# AP Physics C: Electricity & Magnetism The Biot-Savart Law

Section 3: Magnetism: Lecture 5 | 21:50 min

Much like in the electricity portion right before introducing Gauss’s Law, there is a difficult brute force method of finding the influence of the magnetic field at a point due to a current-carrying wire. Also like in the electricity portion, there is a much easier law to use for solving problems like these, which will be taught in the next video. Nevertheless, this law is important in the world of magnetism, and it never hurts to know it. Because this is a rather difficult method to understand, if asked to do a problem using this law on an exam, you can always refer to Ampere’s Law as a way of checking your work once you have an answer.

Table of Contents

30m 48s

- Intro0:00
- Objective0:15
- Electric Charges0:50
- Matter is Made Up of Atoms0:52
- Most Atoms are Neutral1:02
- Ions1:11
- Coulomb1:18
- Elementary Charge1:34
- Law of Conservation of Charge2:03
- Example 12:39
- Example 23:42
- Conductors and Insulators4:41
- Conductors Allow Electric Charges to Move Freely4:43
- Insulators Do Not Allow Electric Charges to Move Freely4:50
- Resistivity4:58
- Charging by Conduction5:32
- Conduction5:37
- Balloon Example5:40
- Charged Conductor6:14
- Example 36:28
- The Electroscope7:16
- Charging by Induction7:57
- Bring Positive Rod Near Electroscope8:08
- Ground the Electroscope8:27
- Sever Ground Path and Remove Positive Rod9:07
- Example 49:39
- Polarization and Electric Dipole Moment11:46
- Polarization11:54
- Electric Dipole Moment12:05
- Coulomb's Law12:38
- Electrostatic Force, Also Known as Coulombic Force12:48
- How Force of Attraction or Repulsion Determined12:55
- Formula13:08
- Coulomb's Law: Vector Form14:18
- Example 516:05
- Example 618:25
- Example 719:14
- Example 823:21

1h 19m 22s

- Intro0:00
- Objectives0:09
- Electric Fields1:33
- Property of Space That Allows a Charged Object to Feel a Force1:40
- Detect the Presence of an Electric Field1:51
- Electric Field Strength Vector2:03
- Direction of the Electric Field Vector2:21
- Example 13:00
- Visualizing the Electric Field4:13
- Electric Field Lines4:56
- E Field Due to a Point Charge7:19
- Derived from the Definition of the Electric Field and Coulomb's Law7:24
- Finding the Electric Field Due to Multiple Point Charges8:37
- Comparing Electricity to Gravity8:51
- Force8:54
- Field Strength9:09
- Constant9:19
- Charge Units vs. Mass Units9:35
- Attracts vs. Repel9:44
- Example 210:06
- Example 317:25
- Example 424:29
- Example 525:23
- Charge Densities26:09
- Linear Charge Density26:26
- Surface Charge Density26:30
- Volume Charge Density26:47
- Example 627:26
- Example 737:07
- Example 850:13
- Example 954:01
- Example 101:03:10
- Example 111:13:58

52m 53s

- Intro0:00
- Objectives0:07
- Electric Flux1:16
- Amount of Electric Field Penetrating a Surface1:19
- Symbol1:23
- Point Charge Inside a Hollow Sphere4:31
- Place a Point Charge Inside a Hollow Sphere of Radius R4:39
- Determine the Flux Through the Sphere5:09
- Gauss's Law8:39
- Total Flux8:59
- Gauss's Law9:10
- Example 19:53
- Example 217:28
- Example 322:37
- Example 425:40
- Example 530:49
- Example 645:06

