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UT Arlington PHYS 1444 - Maxwell’s Equations

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PHYS 1444 – Section 003 Lecture #22AnnouncementsMaxwell’s EquationsAmpere’s LawExpanding Ampere’s LawModifying Ampere’s LawSlide 7Example 32 – 1Slide 9Displacement CurrentGauss’ Law for MagnetismSlide 12Slide 13Maxwell’s Amazing Leap of FaithProduction of EM WavesSlide 16Properties of Radiation FieldsSlide 18EM WavesHappy Thanksgiving!Wendesday, Nov. 23, 2005 PHYS 1444-003, Fall 2005Dr. Jaehoon Yu1PHYS 1444 – Section 003Lecture #22Wednesday, Nov. 23, 2005Dr. Jaehoon Yu•Achievements of Maxwell’s Equations•Extension of Ampere’s Law•Gauss’ Law of Magnetism•Maxwell’s Equations•Production of Electromagnetic WavesWendesday, Nov. 23, 2005 PHYS 1444-003, Fall 2005Dr. Jaehoon Yu2Announcements•Quiz results–Average: 61.2•Previous averages: 71 and 54–Top score: 80•Final term exam –Time: 11am – 12:30pm, Monday Dec. 5–Location: SH103–Covers: 29.3 – which ever chapter we finish next, Wednesday, Nov. 30–Please do not miss the exam–Two best of the three exams will be used for your gradesWendesday, Nov. 23, 2005 PHYS 1444-003, Fall 2005Dr. Jaehoon Yu3Maxwell’s Equations•The development of EM theory by Oersted, Ampere and others was not done in terms of EM fields–The idea of fields was introduced somewhat by Faraday•Scottish physicist James C. Maxwell unified all the phenomena of electricity and magnetism in one theory with only four equations (Maxwell’s Equations) using the concept of fields–This theory provided the prediction of EM waves–As important as Newton’s law since it provides dynamics in electromagnetism–This theory is also in agreement with Einstein’s special relativity•The biggest achievement of 19th century electromagnetic theory is the prediction and experimental verification that the electromagnetic waves can travel through the empty space–What do you think this accomplishment did?•Open a new world of communication•It also yielded the prediction that the light is an EM wave•Since all of Electromagnetism is contained in the four Maxwell’s equations, this is considered as one of the greatest achievements of human intellectWendesday, Nov. 23, 2005 PHYS 1444-003, Fall 2005Dr. Jaehoon Yu4Ampere’s Law•Do you remember the mathematical expression of Oersted discovery of a magnetic field produced by an electric current, given by Ampere?•We’ve learned that a varying magnetic field produces an electric field•Then can the reverse phenomena, that a changing electric producing a magnetic field, possible?–If this is the case, it would demonstrate a beautiful symmetry in nature between electricity and magnetismB dl� =�rr�0menclIWendesday, Nov. 23, 2005 PHYS 1444-003, Fall 2005Dr. Jaehoon Yu5Expanding Ampere’s Law•Let’s consider a wire carrying a current–The current that is enclosed in the loop passes through the surface # 1 in the figure–We could imagine a different surface 2 that shares the same enclosed path but cuts through the wire in a different location. What is the current that passes through the surface?•Still .–So the Ampere’s law still works•We could then consider a capacitor being charged up or being discharged.–The current  enclosed in the loop passes through the surface #1–However the surface #2 that shared the same closed loop do not have any current passing through it. •There is magnetic field present since there is current  In other words there is a changing electric field in between the plates•Maxwell resolved this by adding an additional term to Ampere’s law involving the changing electric fieldWendesday, Nov. 23, 2005 PHYS 1444-003, Fall 2005Dr. Jaehoon Yu6Modifying Ampere’s Law•To determine what the extra term should be, we first have to figure out what the electric field between the two plates is–The charge Q on the capacitor with capacitance C is Q=CV•Where V is the potential difference between the plates–Since V=Ed•Where E is the uniform field between the plates, and d is the separation of the plates–And for parallel plate capacitor C=0A/d–We obtainQ =CV =0Ade� �� �� �0AEeEd =Wendesday, Nov. 23, 2005 PHYS 1444-003, Fall 2005Dr. Jaehoon Yu7Modifying Ampere’s Law–If the charge on the plate changes with time, we can write–Using the relation between current and charge we obtain•Where =EA is the electric flux through the surface between the plates–So in order to make Ampere’s law work for the surface 2 in the figure, we must write it in the following form–This equation represents the general form of Ampere’s law•This means that a magnetic field can be caused not only by an ordinary electric current but also by a changing electric fluxdQdt=I =0 0 0EencldB dl Idtm meF� = +�rr�0dEAdtedQdt=0dEAdte =( )0d AEdte =0EddteFExtra term by MaxwellWendesday, Nov. 23, 2005 PHYS 1444-003, Fall 2005Dr. Jaehoon Yu8Example 32 – 1 Charging capacitor. A 30-pF air-gap capacitor has circular plates of area A=100cm2. It is charged by a 70-V battery through a 2.0- resistor. At the instant the battery is connected, the electric field between the plates is changing most rapidly. At this instant, calculate (a) the current into the plates, and (b) the rate of change of electric field between the plates. (c) Determine the magnetic field induced between the plates. Assume E is uniform between the plates at any instant and is zero at all points beyond the edges of the plates. Since this is an RC circuit, the charge on the plates is: For the initial current (t=0), we differentiate the charge with respect to time.The electric field is Q =E =0I =dEdt=Change of the electric field is 0CV( )1t RCe--0tdQdt==00t RCtCVeRC-==0VR=70352.0VA=W�0se=0Q Ae0dQ dtAe=( ) ( )1412 2 2 2 2354.0 108.85 10 1.0 10AV m sC N m m- -= � �״��Wendesday, Nov. 23, 2005 PHYS 1444-003, Fall 2005Dr. Jaehoon Yu9Example 32 – 1 (c) Determine the magnetic field induced between the plates. Assume E is uniform between the plates at any instant and is zero at all points beyond the edges of the plates. The magnetic field lines generated by changing electric field is perpendicular to E and is circular due to symmetry Whose law can we use to determine B?We choose a circular path of radius r, centered at the center of the plane, following the B. EF =B dl� =�rr�Extended Ampere’s Law w/ encl=0!For r<rplate, the electric flux is since E is uniform throughout the plate So from Ampere’s law, we obtain (


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