraday Cage.

### Conductors

Lecture Slides are screen-captured images of important points in the lecture. Students can download and print out these lecture slide images to do practice problems as well as take notes while watching the lecture.

• Intro 0:00
• Objectives 0:08
• Charges in a Conductor 0:32
• Charge is Free to Move Until the
• All Charge Resides at Surface
• Field Lines are Perpendicular to Surface
• Electric Field at the Surface of a Conductor 3:04
• Looking at Just the Outer Surface
• Large Electric Field Where You Have the Largest Charge Density
• Hollow Conductors 4:22
• Draw Hollow Conductor and Gaussian Surface
• Applying Gaussian Law
• Any Hollow Conductor Has Zero Electric Field in Its Interior
• Electric Field and Potential Due to a Conducting Sphere 6:03
• Example 1 7:31
• Example 2 12:39

### Transcription: Conductors

Hello, everyone, and welcome back to www.educator.com.0000

I'm Dan Fullerton and in this lesson we are going to talk about conductors.0003

To begin with, we are going to try and understand the nature of electric fields and electric potential in a round conductor.0009

We are going to explain how the all the axis charge on a conductor resides on the conductor's surface not in its bulk.0015

Explain why a conductor must be an equal potential.0022

And finally, graph the electric field and electric potential inside and outside a charge conducting sphere.0025

As we start talking about charges in a conductor, we have to remember that charge in a conductor is free to move until there is no E field.0031

Charge is free to move until the electric field is equal to 0.0041

If we have all these free charges in a conductor and they are freedom to move,0052

they are going to migrate to the area of the lowest possible energy state,0055

and as far apart as possible and that usually puts them on the surface.0060

If they are moving, they are no longer in equilibrium.0064

If there are electric fields inside the conductor, the charges would move.0066

When you have a conductor that is in equilibrium, all the charges have already distributed themselves so that they are not going to move anymore.0069

There is no net internal electric field, otherwise they will keep moving.0075

The electric field inside a conductor is 0.0081

You can think about that if we draw some sort of random conducting shape here.0084

If we have some charge + Q that we put somewhere on that surface, the charge is are going to repel each other0089

and they are going to go and move as far apart as they possibly can, in order to minimize their potential energy that puts them all on the outer surface.0095

We can also go and draw a Gaussian surface inside a conductor.0109

Let us put our Gaussian surface something like that.0116

As we draw our Gaussian surface, we can look at it from Gausses law perspective of the integral / the close surface of E ⋅ DA.0118

It is the enclosed charge ÷ ε₀, but we just said that the electric field must be 0.0128

Therefore, the enclosed charge is going to be 0.0135

All the charge has to reside at the surface not in the bulk of the conductor.0139

Now field lines are all going to run perpendicular to the surface.0152

If they are not perpendicular to the surface, there will be some component that is parallel to the surface and the charges would then move.0163

As we just said, the charges already moved in the conductor0173

until the electric field is 0 so electric field lines have to intersect that conductor at 90° angles.0176

They have to be perpendicular.0180

Let us look some more at the electric field at the surface of a conductor.0184

If we look at just the outer surface, let us make a note of that.0187

Just the outer surface, let us take, using the Gausses law, the integral / the closed loop of E ⋅ DA is our enclosed charge ÷ ε₀.0197

Let us assume a surface where we have some symmetry.0212

We will say that our charge is σ A, therefore, electric field × the area is going to be our enclosed charge σ A / ε₀.0215

Or the electric field is equal to the surface charge density σ ÷ the constant ε₀.0229

What this really means that, is you are going to have the largest electric field where you have the largest charge density,0238

which is common sense as well but probably we are stating explicitly here.0254

Let us talk for a minute about hollow conductors.0260

In a hollow conductor, you can determine the location of charge by utilizing Gauss’s law.0264

Choosing Gaussian surfaced in the metal of a hollow conductor.0269

Making out that the electric field inside that conductor must be 0.0273

Let us start by drawing our hollow conductor, there it is.0277

And inside of that we are going to draw our Gaussian surface.0283

We know the electric field inside there must be 0.0288

Applying Gauss’s law, the integral / the closed surface of E ⋅ DA is our enclosed charge / ε 0.0291

Again, we know the electric field must be 0 so our enclosed charge is going to be 0.0304

Therefore, the charge must remain on that outer surface,0312

the entire conductor is at equal potential with the same electric potential, the same voltage,0315

and field lines must run perpendicular to that conducting sphere.0320

Any hollow conductor has 0 electric field in its interior.0324

That allows you to use hollow conductors as a way to isolate regions completely from electric fields.0329