1h 14m 3s

- Intro0:00
- Objectives0:08
- Electric Potential Energy0:58
- Gravitational Potential Energy1:02
- Electric Potential Energy1:11
- Electric Potential1:19
- Example 11:59
- Example 23:08
- The Electron-Volt4:02
- Electronvolt4:16
- 1 eV is the Amount of Work Done in Moving an Elementary Charge Through a Potential Difference of 1 Volt4:26
- Conversion Ratio4:41
- Example 34:52
- Equipotential Lines5:35
- Topographic Maps5:36
- Lines Connecting Points of Equal Electrical Potential5:47
- Always Cross Electrical Field Lines at Right Angles5:57
- Gradient of Potential Increases As Equipotential Lines Get Closer6:02
- Electric Field Points from High to Low Potential6:27
- Drawing Equipotential Lines6:49
- E Potential Energy Due to a Point Charge8:20
- Electric Force from Electric Potential Energy11:59
- E Potential Due to a Point Charge13:07
- Example 414:42
- Example 515:59
- Finding Electric Field From Electric Potential19:06
- Example 623:41
- Example 725:08
- Example 826:33
- Example 929:01
- Example 1031:26
- Example 1143:23
- Example 1251:51
- Example 1358:12

1h 1m 28s

- Intro0:00
- Objectives0:10
- Potential Due to a Charged Ring0:27
- Potential Due to a Uniformly Charged Desk3:38
- Potential Due to a Spherical Shell of Charge11:21
- Potential Due to a Uniform Solid Sphere14:50
- Example 123:08
- Example 230:43
- Example 341:58
- Example 451:41

20m 35s

- Intro0:00
- Objectives0:08
- Charges in a Conductor0:32
- Charge is Free to Move Until the0:36
- All Charge Resides at Surface2:18
- Field Lines are Perpendicular to Surface2:34
- Electric Field at the Surface of a Conductor3:04
- Looking at Just the Outer Surface3:08
- Large Electric Field Where You Have the Largest Charge Density3:59
- Hollow Conductors4:22
- Draw Hollow Conductor and Gaussian Surface4:36
- Applying Gaussian Law4:53
- Any Hollow Conductor Has Zero Electric Field in Its Interior5:24
- Faraday Cage5:35
- Electric Field and Potential Due to a Conducting Sphere6:03
- Example 17:31
- Example 212:39

41m 23s

- Intro0:00
- Objectives0:08
- What is a Capacitor?0:42
- Electric Device Used to Store Electrical Energy0:44
- Place Opposite Charges on Each Plate1:10
- Develop a Potential Difference Across the Plates1:14
- Energy is Stored in the Electric Field Between the Plates1:17
- Capacitance1:22
- Ratio of the Charge Separated on the Plates of a Capacitor to the Potential Difference Between the Plates1:25
- Units of Capacitance1:32
- Farad1:37
- Formula1:52
- Calculating Capacitance1:59
- Assume Charge on Each Conductor2:05
- Find the Electric Field2:11
- Calculate V by Integrating the Electric Field2:21
- Utilize C=Q/V to Solve for Capitance2:33
- Example 12:44
- Example 25:30
- Example 310:46
- Energy Stored in a Capacitor15:25
- Work is Done Charging a Capacitor15:28
- Solve For That15:55
- Field Energy Density18:09
- Amount of Energy Stored Between the Plates of a Capacitor18:11
- Example18:25
- Dielectrics20:44
- Insulating Materials Place Between Plates of Capacitor to Increase The Devices' Capacitance20:47
- Electric Field is Weakened21:00
- The Greater the Amount of Polarization The Greater the Reduction in Electric Field Strength21:58
- Dielectric Constant (K)22:30
- Formula23:00
- Net Electric Field23:35
- Key Take Away Point23:50
- Example 424:00
- Example 525:50
- Example 626:50
- Example 728:53
- Example 830:57
- Example 932:55
- Example 1034:59
- Example 1137:35
- Example 1239:57

