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ROCHESTER PHY 121 - Lecture 21 Notes - Maxwell's Equations

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Slide 1Slide 2“Gauss’s law” for magnetic field“Gauss’s law” for magnetic fieldChanging FB generates EAmpere’s LawEM theoryDisplacement “currents”Modified Ampere’s LawMaxwell’s equationsMaxwell’s equations in vacuumEM waveEM waveEnergy in EM waveIntensity of wavesAverage intensityEM spectrumRadiation from an AC antennaTransmission and receptionModulations01/14/2019 Lecture XXIV 1Physics 122Maxwell’s equations01/14/2019 Lecture IV 2Gauss’s lawthe number of field lines that go in = the number of field lines that go out, unless there are sinks (negative charges) or sources (positive charges)0 QAdEsurfaceclosedE“Gauss’s law” for magnetic field•Can you guess:01/14/2019 Lecture XXIV 3? surfaceclosedBAdB“Gauss’s law” for magnetic field•You can never separate north end of the magnet from south end  There are no magnetic charges (monopoles) - no sinks or sources  magnetic field lines are always closed  magnetic flux through any closed surface is always zero01/14/2019 Lecture XXIV 40 surfaceclosedBAdB•Magnetic flux is increasing  induced emf:01/14/2019 Lecture XXI 5dtdBΕdtdllEB1Ε•Electric current in the circuit, but What makes stationary charges moves? •Induced electric field:Changing FB generates EdtdldEBIntegral of the electric field over a closed path is equal to negative derivative of the magnetic flux over time:01/14/2019 Lecture XII 6Ampere’s Law•Current = moving electric charges produce magnetic field•Integral of the magnetic field over a closed path is equal to the enclosed currentenclIldB001/14/2019 Lecture XXI 7EM theoryKnown phenomena•Currents produce magnetic field•Changing magnetic field produces electricity (induction)Assumptions•Changing electric field is equivalent to current and also produces magnetic field Unification – James Clark Maxwell (1831-1879): one field – electromagnetic (EM).01/14/2019 Lecture XXI 8Displacement “currents”•Moving charged particles (current) create magnetic field•Charged particles create electric field•Electric field is changing when particles move•Changing electric field is equivalent to current – displacement “current” ID•No charges jump across the capacitor  No charges are actually moving in the displacement “current”EAEEddACVQ 000))((dtddtdQIED0""IID01/14/2019 Lecture XII 9Modified Ampere’s Law•Current = moving electric charges + displacement “current” (changing electric field)produce magnetic fielddtdIldBIIldBEencldisplencl0000)(dtddtdQIED0""01/14/2019 Lecture XXIV 10Maxwell’s equations•Gauss’s Law•Gauss’s law for magnetic field == there no magnetic charges == magnetic filed lines are always closed•Faraday’s law•Changing magnetic field can create electric field•Amper’s law + extra term introduced by Maxwell – displacement current–Changing electric field can create magnetic fielddtdIldBdtdldEAdBQAdEEpathclosedBpathclosedsurfaceclosedsurfaceclosed000 0 001/14/2019 Lecture XXIV 11Maxwell’s equations in vacuum •No charges, no currents•Changing magnetic field creates electric field•Changing electric field creates magnetic fielddtdldBdtdldEAdBAdEEpathclosedBpathclosedsurfaceclosedsurfaceclosed00 0001/14/2019 Lecture XXIV 12EM wavedtdldBdtdldEEpathclosedBpathclosed00 dtdBdxdEdxdydtdBdtddEdyEdydydEEldEBpathclosed )( dtdEdxdBdxdzdtdEdtddBdzldBEpathclosed000000 smv /100.31041085.811871200tx2200221xEtE22222xyvty01/14/2019 Lecture XXIV 13EM wavevBEvvtkxBBtkxEExzy/)sin()sin(00vBvEBEvkffk2201/14/2019 Lecture XXIV 14Energy in EM wave•EM waves transport energy•Energy density:•Poynting vector (energy transported by EM wave per unit time per unit area)•Average energy per unit time per unit area20202121BEuBES01rmsrmsBES 0101/14/2019 Lecture XXIV 15Intensity of wavesrE1•Intensity – I, W/m2•Intensity I is proportional to amplitude squared E2, inversely proportional to r2:•Energy of oscillation U is proportional to amplitude squared E22EU areapowerareatimeenergyI /2EI 21rI 01/14/2019 Lecture XXIV 16Average intensity•Electric fields E follows harmonic oscillation:•Intensity (brightness for light) I is proportional to electric field squared•Average over time (one period of oscillation) I:)sin(0tEE)(sin202tIIEI2)2cos1(2121sin21)(sin1)(sin102002020020020IdxxIxdxItdtTIdttTIITT01/14/2019 Lecture XXIV 17EM spectrum•c – speed of light (m/s)•f – frequency (Hz=1/s)•l – wavelength (m)fc f01/14/2019 Lecture XXIV 18Radiation from an AC antenna•Changing electric field creates magnetic field•Changing magnetic field creates electric field•Change propagates with a finite velocity •Electromagnetic wave – proof of unification01/14/2019 Lecture XXIV 19Transmission and reception•Antennas are used to transmit and to receive EM waves•Rod antennas – transmit and receive E componentE || to rod •Loop antennas – B component (use induction) loopB 01/14/2019 Lecture XXIV 20Modulations•Amplitude modulation (AM)•Frequency modulation


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ROCHESTER PHY 121 - Lecture 21 Notes - Maxwell's Equations

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