When you do that, the device we are using a hollow conductor to isolate from electric fields is known as Faraday cage.0335

Hollow conductor completely closed so great way to isolate items from electric fields.0352

Alright, let us take a look at the electric field and electric potential due to a conducting sphere.0359

If this is our conducting sphere down at the bottom and0364

we want to plot the electric field as a function of position, we know inside our conductor the electric field must be 0.0367

Outside, right at the surface we said the electric field is equal to the surface charge density ÷ ε 0.0379

And then that is going to fall off our proportional to 1/ R²0387

as if we are treating it all as a point charge located at the center, when we are outside that sphere.0393

We have done this derivation a few times.0398

But looking at the potential, there is something you have to be careful of.0400

Just because the electric field is 0 inside the conductor does not mean the potential is 0.0403

What it means is the potential is constant, it is not changing.0409

Our potential graph from the outside as we come from infinity toward the edge, we have Q / 4 π ε₀ R.0412

Again, the same as if we had a point charge located in the center of our sphere.0420

Once we get to touching the sphere however, the entire sphere is in equal potential so we have a constant potential0424

that is the same value we had right at the radius of the sphere, which we are calling here R.0429

Take note that potential inside a conducting sphere does not have to be 0.0439

It is what you have right at the edge to the entire sphere is in equal potential.0444

Let us see if we can do a problem or two with this.0451

We have two conducting spheres A and B, they are placed at large distance from each other.0454

By large distance, what we are saying is that the electric field from one does not affect the electric field from the other.0459

The radius of sphere A is 5 cm and the radius of sphere B over here is 20 cm, that is 4 × larger radius.0465

A charge Q of 200 nC is placed on sphere A, our sphere B is uncharged.0472

The spheres are then connected by a wire.0478

Find the charge on each sphere after that wire is connected.0480

A couple of things that may help us here.0484

First, I note that the total charge Q must be equal to QA + QB because we have a closed system.0486

We have a total charge that is going to remain constant.0494

It is going to be 200 nC.0497

Finally, if we rearrange this a little bit, QB = Q – QA.0500

That will be helpful as well.0507

Once we connect these by a wire, we have one large single conducting object.0510

Once everything rearranges itself, all the charges move to wherever they are going to.0515

The entire thing is it in equal potential.0519

We know that the potential in A and the potential at B must be the same once they are connected by the wire.0522

The sum of the charges on each sphere must equal the total charge it had initially by the law of conservation of charge which is going to be 200 nC.0529

Let us see if we can solve this.0538

Looking at sphere A first, let us write that the potential at A we know is going to be the charge on A ÷ 4 π ε₀ RA.0542

We can also look at B and say that the potential at B is going to be the charge on B QB/ 4 π ε₀ radius of B.0557

And because they must be at equal potential, VA = VB.0569

We can state then that QA/ 4 π ε₀ RA must be equal to QB/ 4 π ε₀ RB.0575

We can do a little simplification here.0598

We have got 4 π ε₀ in the denominator on both sides and let us just simplify this with one more step.0600

We can rewrite this now as QA/ RA must equal QB/ RB.0605

We also said on the previous slide that QB must equal Q -QA by the law of conservation of charge.0616

We can write this again as QA/ RA will equal Q –QA, substituting that in for QB, ÷ RB.0626

And now with a little bit of algebra, we can start to solve for our charges.0638

Let us do some cross multiplication QA RB must equal Q RA-QA RA.0643

Or rearranging these, QA RA + QA RB will equal Q RA.0655

If I factor out a QA from this left hand side, QA × RA + RB must equal Q RA.0674

Therefore, solving for just QA, QA will equal Q × RA/(RA+ RB).0688

Now I can substitute in the values that I know.0700

Q is 200 nC, radius of A we said was 5 cm and RA is 5 cm again + RB 20 cm.0703

That is going to be 5/ 25 or 20% of 200 nC is just going to be 40 nC.0718

If that is the charge on the sphere A and our total is 200, that means that the charge on B must be 200 -40 or 160 nC by the law of conservation of charge.0732

QA is 40 nC, QB must be 160 nC, our total charge is still 200 nC.0748

Alright let us take a look at an AP style problem, going back to the 2004 APC E and M exam free response number 1.0758

Take a minute, you can pull up off the site, find it from the Internet, download it, print it out.0768

Take a minute to give it a try and we will see what happens when we try and do it together here.0773