17m 59s

- Intro0:00
- Objectives0:08
- Electric Current0:44
- Flow Rate of Electric Charge0:45
- Amperes0:49
- Positive Current Flow1:01
- Current Formula1:19
- Drift Velocity1:35
- Constant Thermal Motion1:39
- Net Electron Flow1:43
- When Electric Field is Applied1:49
- Electron Drift Velocity1:55
- Derivation of Current Flow2:12
- Apply Electric Field E2:20
- Define N as the Volume Density of Charge Carriers2:27
- Current Density4:33
- Current Per Area4:36
- Formula4:44
- Resistance5:14
- Ratio of the Potential Drop Across an Object to the Current Flowing Through the Object5:19
- Ohmic Materials Follow Ohm's Law5:23
- Resistance of a Wire6:05
- Depends on Resistivity6:09
- Resistivity Relates to the Ability of a Material to Resist the Flow of Electrons6:25
- Refining Ohm's Law7:22
- Conversion of Electric Energy to Thermal Energy8:23
- Example 19:54
- Example 210:54
- Example 311:26
- Example 414:41
- Example 515:24

29m 8s

- Intro0:00
- Objectives0:08
- Ohm's Law Revisited0:39
- Relates Resistance, Potential Difference, and Current Flow0:39
- Formula0:44
- Example 11:09
- Example 21:44
- Example 32:15
- Example 42:56
- Electrical Power3:26
- Transfer of Energy Into Different Types3:28
- Light Bulb3:37
- Television3:41
- Example 53:49
- Example 64:27
- Example 75:12
- Electrical Circuits5:42
- Closed-Loop Path Which Current Can Flow5:43
- Typically Comprised of Electrical Devices5:52
- Conventional Current Flows from High Potential to Low Potential6:04
- Circuit Schematics6:26
- Three-dimensional Electrical Circuits6:37
- Source of Potential Difference Required for Current to Flow7:29
- Complete Conducting Paths7:42
- Current Only Flows in Complete Paths7:43
- Left Image7:46
- Right Image7:56
- Voltmeters8:25
- Measure the Potential Difference Between Two Points in a Circuit8:29
- Can Remove Voltmeter from Circuit Without Breaking the Circuit8:47
- Very High Resistance8:53
- Ammeters9:31
- Measure the Current Flowing Through an Element of a Circuit9:32
- Very Low Resistance9:46
- Put Ammeter in Correctly10:00
- Example 810:24
- Example 911:39
- Example 1012:59
- Example 1113:16
- Series Circuits13:46
- Single Current Path13:49
- Removal of Any Circuit Element Causes an Open Circuit13:54
- Kirchhoff's Laws15:48
- Utilized in Analyzing Circuits15:54
- Kirchhoff's Current Law15:58
- Junction Rule16:02
- Kirchhoff's Voltage Law16:30
- Loop Rule16:49
- Example 1216:58
- Example 1317:32
- Basic Series Circuit Analysis18:36
- Example 1422:06
- Example 1522:29
- Example 1624:02
- Example 1726:47

39m 9s

- Intro0:00
- Objectives0:16
- Parallel Circuits0:38
- Multiple Current Paths0:40
- Removal of a Circuit Element May Allow Other Branches of the Circuit to Continue Operating0:44
- Draw a Simple Parallel Circuit1:02
- Basic Parallel Circuit Analysis3:06
- Example 15:58
- Example 28:14
- Example 39:05
- Example 411:56
- Combination Series-Parallel Circuits14:08
- Circuit Doesn't Have to be Completely Serial or Parallel14:10
- Look for Portions of the Circuit With Parallel Elements14:15
- Lead to Systems of Equations to Solve14:42
- Analysis of a Combination Circuit14:51
- Example 520:23
- Batteries28:49
- Electromotive Force28:50
- Pump for Charge29:04
- Ideal Batteries Have No Resistance29:10
- Real Batteries and Internal Resistance29:20
- Terminal Voltage in Real Batteries29:33
- Ideal Battery29:50
- Real Battery30:25
- Example 631:10
- Example 733:23
- Example 835:49
- Example 938:43

