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Wave Dispersion EM radiation Maxwell’s Equations 1 Wave Dispersion - Simply stated, a dispersion relation is the function ω(k) for an harmonic wave. - A dispersion relation connects different properties of the wave such as its energy, frequency, wavelength and wavenumber. - From these relations the phase velocity and group velocity of the wave can be found and thereby refractive index of the medium can be determine. 2 Non dispersive: All colours moving with same speed t0 t1 t2 Dispersive: Red moving faster than blue 3 Normal dispersion of visible light Shorter (blue) wavelengths refracted more than long (red) wavelengths. Refractive index of blue light > red light. A medium in which phase velocity is frequency dependent is known as a dispersive medium, and a dispersion relation expresses the variation of as a function of k. If a group contains number of components of frequencies which are nearly equal, then: Group velocity Phase and Group velocity vg P’ 6 Non dispersive waves vp = Constant Signal is propagated without distortion (or k) More generally vp is a function of vpk 7 Usually, dv p is positive, so that vg < vp d dv p When, is negative, so that vg > vp d When, is constant, so that vg = vp k Normal Dispersion Anomalous Dispersion Non-Dispersive medium (Ex: Free space) More in electromagnetic waves Wave packet (without Dispersion) Wave packet (with Dispersion) Non-dispersive Dispersive Wikipedia.org Wave Packets Superposition of waves and wave packet formation y(t) = Sin t 0 < t < 200 12 y(t) = [Sin t + Sin (1.08 t)]/2 0 < t < 200 13 Suppose we have group of many frequency components lying within the narrow frequency range … y(t) = [Sin t + Sin(1.04 t) + Sin (1.08 t)]/3 0 < t < 400 15 y(t) = [Sin t + Sin(1.02 t) + Sin (1.04 t) + Sin(1.06 t) + Sin (1.08 t)]/5 0 < t < 400 16 y(t) = [Sin t + Sin(1.01 t) + Sin (1.02 t) + Sin(1.03 t) + Sin (1.04 t) + Sin (1.05 t) + Sin (1.06 t) + Sin (1.07 t) + Sin (1.08 t)]/9 0 < t < 800 17 y(t) = [Sin t + Sin(1.01 t) + Sin (1.02 t) + Sin(1.03 t) + Sin (1.04 t) + Sin (1.05 t) + Sin (1.06 t) + Sin (1.07 t) + Sin (1.08 t)]/9 0 < t < 400 18 19 http://en.wikipedia.org/wiki/Coherence_%28physics%29 Electromagnetic Radiation Let‘s first develop the understanding by taking the example of oscillating charge and/or dipole oscillator 21 For stationary charges the electric force field 1 2 r Coulomb’s law © 2005 Pearson Prentice Hall, Inc © 2005 Pearson Prentice Hall, Inc Coulomb’s law What is the electric field produced at a point P by a charge q located at a distance r? q eˆr E 2 4 0 r where er is an unit vector from P to the position of the charge © 2005 Pearson Prentice Hall, Inc If a charge moves non-uniformly, it radiates The electric field of a moving point charge http://www.cco.caltech.edu/~phys1/java/phys1/MovingCharge/MovingCharge .html Electric field q E t 4 0 er r d er 1 d r 2 c dt r 2 c 2 dt 2 er 2 er :unit vector directed from q to P at earlier time q - P The correct formula for the electric field _ Important features 1. No information can propagate instantaneously 2. The electric field at the time t is determined by the position of the charge at an earlier time, when the charge was at r’, the retarded position. 3. First two terms falls off as 1/r’2 and hence are of no interest at large distances Correct Expression (at large distances) This is electro-magnetic radiation or simply radiation. It is also to be noted that only accelerating charges produce radiation. Electric Dipole Oscillator © SPK/SB Radio-wave transmission TV Antenna Car Antenna “Let there be electricity and magnetism and there is light” J.C. Maxwell Vector Analysis (Refresh) GRADIENT For a scalar function T of three variable T(x,y,z), the gradient of T is a vector quantity given by: T T T T xˆ yˆ zˆ x y z - The gradient points in the direction of the greatest rate of increase of the function, - and its magnitude is the slope (rate of increase) of the graph in that direction. DIVERGENCE For a vector T the divergence of T is given by: T xˆ yˆ zˆ Tx xˆ Ty yˆ Tz zˆ y z x Tx Ty Tz y z x It is a measure of how much the vector T diverges / spreads out from the point in question. CURL For a vector T the Curl of T is given by: T xˆ y x xˆ yˆ x y T T y x yˆ z zˆ Tx xˆ Ty yˆ Tz zˆ zˆ z Tz It is a measure of how much the vector T curls around the point in question. DIVERGENCE THEOREM / Green’s Theorem / Gauss’s Theorem ( E ) d E d a V S Integral of a derivative (in this case the divergence) over a volume is equal to the value of the function at the surface that bounds the volume. (Faucets within th e volume) (Flow out throug h the surface) V S STOKES’ THEOREM ( E ) d a E d l S P Integral of a derivative (in this case the curl) over a patch of surface is equal to the value of the function at the boundary (perimeter of the patch). What we know from previous classes? 