Physics 11bLecture #17Electromagnetic RadiationS&J Chapter 34What We Did Last Time Impedance C and L are frequency dependent RLC circuit Resonance at , with quality factor  TransformerRRRVZRI∆==1CCCVZICω∆==LLLVZLIω∆==Rv∆RLv∆LCv∆C221ZR LCωω⎛⎞=+−⎜⎟⎝⎠01LCω=00LQRωωω==∆2211NN=EE2121NIIN=21.2equivNRRN⎛⎞=⎜⎟⎝⎠N1N2∆vRToday’s Goals Maxwell’s Equations Complete the “fundamental E&M equations” Predict electromagnetic radiation That’s light, radio waves, etc. Electromagnetic waves Study the plane-wave solution Energy and momentum carried by EM wavesFour E&M Equations We’ve summarized them in Lecture #13 Gauss’s law for E Gauss’s law for B Faraday’s law Ampère’s law Maxwell (1831-1879) noticed something was wrong0dIµ⋅=∫BsvBdddtΦ⋅=−∫Esv0Sqdε⋅=∫EAv0Sd⋅=∫BAvSurface integral istaken over a closedsurface SLine integral istaken around aclosed loopDifference Between E & M Electricity and magnetism appear very similar except for one thing: there is no magnetic charge This explains some differences between the equations The symmetry would be perfect if it weren’t for the Maxwell: “I’ll add an extra term just so that theequations become symmetric!”0dIµ⋅=∫BsvBdddtΦ⋅=−∫Esv0Sqdε⋅=∫EAv0Sd⋅=∫BAvbutno qbutno IWhat happened to this one?BddtΦ−Maxwell’s Equations The extra term is called the displacement current Name is historic – Don’t worry about it Maxwell’s reasoning is explained in textbook 30.7 The extra term opens a new way of producing B field by time-variation of the E field We already know induction: varying B field creates E field Can E field and B field exist without charge or current?000EddIdtµµεΦ⋅= +∫BsvBdddtΦ⋅=−∫Esv0Sqdε⋅=∫EAv0Sd⋅=∫BAvE&M Fields in Free Space Consider a simple case: Empty space Let’s assume E and B depends on x and t But not y and z Draw a small loop in x-y planeand calculate the integrals Assume ∆x and ∆y are very small00EdddtµεΦ⋅=∫BsvBdddtΦ⋅=−∫Esv0Sd⋅=∫EAv0Sd⋅=∫BAvx + ∆xxy + ∆yyabcdE&M Fields in Free Space Faraday’s lawBdddtΦ⋅=−∫Esvx + ∆xxy + ∆yy()()()bcdaxyxyabcdyyyd E dx E dy E dx E dyEx x Ex yExyx⋅= + + +=+∆− ∆∂=∆∆∂∫∫∫∫∫Esvabcd()()()bcdaxyxyabcdyyyd E dx E dy E dx E dyEx x Ex yExyx⋅= + + +=+∆− ∆∂≈∆∆∂∫∫∫∫∫Esvcancel each other0()()limxffxxfxxx∆→∂+∆ −≡∂∆RememberyzEBxt∂∂=−∂∂BzBxyΦ=∆∆E&M Fields in Free Space Ampere’s law works the same way Just the sign plus the constant From the loop in the x-y plane Æ Use a loop in the x-z plane We have a pair of equations00EdddtµεΦ⋅=∫BsvyzEBxt∂∂=−∂∂00yzBExtµε∂∂=∂∂00yzEBxtµε∂∂=−∂∂Minus sign comes from the fact thatbutˆˆ ˆ×=xy zˆˆ ˆ×=−xz y00yzEBxtµε∂∂=−∂∂E&M Fields in Free Space Eliminate BzÆ Eliminate EyÆ Solutions are  Plug into the wave equationsyzEBxt∂∂=−∂∂00yzEBxtµε∂∂=−∂∂220022yyEExtµε∂∂=∂∂220022zzBBxtµε∂∂=∂∂These are general waveequations, for which thesimplest solutions aresinusoidal wavesmaxcos( )yEE kxtω=−maxcos( )zBB kxtω=−220022cos( ) cos( )kx t kx txtωωµε∂− ∂−=∂∂2200kµεω=Wave Solution How does cos(kx –ωt) looks like? Sinusoidal waves at anymoment Wave length λis given by k is the wave number The waves move toward+x direction with timeωis the angular frequencyas usualxtπλ2=kkπλ2=Speed of the Waves We can calculate the speed of the waves We’ve found Maxwell’s discovery: Electromagnetic fields can exist in free space as waves propagating with the speed of light Light must be electromagnetic wavesvTλ= Tπω2=kπλ2= vkω=2200kµεω=00812.998 10 m/svkωµε===×7012 2 20410Tm/A8.854 10 C /N mµπε−−=× ⋅=× ⋅Closer Look on the Waves Plug into  At any point on the waves, E and B arealways proportional to each other, and the ratio is cmaxcos( )yEE kxtω=−maxcos( )zBB kxtω=−yzEBxt∂∂=−∂∂max maxkE Bω=maxmaxEcBkω==EBxPlane Waves The EM waves discussed so far are plane waves It depends on x, but uniformin the y-z plane It moves along +x Note that E, B, and +xmake a right-hand i.e., E×B points +x Obviously there is anotherindependent solution:maxcos( )yEE kxtω=−maxcos( )zBB kxtω=−maxcos( )zEE kxtω=−maxcos( )yBB kxtω=− −Energy in the Waves E and B fields hold energies Energy densities are given by In the EM waves, E = cB Equal energies in E and B Total energy density is E and B vary with position and time Average energy density is202EEuε=202BBuµ=2200()22EBcBBuuεµ===Remember 001cµε=2200EBBuu u Eεµ=+= =220max maxave022EBuεµ==Energy Carried by the Waves EM waves hits an area A Æ How much energy? Each second, energy in thebox of A×c arrives Define intensity as the energyarriving in a unit area per second Unit: W/m2 Example: direct sunlight on earth is about 1kW/m2What’s Emaxand Bmax?AcavePower PuAc=2max max maxave0022cB E BPIucAµµ== = =Poynting Vector In EM waves, E and B fields are perpendicular to the direction of propagation, which is given by Define Poynting vector: It points the directionof propagation Magnitude is theintensity = energyflow per area persecond Time-average isEB×EB0µ×=EBSSmax maxave02µ×=EBSMomentum and Pressure EM waves carry momentum as well as energy Object absorbing light is pushed by the light pressure Object emitting light is pushed back If power of the light is P, force F is e.g. a 100 MW laser pulse produces a force ofenergymomentum pc=lightFFPFc=(N) 0.3(m/s) 10(W)/3 10188=××Spherical Waves Real-world EM waves are (often) not plane waves Light from small sources, radio waves from an antenna, etc. They usually spread spherically Spherical waves behave pretty muchthe same as plane waves, except: It gets weaker as it goes All what we discussed aboutplane waves applies tospherical waves21Ir∝1Er∝1Br∝This is obvious because of energy conservationSummary Maxwell’s Eqns. In free space, predict electromagnetic waves Speed  Poynting vector gives the energy flow Energy and momentum related by000EddIdtµµεΦ⋅=