34m 3s

- Intro0:00
- Objectives0:17
- Capacitors in Parallel0:51
- Store Charge on Plates0:52
- Can Be Replaced with an Equivalent Capacitor0:56
- Capacitors in Series1:12
- Must Be the Same1:13
- Can Be Replaced with an Equivalent Capacitor1:15
- RC Circuits1:30
- Comprised of a Source of Potential Difference, a Resistor Network, and Capacitor1:31
- RC Circuits from the Steady-State Perspective1:37
- Key to Understanding RC Circuit Performance1:48
- Charging an RC Circuit2:08
- Discharging an RC Circuit6:18
- The Time Constant8:49
- Time Constant8:58
- By 5 Time Constant9:19
- Example 19:45
- Example 213:27
- Example 316:35
- Example 418:03
- Example 519:39
- Example 626:14

1h 1m 7s

- Intro0:00
- Objectives0:13
- Charging an RC Circuit1:11
- Basic RC Circuit1:15
- Graph of Current Circuit1:29
- Graph of Charge2:17
- Graph of Voltage2:34
- Mathematically Describe the Charts2:56
- Discharging an RC Circuit13:29
- Graph of Current13:47
- Graph of Charge14:08
- Graph of Voltage14:15
- Mathematically Describe the Charts14:30
- The Time Constant20:03
- Time Constant20:04
- By 5 Time Constant20:14
- Example 120:39
- Example 228:53
- Example 327:02
- Example 444:29
- Example 555:24

8m 38s

- Intro0:00
- Objectives0:08
- Magnetism0:35
- Force Caused by Moving Charges0:36
- Dipoles0:40
- Like Poles Repel, Opposite Poles Attract0:53
- Magnetic Domains0:58
- Random Domains1:04
- Net Magnetic Field1:26
- Example 11:40
- Magnetic Fields2:03
- Magnetic Field Strength2:04
- Magnets are Polarized2:16
- Magnetic Field Lines2:53
- Show the Direction the North Pole of a Magnet Would Tend to Point if Placed on The Field2:54
- Direction3:25
- Magnetic Flux3:41
- The Compass4:05
- Earth is a Giant Magnet4:07
- Earth's Magnetic North Pole4:10
- Compass Lines Up with the Net Magnetic Field4:48
- Magnetic Permeability5:00
- Ratio of the magnetic Field Strength Induced in a Material to the Magnetic Field Strength of the Inducing Field5:01
- Free Space5:13
- Permeability of Matter5:41
- Highly Magnetic Materials5:47
- Magnetic Dipole Moment5:54
- The Force That a Magnet Can Exert on Moving Charges5:59
- Relative Strength of a Magnet6:04
- Example 26:26
- Example 36:52
- Example 47:32
- Example 57:57

29m 7s

- Intro0:00
- Objectives0:08
- Magnetic Fields0:57
- Vector Quantity0:59
- Tesla1:08
- Gauss1:14
- Forces on Moving Charges1:30
- Magnetic Force is Always Perpendicular to the Charged Objects Velocity1:31
- Magnetic Force Formula2:04
- Magnitude of That2:20
- Image2:29
- Direction of the Magnetic Force3:54
- Right-Hand Rule3:57
- Electron of Negative Charge4:04
- Example 14:51
- Example 26:58
- Path of Charged Particles in B Fields8:07
- Magnetic Force Cannot Perform Work on a Moving Charge8:08
- Magnetic Force Can Change Its Direction8:11
- Total Force on a Moving Charged Particle9:40
- E Field9:50
- B Field9:54
- Lorentz Force9:57
- Velocity Selector10:33
- Charged Particle in Crosses E and B Fields Can Undergo Constant Velocity Motion10:37
- Particle Can Travel Through the Selector Without Any Deflection10:49
- Mass Spectrometer12:21
- Magnetic Fields Accelerate Moving Charges So That They Travel in a Circle12:26
- Used to Determine the Mass of An Unknown Particle12:32
- Example 313:11
- Example 415:01
- Example 516:44
- Example 617:33
- Example 719:12
- Example 819:50
- Example 924:02
- Example 1025:21

17m 52s

- Intro0:00
- Objectives0:08
- Forces on Current-Carrying Wires0:42
- Moving Charges in Magnetic Fields Experience Forces0:45
- Current in a Wire is Just Flow of Charges0:49
- Direction of Force Given by RHR4:04
- Example 14:22
- Electric Motors5:59
- Example 28:14
- Example 38:53
- Example 410:09
- Example 511:04
- Example 612:03