As you look at this problem, we are given a hollow conductor0780

that has a line of linear charge density parallel to the axis but it is a little bit off center.0785

We are asked to sketch the electric field lines and use ± signs to show any charge that is used on the conductor.0792

Let us try drawing it here first.0799

Looks like it is kind of something like that and then we have got our line of charge + λ there.0805

And we know that we have, this is a positive charge + λ.0812

The first thing is to sketch the electric field lines in these different regions.0816

Let us start here at the middle.0820

As we draw electric field lines, because this is closer to the edge over here, we are going to have more field lines over here.0822

As I draw these, I'm trying very hard to draw them so that they intersect the conductor at a 90° angle.0834

They have to remain perpendicular.0840

We are going to have more of these lines over on the right hand side than the left hand side because we are close to the conductor there.0842

Now inside the conductor, we are not going to have any electric field.0854

The electric field inside the conductor is 0.0859

But outside in region 3, now we can draw our electric field lines here.0861

Of course, we are going to have them roughly equally spaced going radially outward.0866

Looking at the charges, if this is a + λ then we are going to have negative charges attracted right at the surface of the conductor.0880

By conservation of charge, we have to have the same opposite charge on the outside of our hollow conductor.0892

That I think would do pretty well for getting us through part A.0904

For part B, it asks us to rank the electric potentials at A, B, C, D, and E from the highest to the lowest.0909

Without a doubt, as I look at our diagram there, it looks to me like the highest has to be D because it is going to be closest to the charge.0918

I would say that we are going to have, VD is going to be the highest.0926

And next, it looks like we have VC, and then we get into B and E, that are in the conductor themselves.0931

B and E because they are in the conductor must be at equal potential.0943

We can write that we have VB equal VE.0947

All of those must be greater than our lowest potential which is going to be VA, way out there in region 3,0951

the farthest from our line of charge.0959

That would be how I would rank the electric potentials.0962

Going on to part C, let us give ourselves more room here on the next page.0967

For part C, we are replacing the shell by another cylindrical shelling that has the same dimensions0974

but it is non conducting and has some uniform volume charge density +ρ.0978

The infinite line of charge is still there but now it is in the center of the shell.0984

We have got something nice and symmetric.0988

We are asked to use Gauss’s law to find the magnitude of the electric field0990

as a function of distance from the center of the shell for the different regions.0994

Alright let us start as we look inside our shell.0998

As we go in that region, we will use Gauss’s law, integral / the close surface of E ⋅ DA = Q enclosed / ε₀.1002

The left hand side is going to become EA again, where A if we are for careful about how we choose this,1017

we are going to pick a cylindrical shape for our Gaussian surface.1024

That is going to be 2 π R, the circumference of our Gaussian cylinder × its length, let us call that L.1027

Our charge enclosed is just going to be the linear charge density λ × its length L ÷ ε₀.1038

Therefore, our electric field strength is going to be λ L / 2 π RL ε₀ or just λ / 2 π ε₀ R.1047

Moving onto region 2, between R1 and R2, find the electric field in the region between R1 and R2.1069

As we look there between R1 and R2, the electric field is going to be.1085

How are we going to do that?1091

Let us take a look, use Gauss’s law again, integral / the close surface of E ⋅ DA = Q enclosed/ ε₀,1093

which implies then that E × 2 π RL =, as we look at our charge enclosed, we are going to have the charge from that line λ L ,1108

+ we have to take into account the volume that we have in our cylinder, the portion that is inside our Gaussian surface.1121

That is going to be π R² L × ρ - π R1 1² L × that volume charge density / ε₀.1131

That is what is going to give us our enclosed charge when we are in that region between R1 and R2.1151

A little bit of math here, our electric field then is going to be.1157

We have got, we can take the L out, we can factor that of all the sides.1163

We will have λ + ρ π R² – ρ π R1² all over 2 π ε₀ R.1167

Or just factoring this a little bit, let us call that λ/ 2 π ε₀ R.1183

The contribution we have from that internal line + factoring out a ρ/ 2 ε₀ R² – R1².1190

And that gives you the electric field in the region between R1 and R2.1212

The region outside R2 by now should be pretty straightforward to use.1218

I’m going to leave that one to you, for you guys do on your own.1222

At the same place where you found this problem, you can check your answer there.1226

Thank you so much for your time and for watching www.educator.com.1229

We will see you at the next lesson.1232

Make it a great day everyone.1234

OR

### Start Learning Now

Our free lessons will get you started (Adobe Flash® required).