1) Oscillating magnetic field generates electric field (Faraday´s law) and vice-versa (modified Ampere´s Law). 2) Reciprocal production of electric and magnetic fields leads to the propogation of EM waves with the speed of light. Question: WAVES?????? How do we show that a wave is obtained? 40 Our Attempt: To derive the relevant wave equation 41 Consider an oscillating electric field Ey y Ey This will generate a magnetic field along the zaxis x Bz z May 3, 2017 42 Faraday’s Law Y Ey(x) Ey(x+x) The induced electromotive force in any closed circuit is equal to the negative of the time rate of change of the magnetic flux through the circuit. BA Voltage generated N t C x N: Number of turns B: External magnetic field A: Area of coil We know that Faraday´s law in the integral form in given as: Z E.dl B.ds t C s where C is the rectangle in the XY plane of length l width x, and S is the open surface spanning the contour C May 3, 2017 43 Using the Faraday´s law on the contour C, we get: E.dl C B.ds t s Bz E y ( x x) E y ( x) l lx t this implies... E B x t y z Keep this in mind... 44 We know that the Ampere´s law with displacement current term can be written as: B.dl E.ds t o C o Y Ey s x C/ x Bz(x) z Bz(x+x) 45 Using the Ampere´s law, for the Contour C/, we get: B.dl E.ds t o o C s E B ( x x) B ( x)l lx t y z z 0 0 this implies... E B x t z 0 y 0 46 Using the eq. obtained earlier i.e., E B x t E B x t y z z 0 Ey 2 t 0 Ey 2 c 2 c2 y 2 where 1 0 0 x 2 Form of wave equation Note: Similar Equation can be derived for Bz 47 Solution of EM Wave equation Electromagnetic waves for E field for B field In general, electromagnetic waves 1 2 2 c t Where represents E or B 2 2 or their components # A plane wave satisfies wave equation in Cartesian coordinates # A spherical wave satisfies wave equation in spherical polar coordinates # A cylindrical wave satisfies wave equation in cylindrical coordinates Solution of 3D wave equation In Cartesian coordinates 1 2 2 2 2 2 x y z c t 2 2 2 2 2 Separation of variables ( x, y, z, t ) X ( x)Y ( y ) Z ( z )T (t ) Substituting for we obtain 2 1 X 1Y1 Z 1 1T 2 2 2 2 2 X x Y y Z z c T t 2 2 2 Variables are separated out Each variable-term independent And must be a constant So we may write 1 X k 2 ; 1 Y k 2 ; x y 2 2 X x Y y 2 2 2 1 Z k ; 1 T 2 2 2 Z z T t 2 2 z where we use c k k k k 2 2 2 x 2 y 2 z 2 Solutions are then X ( x) e Z ( z) e ik x x ik z z ; Y ( y) e ; T (t ) e ik y y ; i t Total Solution is ( x, y, z, t ) X ( x)Y ( y ) Z ( z )T (t ) i[t ( k x x k y y k z z )] i[t k .r ] Ae Ae plane wave Traveling 3D plane wave Spherical coordinates (r, θ, φ): radial distance r, polar angle θ (theta), and azimuthal angle φ (phi) spherical waves x r sin cos y r sin sin z r cos Spherical waves 2 1 cos 1 2 2 2 2 2 2 2 r r r r sin r sin r 2 2 2 2 f f 2 Alternatively 2 2 1 2 2 2 2 r r r r r r r The wave equation becomes 1 r 2 2 r r r 1 2 2 c t 2 1 r 2 r 2 r r 2 1 2 2 c t u (r ) Put (r ) r 1 u u 2 Then 2 r r u u r r r r r r 2 Hence r r r u r u r r 2 u r u2 u r r r 1 r 2 r 2 r r (r ) Therefore 1 r 2 2 r r r 1 2u 2 r r Wave equation transforms to 1 u 1 1 u 2 2 2 r r c r t 2 2 1 2 2 c t 2 u 1 u 2 2 2 r c t 2 2 u (r ) r Separation of variables u (r , t ) R(r )T (t ) Which follows that 1 R 1 1 T k 2 2 2 2 R r c T t 2 2 Solutions are ikr R(r ) e ; T (t ) e Total solution is u (r ) e i t i ( t kr ) kc Final form of solution i ( t kr ) 1 (r ) e spherical wave r General solution i ( t kr ) i ( t kr ) 1 1 (r ) e e r r outgoing waves incoming waves Cylindrical waves Cylindrical Coordinate Surfaces(ρ, φ, z). The three orthogonal components, ρ (green), φ (red), and z (blue), each increasing at a constant rate. The point is at the intersection between the three coloured surfaces. 1 2 2 c t 2 2 1 1 2 2 2 2 r r r r z 2 2 2 2 with angular and azimuthal symmetry, the Laplacian simplifies and the wave equation 1 r r r r 1 2 2 v t 2 The solutions are Bessel functions. For large r, they are approximated as A (r , t ) cos( kr t ) r Maxwell’s equations II Use B in Divergence Theorem ( E )d E da V S No magnetic monopoles III Use E in Stokes’ Theorem ( E) da E dl S From Faraday’s Law P IV Use B in Stokes’ Theorem From Ampere’s Law B dl o I C Charge conservation is a fundamental law of Physics which is written as a continuity equation IV Maxwell’s equations Plane EM waves in vacuum Wave vector k is perpendicular to E Wave vector k is perpendicular to B B is perpendicular to E B, k and E make a right handed Cartesian co-ordinate system Plane EM waves in vacuum