24m 43s

- Intro0:00
- Objectives0:08
- Force on a Current-Carrying Wire0:38
- Magnetic Fields Cause a Force on Moving Charges0:40
- Current Carrying Wires0:44
- How to Find the Force0:55
- Direction Given by the Right Hand Rule1:04
- Example 11:17
- Example 22:26
- Magnetic Field Due to a Current-Carrying Wire4:20
- Moving Charges Create Magnetic Fields4:24
- Current-Carrying Wires Carry Moving Charges4:27
- Right Hand Rule4:32
- Multiple Wires4:51
- Current-Carrying Wires Can Exert Forces Upon Each Other4:58
- First Right Hand Rule5:15
- Example 36:46
- Force Between Parallel Current Carrying Wires8:01
- Right Hand Rules to Determine Force Between Parallel Current Carrying Wires8:03
- Find Magnetic Field Due to First Wire, Then Find Direction of Force on 2nd Wire8:08
- Example8:20
- Gauss's Law for Magnetism9:26
- Example 410:35
- Example 512:57
- Example 614:19
- Example 716:50
- Example 818:15
- Example 918:43

21m 50s

- Intro0:00
- Objectives0:07
- Biot-Savart Law0:24
- Brute Force Method0:49
- Draw It Out0:54
- Diagram1:35
- Example 13:43
- Example 27:02
- Example 314:31

26m 31s

- Intro0:00
- Objectives0:07
- Ampere's Law0:27
- Finds the Magnetic Field Due to Current Flowing in a Wire in Situations of Planar and Cylindrical Symmetry0:30
- Formula0:40
- Example1:00
- Example 12:19
- Example 24:08
- Example 36:23
- Example 48:06
- Example 511:43
- Example 613:40
- Example 717:54

7m 24s

- Intro0:00
- Objectives0:07
- Magnetic Flux0:31
- Amount of Magnetic Field Penetrating a Surface0:32
- Webers0:42
- Flux1:07
- Total Magnetic Flux1:27
- Magnetic Flux Through Closed Surfaces1:51
- Gauss's Law for Magnetism2:20
- Total Flux Magnetic Flux Through Any Closed Surface is Zero2:23
- Formula2:45
- Example 13:02
- Example 24:26

1h 4m 33s

- Intro0:00
- Objectives0:08
- Faraday's Law0:44
- Faraday's Law0:46
- Direction of the Induced Current is Given by Lenz's Law1:09
- Formula1:15
- Lenz's Law1:49
- Lenz's Law2:14
- Lenz's Law2:16
- Example2:30
- Applying Lenz's Law4:09
- If B is Increasing4:13
- If B is Decreasing4:30
- Maxwell's Equations4:55
- Gauss's Law4:59
- Gauss's Law for Magnetism5:16
- Ampere's Law5:26
- Faraday's Law5:39
- Example 16:14
- Example 29:36
- Example 311:12
- Example 419:33
- Example 526:06
- Example 631:55
- Example 742:32
- Example 848:08
- Example 955:50

6m 41s

- Intro0:00
- Objectives0:08
- Self Inductance0:25
- Ability of a Circuit to Oppose the Magnetic Flux That is Produced by the Circuit Itself0:27
- Changing Magnetic Field Creates an Induced EMF That Fights the Change0:37
- Henrys0:44
- Function of the Circuit's Geometry0:53
- Calculating Self Inductance1:10
- Example 13:40
- Example 25:23

42m 17s

- Intro0:00
- Objectives0:11
- Inductors in Circuits0:49
- Inductor Opposes Current Flow and Acts Like an Open Circuit When Circuit is First Turned On0:52
- Inductor Keeps Current Going and Acts as a Short1:04
- If the Battery is Removed After a Long Time1:16
- Resister Dissipates Power, Current Will Decay1:36
- Current in RL Circuits2:00
- Define the Diagram2:03
- Mathematically Solve3:07
- Voltage in RL Circuits7:51
- Voltage Formula7:52
- Solve8:17
- Rate of Change of Current in RL Circuits9:42
- Current and Voltage Graphs10:54
- Current Graph10:57
- Voltage Graph11:34
- Example 112:25
- Example 223:44
- Example 334:44

