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Physics 203/204 1: Electromagnetic Waves •Maxwells Equations •Integral form •Differential form •Solutions to Maxwells equation •Transport of Energy in em waves •Antennas. Maxwells Equations •Gauss’ Law •For Electric Fields: Q E dA E 0 BdA 0B surface enclosing electric charge •Gauss’ Law •For Magnetic Fields: surface enclosing magnetic cha Ampere Maxwell Law dE Bdsm0I+0m0 dt path enclosing current I and changing electric field Faradays Law Eds dB dt path enclosing changing magnetic flux Changing Magnetic Flux Produces Induced Electric Field Changing Electric Flux Produces Induced Magnetic Field Changing Electric Field Fluctuating electric and magnetic fields electromagnetic Radiation Changing Magnetic Field 1 c 0m 0 Differential Form of Maxwells equations Maxwells Equations r E B 0 B E t E B m J+ t Integral to Differential E dA B dA closed surface closed surface r dV enclosed volume 0 B dA t E ds closed path Area enclosed by path as B B dA B ds closed path E m J+ dA t area enclosed by path as I J dA Vector Calculus Gauss Divergence Theorem F dA closed surface F dV volume enclosed by surface i + j + k x z y closed path Stokes F ds Theorem F d A area enclosed by path EdA Eds closed path area enclosed by path B dA t Area enclosed by path B E t Bds closed path BdA area enclosed by path E m J+ dA t area enclosed by path E Bm J+ t EdA closed surface EdA enclosed volume r dV enclosed volume r E B dA closed surface BdA 0 enclosed volume B 0 Maxwells Equations in empty space : = 0; J 0 gives 2E 1 2E c2 t2 Wave Equation for light 2B 1 2B 2 2 c t where the Laplacian 2 2 2 2 2 + 2 + 2 x y z These fields can be expanded as Ex, y, z,t Ex x, y, z,t i + Ey x, y, z,t j + Ez x, y, z,t k Bx, y, z,t B x x, y, z,t i + B y x, y, z,t j + B z x, y, z,t k and we get general linear wave equations for the components, e.g. 2E x 2E x 2E x 1 2E x + 2 + 2 2 2 2 x y z c t The solution of these equations MUSTalso satisfy Maxwells equations. The simplest solution has the form Er, t E x, t ; The field has the same, constant,value for all y, z at any time t and propagates in the positive- x direction. Then E x, t Emax sinkx wt B x, t Bmax sinkx wt E2 c 2B2 ; 2p wave number k ; angular frequency w = 2pn l w ln = c c k One set of solutions to Maxwells Equations are Er, t E y max j sin kx wt Br, t Bz max k sin kx wt Which are PLANE waves propagating in the xdirection z y E x B •Waves are made up of propagating vibrations in some medium. In the case of em waves: •the medium is NON MATERIAL i.e. does not have MASS •the medium is electric and magnetic fields •the vibrations are fluctuations of the electric and magnetic fields •the medium creates itself as •changing magnetic fields produce •changing electric fields etc.... •Energy is propagated through the medium •Energy Density in : •Electric Field 1 2 uE 0 E 2 •Magnetic Field 1 2 uB B 2m0 0 2 0 2 2 0c 2 uE E c B 2m 0 uB 2 2 2 1 2 as uB B 2m0 1 uE 0 c m 0 uB 0 m 0 uB uB 0m 0 2 thus total energy density is 2 B u uE + uB 0 E m0 2 In order to describe the energy propagated we need to consider the transport of energy per unit time across a unit area. energy per unit time is power During a very small time interval D t, A ( c D t), the energy contained in the volume will cross the area A , this energy is A ( c D t) u given Thus the energy and per unit area per unit time is by S A ( c D t) u EB = uc m0 ADt energy flow is in direction of propagation of wave, the vector describing this is called the Poynting vector S 1 E B m The average power transported over each cycle for a sinusodial plane em wave is called the WAVE INTENSITY, denoted by I. 2 2 Emax Bmax Emax c Bmax I Sav = = = = cuav 2m 0 2m 0 c 2m 0 The SI units of S are S = W m 2 Electromagnetic waves transport linear momentum as well as energy, therefore produce pressure *If the radiation makes a perfect inelastic collision with a surface (i.e. all the radiation is absorbed - -a black body surface) then all the momentum of the radiation to surface) is transferred to the surface, (consider component normal then momentum delivered to the surface U Dt DP = ; where U is the total energy transported during time c *If all the radiation is reflected the total momentum 2U DP = Change of surface during this time will be c Rate of momentum transfer = Force dP Newtons 2nd Law F= dt F p , pressure Force per unit area A dP \ radiation pressure p = dt A 1 dU S * for perfect absorber p = Ac dt c 1 2dU 2S * for perfect reflector p = Ac dt c for sunlight p 5 10 6 N m 2 Accelerating charge produces em waves e.g. oscillating charge varying current produces varying magnetic field varying position of charge produces varying electric field radiation “far” from “dipole antenna”