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LSU PHYS 2102 - Maxwell’s equations

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Lecture 31: MON 30 MARLecture 31: MON 30 MAR Ch.32.1Ch.32.1––3: Maxwell3: Maxwell’’s equationss equationsJames Clerk Maxwell (1831-1879) Physics 2102Jonathan DowlingMaxwell I: GaussMaxwell I: Gauss’’ Law for E-Fields: Law for E-Fields:charges produce electric fields,field lines start and end in charges!=•SqdAE0/"SSSSMaxwell II: GaussMaxwell II: Gauss’’ law for B-Fields: law for B-Fields:field lines are closedor, there are no magnetic monopoles!=•SdAB 0SSSSSSMaxwell III: AmpereMaxwell III: Ampere’’s law:s law:electric currents produce magnetic fields!=•CidsB0µCMaxwell IV: FaradayMaxwell IV: Faraday’’s law:s law:changing magnetic fields produce (“induce”) electric fields!!•"=•SCdABdtddsEMaxwell Equations I Maxwell Equations I –– IV: IV:!=•SqdAE0/"!=•SdAB 0!=•CidsB0µ!!•"=•SCdABdtddsE!=•SdAB 0!=•CdsB 0!!•"=•SCdABdtddsEIn Empty Space with No Charge or CurrentIn Empty Space with No Charge or Current…very suspicious…!?qq=0=0ii=0=0 E • dA = 0S!MaxwellMaxwell’’s Displacement Currents Displacement CurrentIf we are charging a capacitor, there is acurrent left and right of the capacitor.Thus, there is the same magnetic field rightand left of the capacitor, with circular linesaround the wires.But no magnetic field inside the capacitor?With a compass, we can verify there isindeed a magnetic field, equal to the fieldelsewhere.But there is no current producing it! ?EEBBBBThe missingMaxwellEquation!Maxwell’s FixdtddtEAddtEdddAdtdVCdtCVddtdqiE!======000)()()("""We can write the current as:We calculate the magnetic field produced by thecurrents at left and at right using Ampere’s law :!=•CidsB0µq=CVV=EdC=ε0A/dΦE=∫E•dA=EA EEid=ε0dΦ/dt!!•=•SCdAEdtddsB00"µB !EiBiBDisplacement CurrentDisplacement CurrentMaxwell proposed it based onsymmetry and math — no experiment!!"•CdsB 0MaxwellMaxwell’’s Equations I s Equations I –– V:V:!=•SqdAE0/"!=•SdAB 0idAEdtddsBSC000µ!µ+•=•""!!•"=•SCdABdtddsEIIIIIIIIIIIIIVIVV!=•SdAE 0!=•SdAB 0!!•=•SCdAEdtddsB00"µ!!•"=•SCdABdtddsEMaxwell Equations in Empty Space:Maxwell Equations in Empty Space:Fields withoutsources?A solution to the Maxwell equations in empty space isa “traveling wave”…c =1µ0!0= 3 " 108m/sThe “electric” waves travelat the speed of light!?Light itself is a wave of electricity and magnetism!?Maxwell, Waves and LightMaxwell, Waves and Lightd2Edx2= !µ0"0d2Edt2# E = E0sin k(x ! ct )electric and magnetic “forces” can travel! !!•=•SCdAEdtddsB00"µ!!•"=•SCdABdtddsEFirst person to use electromagnetic waves for communications:Guglielmo Marconi (1874-1937), 1909 Nobel Prize (first transatlantic commercial wirelessservice, Nova Scotia, 1909)Electromagnetic wavesElectromagnetic wavesFirst person to prove that electromagnetic waves existed:Heinrich Hertz (1875-1894)Electromagnetic Waves: Electromagnetic Waves: One Velocity, Many Wavelengths!One Velocity, Many Wavelengths!with frequencies measured in “Hertz” (cycles per second)and wavelength in meters.http://imagers.gsfc.nasa.gov/ems/http://www.astro.uiuc.edu/~kaler/sow/spectra.htmlHow do E&M Waves Travel?How do E&M Waves Travel?Is there an “ether” they ride on? Michelsonand Morley looked and looked, and decidedit wasn’t there. How do waves travel???Electricity and magnetism are “relative”:Whether charges move or not depends onwhich frame we use…This was how Einstein beganthinking about his “theory ofspecial relativity”…We’ll leave that theory for later.SummarySummary• Changing electric fields produce (induce) magneticfields: displacement currents.•Maxwell’s laws allow us to calculate electric andmagnetic fields everywhere in space if we are giventhe sources: electric charges and currents.• If there are no sources, we can still have electricand magnetic fields as electromagnetic waves, whichtravel at the speed of


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LSU PHYS 2102 - Maxwell’s equations

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