9m 47s

- Intro0:00
- Objectives0:08
- LC Circuits0:30
- Assume Capacitor is Fully Charged When Circuit is First Turned On0:38
- Interplay of Capacitor and Inductor Creates an Oscillating System0:42
- Charge in LC Circuit0:57
- Current and Potential in LC Circuits7:14
- Graphs of LC Circuits8:27

3m 38s

- Intro0:00
- Objectives0:07
- Maxwell's Equations0:19
- Gauss's Law0:20
- Gauss's Law for Magnetism0:44
- Faraday's Law1:00
- Ampere's Law1:18
- Revising Ampere's Law1:49
- Allows Us to Calculate the Magnetic Field Due to an Electric Current1:50
- Changing Electric Field Produces a Magnetic Field1:58
- Conduction Current2:33
- Displacement Current2:44
- Maxwell's Equations (Complete)2:58

32m 33s

- Intro0:00
- 1998 AP Practice Exam Link0:11
- Multiple Choice 360:36
- Multiple Choice 372:07
- Multiple Choice 382:53
- Multiple Choice 393:32
- Multiple Choice 404:37
- Multiple Choice 414:43
- Multiple Choice 425:22
- Multiple Choice 436:00
- Multiple Choice 448:09
- Multiple Choice 458:27
- Multiple Choice 469:03
- Multiple Choice 479:30
- Multiple Choice 4810:19
- Multiple Choice 4910:47
- Multiple Choice 5012:25
- Multiple Choice 5113:10
- Multiple Choice 5215:06
- Multiple Choice 5316:01
- Multiple Choice 5416:44
- Multiple Choice 5517:10
- Multiple Choice 5619:08
- Multiple Choice 5720:39
- Multiple Choice 5822:24
- Multiple Choice 5922:52
- Multiple Choice 6023:34
- Multiple Choice 6124:09
- Multiple Choice 6224:40
- Multiple Choice 6325:06
- Multiple Choice 6426:07
- Multiple Choice 6527:26
- Multiple Choice 6628:32
- Multiple Choice 6729:14
- Multiple Choice 6829:41
- Multiple Choice 6931:23
- Multiple Choice 7031:49

29m 55s

- Intro0:00
- 1998 AP Practice Exam Link0:14
- Free Response 10:22
- Free Response 210:04
- Free Response 316:22

For more information, please see full course syllabus of AP Physics C: Electricity & Magnetism

1 answer

Last reply by: Professor Dan Fullerton

Tue Apr 17, 2018 7:29 AM

Post by Kevin Fleming on April 14, 2018

Prof. Fullerton,

For the purposes of the Electricity and Magnetism course, will (dl * r^) always equals dlsin(theta)? Thanks

4 answers

Last reply by: Professor Dan Fullerton

Wed Oct 5, 2016 11:55 AM

Post by Jessie Sun on March 17, 2016

Hello Professor Fullerton,

Can you please explain why dl cross r hat is equal to dx?

And why dl cross r hat in the previous example is just dlsin(theta)?

1 answer

Last reply by: Professor Dan Fullerton

Wed Apr 15, 2015 9:38 AM

Post by Luvivia Chang on April 15, 2015

Hello Professor Dan Fullerton

Will I be provided with a Table of Integral during an AP physics C test?

If not, what can I do if I derive the functions but cannot finish the integral part? Such as the difficult integral in example 2.

1 answer

Last reply by: Professor Dan Fullerton

Tue Mar 24, 2015 6:04 AM

Post by Arjun Srivatsa on March 24, 2015

How often is the Biot-Savart Law Tested on AP? Is Ampere's Law used much more frequently?