Maxwell's Equations


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Video Clips

Video RealVideo®
6:03 minutes (44:57 - 51:00)

Statement of all four of Maxwell's Equations in completed form.

Prior Knowledge: Relative Permeability (30:36 of video lecture 21)
Instructor: Prof. Walter Lewin
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Video RealVideo®
1:37 minutes (15:17 - 16:54)

Statement of the second of Maxwell's Equations, that there is no net magnetic flux through a closed surface.

Prior Knowledge: None
Instructor: Prof. Walter Lewin
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Video RealVideo®
9:58 minutes (0:00 - 9:58)

Definition, motivated by trying to find the magnetic field near a charging capacitor using Ampere's Law.

Prior Knowledge: Ampere's Law (beginning of video lecture 15)
Instructor: Prof. Walter Lewin
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Video RealVideo®
4:27 minutes (9:58 - 14:25)

Finding the magnetic field around a charging capacitor using the modified form of Ampere's Law.

Prior Knowledge: Displacement Current (beginning of video lecture 18)
Instructor: Prof. Walter Lewin
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Video RealVideo®
5:10 minutes (14:25 - 19:35)

Using Maxwell-Ampere's Law to find the magnitude of the magnetic field anywhere between the plates of a charging capacitor.

Prior Knowledge: Displacement Current (beginning of video lecture 18)
Instructor: Prof. Walter Lewin
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Video RealVideo®
3:52 minutes (19:35 - 23:27)

Further discussion of Maxwell's discovery of displacement current, as well as his successful prediction of radio waves before their existence was proven by Hertz.

Prior Knowledge: Displacement Current (beginning of video lecture 18)
Instructor: Prof. Walter Lewin
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Lecture Notes

Document PDF
Page 2 to page 7

Inconsistency in Ampere's Law leads to new term; calculation of displacement current; application to RC circuit; complete Maxwell's Equations.

Prior Knowledge: Faraday's Law
Instructor: Prof. Gabriella Sciolla
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Document PDF
Page 7 to page 10

Solutions of E-field and B-field wave equations in vacuum.

Prior Knowledge: Maxwell's Equations
Instructor: Prof. Gabriella Sciolla
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Document PDF
Page 10 to page 12

Proof that light is an EM wave; connection with SI units; measuring c.

Prior Knowledge: Wave Equation Solutions
Instructor: Prof. Gabriella Sciolla
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Page 1

Maxwell's four equations in complete form.

Prior Knowledge: Maxwell's Equations (44:57 of video lecture 22)
Instructor: Prof. Walter Lewin
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Online Textbook Chapter

Document PDF
Page 3 to page 5

Introduces the displacement current in the context of Ampere's Law in a charging capacitor, and states the Ampere-Maxwell Law.

Prior Knowledge: Ampere's Law (OT9.3), Capacitors (OT5.1-5.2)
Instructors: Prof. John Belcher, Dr. Peter Dourmashkin, Prof. Robert Redwine, Prof. Bruce Knuteson, Prof. Gunther Roland, Prof. Bolek Wyslouch, Dr. Brian Wecht, Prof. Eric Katsavounidis, Prof. Robert Simcoe, Prof. Joseph Formaggio, Andy Neely, Matthew Strafuss, Prof. Eric Hudson, Dr. Sen-Ben Liao
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Document PDF
Page 5 to page 5

States Gauss's Law for magnetic field by analogy with Gauss's Law for electric field, using the non-existence of magnetic monopoles.

Prior Knowledge: Gauss's Law (OT4.2)
Instructors: Prof. John Belcher, Dr. Peter Dourmashkin, Prof. Robert Redwine, Prof. Bruce Knuteson, Prof. Gunther Roland, Prof. Bolek Wyslouch, Dr. Brian Wecht, Prof. Eric Katsavounidis, Prof. Robert Simcoe, Prof. Joseph Formaggio, Andy Neely, Matthew Strafuss, Prof. Eric Hudson, Dr. Sen-Ben Liao
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Document PDF
Page 5 to page 6

Lists all of Maxwell's Equations together in both integral and differential forms; also derives the speed of light from Maxwell's Equations in vacuum.

