Electromagnetic Waves

Transcription

1 M- letrmagneti Waves Last semester, we studied Classial Mehanis. The fundamental laws (axims) f Classial Mehanis are alled Newtn's Laws. This semester, we are studying a subjet alled Classial letrmagnetism. There are fur fundamental laws f eletrmagnetism, alled Maxwell's quatins (after the Sttish physiist James Clerk Maxwell). qenlsed () Gauss's Law da = ( -fields are aused by harges. ) ε () Faraday's Law: L d d = da dt (-fields are als aused by hanging -fields.) d (3) mpere-maxwell Law: d µ enlsed µ ε = + da dt L() S (-fields are aused bth by urrents and by hanging -fields.) (4) Gauss's Law fr -fields: da = (There are n magneti mnples). xept fr the last term in equatin (3), all fur f these laws had been disvered experimentally befre Maxwell started his researh in the 85's. S why d we all them Maxwell's quatins? Maxwell realized that mpere's Law, d = µ, was inmplete. He ntied that L () enlsed there are situatins in whih mpere's Law fails t give the (inreasing) rret answer. Fr instane, if a apaitr is being harged up by a steady urrent, then there must be a -field arund the apaitr, aused by the nearby urrents. ut arding t the riginal frm f mpere's law, if we nsider an imaginary lp irling the apaitr (diagram belw), the urrent thrugh this lp is zer. S mpere's Law predits that the -field is alng that lp is zer (sine thru = ). Maxwell ntied that althugh there is n urrent thrugh the lp, there is a hanging -field flux thrugh the lp. He saw Last update: /6/9 Dubsn Phys Ntes, University f Clrad

2 M- that he uld fix the prblem by mdifying mpere's Law with the additin f a new term. The hanging eleti flux in the apaitr leads t a quantity that has the d dimensins f urrent: ε da dt. Ntie that, frm Gauss's Law, the quantity ε da has the dimensins f harge. S, ε d da dt has the dimensins f urrent. imaginary lp L thru =, =? Maxwell alled this new quantity the displaement urrent. y replaing the urrent in d mpere's Law + ε dt da, he was able t reslve the prblem. This new frm f mpere's Law (nw alled the mpere-maxwell Law) appealed t Maxwell's sense f aesthetis. There was nw a pleasing symmetry in the equatins: hanging -fields reate -fields (Faraday's Law) hanging -fields reate -fields (mpere-maxwell Law) Maxwell realized that beause f this symmetry, the equatins predited a peuliar kind f selfsustaining interatin between and fields. Maxwell thught: Suppse yu have a harge q and yu shake it, bak and frth. The q reates an -field, but when yu shake the harge, yu are hanging the -field in the spae arund it. This hanging -field reates a -field. ut nw yu just reated a -field where there was nne befre, s yu have a hanging -field. This hanging -field will reate an -field, and that newly reated -field will reate a -field, whih will reate an, whih will reate a, whih will (the press will g n, frever). Maxwell shwed that the equatins predited the existene f an eletrmagneti wave whih travels utward frm the shaking harge: Last update: /6/9 Dubsn Phys Ntes, University f Clrad

3 M-3 Maxwell mputed the speed f this strange, new eletrmagneti wave and fund that the speed was given by a simple frmula: speed v = = = ε µ 8 3. m/s This number is the same as the speed f light! Maxwell had shwn that light was an eletrmagneti wave! efre Maxwell, sientists had n lear idea what light is. This was a great synthesis, a bringing tgether f previusly separate fields f physis: eletriity, magnetism, and ptis. efre Maxwell, n ne knew what light was. t was knwn that light was sme kind f wave (we will see the evidene fr this later), but n ne knew what kind f wave it was. Maxwell figured it ut.. Light is an eletrmagneti wave whih is reated by aelerating eletri harge. Wave speed is v distane wavelength λ = = = time time fr λ t g by T v λ = = λ T f Fr light waves, speed v =, this is written = λ f M waves are transverse waves: the - and -field vetrs are bth perpendiular t the diretin f the wave. Drawing an M wave in spae is quite diffiult; the and -fields are everywhere and intimately mixed. The figure here shws the -field alng a partiular line, at a mment in time. (up and dwn) wavelength λ speed (in and ut) ll M radiatin is aused by shaking (aelerating) eletri harge. The mre rapidly the harge is shaken (the higher the frequeny f the shake), the shrter the wavelength f the light, sine λ = f. Nw we an understand why all things glw (give ff light) when they get ht. Last update: /6/9 Dubsn Phys Ntes, University f Clrad

