Chapte 3 lectmagnetic Waves 3.1 Maxwell s quatins and ectmagnetic Waves A. Gauss s Law: # clsed suface aea " da Q enc lectic fields may be geneated by electic chages. lectic field lines stat at psitive chages and extend ut eithe t negative chages t infinity. B. Absence f magnetic mnples B " da 0 clsed suface aea Thee ae n magnetic chages in natue. One says thee ae n magnetic mnples in natue. Magnetic field lines fm clsed lps. 1
C. Faaday s Law (1831) $ clsed lp ind # ds d " dt B whee # B B " da pen suface aea bunded by lp lectic fields may be geneated by a time-changing magnetic field. Induced electic field lines fm clsed lps. D. Ampee Maxwell equatin (1873) $ clsed lp B # ds µ I enc + µ " d dt whee # " da pen aea bunded by lp Magnetic fields may be geneated by: (i) electic cuents (mving chages) and by (ii) time-changing electic fields.
3. Plane lectmgnetic Waves and the Speed f Light Cmments n Maxwell s quatins: 1. Maxwell s equatins fm the theetical basis f all electmagnetic phenmena.. Maxwell s equatins pedict the existence f electmagnetic waves that ppagate thugh vacuum at the speed equal t the speed f light. Theefe, light is an electmagnetic wave c 1 8 µ 3x10 m / s " 8.85x10 1 C N m # (pemittivity f fee space). #7 µ " x10 T m A 4 (pemeability f fee space). 3. By mathematically manipulating Faaday s equatin and the Ampee Maxwell equatin, ne finds that bth the electic field and the magnetic field B satisfy a wave equatin These wave equatins ae: 3
4 t x " µ t B x B " µ which, when cmpaed t the geneal fm f the wave equatin 1 t y v x y yields the esult that electmagnetic waves tavel in vacuum at a speed v that satisfies: v µ 1 s m x c v / 10 3 1 8 µ " the speed f light 3.3 Sinusidal lectmagnetic Waves Cmments n Maxwell s equatins cntinued
4. The electic field and the magnetic field B ae t each the, and they ae bth t the diectin f wave ppagatin. 5. F a plane wave ppagating in the x-diectin, the slutins t the electmagnetic wave equatins can be witten as B whee ( kx " t) yˆ cs ( kx " t) zˆ B cs " f (is the angula fequency in adians pe secnd) k " (k is the wave numbe and equals the magnitude f what is called the wave vect k. The wave vect k pints in the diectin that the electmagnetic wave ppagates. Als, is the wavelength). 5
" (i) the speed f ppagatin equals c f and k (ii) the electic field amplitude is elated t the magnetic field amplitude B by the equatin: c B 3.4 negy and in lectmagnetic Waves A. Intensity I lectmagnetic waves cay enegy. The enegy tanspted pe unit time, pe unit aea by an electmagnetic wave is given by what is called the Pynting vect S, whee S B µ Nte that the electmagnetic wave ppagates in the diectin f the css pduct B. 6
The Pynting vect has units f W/m. The aveage value f the Pynting vect is "S whee S 1 µ B whee I use backets, ", t dente the aveage ve a cmplete cycle. Thus S 1 cs cs µ 1 S # B cs kx " t µ ( kx " t) # B ( kx " t) ( ) but cs (") 1 # $ cs (")d" 1 s that # 0 S 1 µ B The magnitude f the aveage value f the Pynting vect, S, is called the intensity I f the electmagnetic wave. Hence 7
I I I S " S S B µ c c B µ since c 1 µ and c B. Intducing the t-mean-squaed (ms) values f the electic and magnetic fields as ms B ms B then ne can expess the aveage intensity S as I S ms B ms " c ms µ c B ms µ 8
B. The Aveage negy Density u B The enegy pe unit vlume u B assciated with an electmagnetic wave is equal t the sum f the enegy density sted in the electic field uand the enegy density sted in the magnetic field u B. That is but u u + u B # # u cs B ( kx " t) 1 u B 1 B B cs µ µ s that u B becmes ( kx " t) u B # cs 1 µ ( kx " t) + B cs ( kx " t) nw use the elatin B c µ in the secnd tem f the ight hand side f the equatin t get ( kx " t) ub # cs 9
u B f the electmagnetic enegy The aveage value density is then ub # u # ( kx " t) ( kx " t) B cs u B Finally, nte that since the intensity S equals I S c then, the intensity f the electmagnetic wave is elated t the aveage enegy density f the wave by I S c u B S is in units f Watts pe squae mete, while u B is in units f Jules pe cubic mete. 10
Pductin f lectmagnetic Waves by an Antenna The fundamental mechanism espnsible f this adiatin is the acceleatin f a chaged paticle. Wheneve a chaged paticle acceleates, it must adiate enegy. 11
The Spectum f M Waves All M waves tavel thugh vacuum with a speed c. Thei fequency f and wavelength ae elated by c f The electmagnetic spectum cnsists f (in de f deceasing wavelengths): 1. Radi waves (a) " 0.3 m (b) geneated by electns acceleated in wies.. Micwaves (a) 1 mm < < 0. 3 m (b) geneated by electn tubes called klystns. 3. Infaed waves (ften times called heat waves) (a) 700 nm < < 1 mm (b) geneated by the tatins and vibatins f mlecules within a mateial. 4. Visible light (a) 400 nm < < 700 nm (b) detected by human eye. (c) pduced by e-aangement f electns in atms and mlecules. 1
5. UV light (a) 10 nm < < 400 nm 6. X-Rays " (a) 10 4 nm < < 10 nm (b) geneated by sudden deceleatin f high speed electns, by electnic tansitins within atms. 7. Gamma ays ( ays) (a) " < 10 4 nm (b) emitted by adiactive atmic nuclei. 13