Prior Knowledge: Gauss's Law for Magnetism (OT13.2), Faraday's Law (OT10.1), Displacement Current (OT13.1)
Instructors: Prof. John Belcher, Dr. Peter Dourmashkin, Prof. Robert Redwine, Prof. Bruce Knuteson, Prof. Gunther Roland, Prof. Bolek Wyslouch, Dr. Brian Wecht, Prof. Eric Katsavounidis, Prof. Robert Simcoe, Prof. Joseph Formaggio, Andy Neely, Matthew Strafuss, Prof. Eric Hudson, Dr. Sen-Ben Liao
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Practice Problems

Document PDF
Problem on page 44 to page 46

A thick wire with a narrow cut builds up charge; find the electric and magnetic fields between the ends of the wire. Solution is included after problem.

Prior Knowledge: Displacement Current, Ampere-Maxwell Law
Instructors: Prof. John Belcher, Dr. Peter Dourmashkin, Prof. Robert Redwine, Prof. Bruce Knuteson, Prof. Gunther Roland, Prof. Bolek Wyslouch, Dr. Brian Wecht, Prof. Eric Katsavounidis, Prof. Robert Simcoe, Prof. Joseph Formaggio, Andy Neely, Matthew Strafuss, Prof. Eric Hudson, Dr. Sen-Ben Liao
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Document PDF
Problem on page 28 to page 32

Find the electric and magnetic fields over time and the voltage at a point within a charging capacitor; use these to find the total stored energy. Solution is included after problem.

Prior Knowledge: Gauss's Law, Potential from Gauss's Law, Ampere-Maxwell Law, Energy Density of Electric Field, Energy Density of Magnetic Field
Instructors: Prof. John Belcher, Dr. Peter Dourmashkin, Prof. Robert Redwine, Prof. Bruce Knuteson, Prof. Gunther Roland, Prof. Bolek Wyslouch, Dr. Brian Wecht, Prof. Eric Katsavounidis, Prof. Robert Simcoe, Prof. Joseph Formaggio, Andy Neely, Matthew Strafuss, Prof. Eric Hudson, Dr. Sen-Ben Liao
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Document PDF
Problem on page 41 to page 42

Given the electric field of a plane electromagnetic wave, find its propagation direction and the corresponding magnetic field. Solution is included after problem.

Prior Knowledge: Electromagnetic Wave in Vacuum
Instructors: Prof. John Belcher, Dr. Peter Dourmashkin, Prof. Robert Redwine, Prof. Bruce Knuteson, Prof. Gunther Roland, Prof. Bolek Wyslouch, Dr. Brian Wecht, Prof. Eric Katsavounidis, Prof. Robert Simcoe, Prof. Joseph Formaggio, Andy Neely, Matthew Strafuss, Prof. Eric Hudson, Dr. Sen-Ben Liao
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Document PDF
Problem on page 1 to page 2

Characterize the circulation of magnetic field on an Amperian loop within a charging or discharging capacitor.

Prior Knowledge: Ampere-Maxwell Law
Instructors: Prof. John Belcher, Dr. Peter Dourmashkin, Prof. Robert Redwine, Prof. Bruce Knuteson, Prof. Gunther Roland, Prof. Bolek Wyslouch, Dr. Brian Wecht, Prof. Eric Katsavounidis, Prof. Robert Simcoe, Prof. Joseph Formaggio, Andy Neely, Matthew Strafuss, Prof. Eric Hudson, Dr. Sen-Ben Liao
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Document PDF
Problem on page 1 to page 2

Find the electric component of an electromagnetic plane wave consistent with the given magnetic component; and vice versa.

Prior Knowledge: EM Radiation in Vacuum
Instructors: Prof. John Belcher, Dr. Peter Dourmashkin, Prof. Robert Redwine, Prof. Bruce Knuteson, Prof. Gunther Roland, Prof. Bolek Wyslouch, Dr. Brian Wecht, Prof. Eric Katsavounidis, Prof. Robert Simcoe, Prof. Joseph Formaggio, Andy Neely, Matthew Strafuss, Prof. Eric Hudson, Dr. Sen-Ben Liao
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Document PDF
Problem 1

How do you know sound is not an EM wave?

Prior Knowledge: None
Instructors: Dr. Peter Dourmashkin, Prof. Gunther Roland
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Exam Questions

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Problem 3

4-part problem; finding direction, λ of wave; deciding whether it can be sound or EM.

Prior Knowledge: None
Instructors: Dr. Peter Dourmashkin, Prof. Gunther Roland
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