4 M-4 When smething is very ht, its atms are jiggling furiusly. tms are made f harges (eletrns and prtns), and the jiggling harges emit M radiatin. Different wavelength ranges are given names: Wavelength λ Name Use/urrene <. nm Gamma-rays Radiativity. nm nm X-ray medial nm 4 nm Ultravilet(UV) Sunburns, "blak" lights 4nm 7 nm Visible Human seeing 7nm mm nfrared (R) "Heat rays" m mirwave Cmmuniatins, mirwave vens m km radi Radi, TV letrmagneti radiatin (light) an have any wavelength. ut ur eyes are sensitive nly t a narrw range f wavelengths between 4 nm and 7 nm. Different wavelengths in this range f visible light rrespnd t different lrs. Wavelength = 7 nm light appears red t us, 4 nm light appears vilet, and the wavelengths in between rrespnd t all the lrs f the rainbw (ROYGV). ll wavelengths utside this narrw band are invisible t human eyes. Sme imprtant fats abut M waves: M waves are transverse: The and -field are perpendiular t eah ther are eah perpendiular t the diretin f prpagatin, like s: Radi reeiving antennas must be riented rretly in rder t funtin: the wire antenna must be parallel t the -field in rder fr the eletrns in the wire t be aelerated alng the wire. The and fields are in phase: reahes max at same time/plae as des. The amplitudes f the and fields are related by =. (Ntie that units are OK: F = q = qv = v) Last update: /6/9 Dubsn Phys Ntes, University f Clrad

5 M-5 M waves arry energy. n an M wave, the energy density f the -field ( u is the same as the energy density f -field ( u = µ ). = ε ) Yu an feel the energy f an M wave hitting yur fae when yu fae the sun. The intensity is defined as the energy per time per area impinging n a surfae: pwer P intensity =, = units[ ] = W / m area ntensity is als alled brightness. The intensity f an M wave is prprtinal t. The intensity is desribed by the Pynting vetr: S =. The instantaneus value f the µ intensity is S = /µ. The average intensity is S rms = rms rms /µ = /(µ ) = /( µ ) amplitude speed diretin f S λ Cnsider a small sure f mnhrmati (single-wavelength) light, emitting energy at a rate P (in J/s). f the sure sends light ut in all diretins unifrmly, then the wavefrnts are expanding spherial shells. small detetr (like an eyeball), very far frm the sure, will sense a small prtin f the wavefrnts. Over the small regin that is deteted, the wavefrnts are nearly flat planes. Suh a beam f light is alled a plane wave.... speed detetr sure λ R Last update: /6/9 Dubsn Phys Ntes, University f Clrad

6 M-6 plane M wave, traveling in the z-diretin, has the mathematial frm: (x) = ˆ sin(kz ωt)x = ˆ sin(kz ωt)y (z) (y) Ntie that and are in phase. xample f intensity alulatin: Suppse a small sure f M radiatin emits pwer P istrpially (unifrmly in all diretins). What is the intensity f the light at a distane R frm the sure? nswer: Cnsider the energy U emitted frm the sure during a very brief time interval t. When that energy is arried by the M wave a distane R frm the sure, the energy is spread ut unifrmly ver a sphere f area = 4πR, and the energy per area is U / (4πR ). U /t P The energy per time per area, the intensity, is then = = 4π R. Ntie that the intensity falls as / (distane). The mre distant a light sure, the dimmer it appears. xample f pwer deteted frm a distant sure: What is the pwer entering an bserver's eye frm a -W tungsten-filament lightbulb a distane R = m away? Only abut 3% f the pwer frm an inandesent light bulb mes ut as visible light (the rest is heat). The diameter f a human eye pupil is abut mm. nswer: The visible light pwer frm the bulb is P = 3 W. t R = m, the intensity is P 3W = = = 4πR 4 π(m) 6. 4 W/m. The pwer entering a detetr f area is pwer P area 4π R (pwer deteted) = (detetr area) = intensity (detetr area) = n this ase, = area f human iris = π r 3 (mm) = 3-6 m. S the pwer deteted is (6-4 W/m ) (3-6 m ) =.8-9 W (nt muh pwer the eye is extremely sensitive!). M vs. FM TV transmissins are in the radi range f wavelengths: λ TV = t 5 m, frequeny f 8 Hz = MHz. On TV, different hannels rrespnds t different frequenies. Fr instane, hannel 6 is alltted the frequeny range f = 8 86 MHz (wavelength range λ = m ) Last update: /6/9 Dubsn Phys Ntes, University f Clrad

7 M-7 The signal (audi and vide infrmatin) is arried by a range f frequenies ( f = "bandwidth") entered n a "arrier frequeny" f. n audi-nly radi transmissins, the signal's infrmatin is ended either as mplitude Mdulatin (M) r Frequeny Mdulaitn (FM) nalg televisin signals (whih are n lnger used as f 9) are always sent as FM. Digital TV signals are sent in an entirely different frmat: the piture is ended as a series f numbers ('s and 's). Plarized Light M wave always has diretin f diretin f travel: bserver sees: Ordinary light (frm the Sun r a lightbulb) is unplarized, a mixture f waves with -field in randm diretins but always diretin f travel. Last update: /6/9 Dubsn Phys Ntes, University f Clrad

8 M-8 's plarized unplarized Plarizer = plarid filter = filter that passes light with the -field alng the "pass axis" f the filter. light passes light stpped f is nt parallel t the pass axis, then nly the mpnent f alng the pass axis gets thrugh. θ ' θ light ming ut f page plarid pass axis ' = sθ The intensity f the light is prprtinal t, s trans =. s θ Fr unplarized light, with is a mixture with θ randm, s θ =, s avg trans =. Last update: /6/9 Dubsn Phys Ntes, University f Clrad