Lecture 22Lecture 22James Clerk Maxwell (1831-1879) Physics 2102Jonathan DowlingMaxwellMaxwell’’s equationss equationsthe dawn of the 20th centuryrevolution in physicsGaussGauss’’ Law: Law:charges produce electric fields,field lines start and end in charges!=•SqdAE0/"SSSSGaussGauss’’ law for magnetism: law for magnetism:field lines are closedor, there are no magnetic monopoles!=•SdAB 0SSSSSSAmpereAmpere’’s law:s law:electric currents produce magnetic fields!=•CidsB0µCFaradayFaraday’’s law:s law:changing magnetic fields produce (“induce”) electric fields!!•"=•SCdABdtddsEAll together:All together:!=•SqdAE0/"!=•SdAB 0!=•CidsB0µ!!•"=•SCdABdtddsE!=•SdAB 0!=•CdsB 0!!•"=•SCdABdtddsENo charges or currentsNo charges or currents::…very suspicious…!?qq=0=0ii=0=0! E • dA = 0S"Something is not rightSomething is not right……If 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 is indeeda magnetic field, equal to the field elsewhere.But there is no current producing it! ?EEBBBBMaybe we can make itright…dtddtEAddtEdddAdtdVCdtCVddtdqiE!======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•A=EAEEid=ε0dΦ/dt!!•=•SCdAEdtddsB00"µB !EiBiBDisplacement currentDisplacement currentMaxwell proposed it, andit was confirmed.!"•CdsB 0““MaxwellMaxwell”” equations: equations:!=•SqdAE0/"!=•SdAB 0idAEdtddsBSC000µ!µ+•=•""!!•"=•SCdABdtddsE!=•SdAE 0!=•SdAB 0!!•=•SCdAEdtddsB00"µ!!•"=•SCdABdtddsEMaxwell equations in freeMaxwell equations in freespace:space:Fields withoutsources?A solution to the Maxwell equations in free space is a “traveling wave”…001!µ=vThe “electric” waves travelat the speed of light!?Light itself is a wave of electricity and magnetism!?Maxwell, waves and lightMaxwell, waves and light=3 108 m/s)(sin220022vtxkEdtEddxEd!"#!=$µelectric 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 frequencies!one velocity, many frequencies!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 waves travel?How do waves travel?Is there an ether they ride on? Michelson andMorley looked and looked, and decided it wasn’tthere. How do waves travel???Electricity and magnetism are “relative”:Whether charges move or not depends on whichframe we use…This was how Einstein began thinkingabout his “theory of specialrelativity”…We’ll leave that theory for later…maybe.SummarySummary• Changing electric fields produce (induce)magnetic fields: 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,which travel at the speed of
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