Electricity and Magnetism-Polarization of E.M.waves

Topics: 􀂄 􀂄 Polarization of EM waves 􀂄 8.022 (E&M) – Lecture 20 Electromagnetic plane waves and their properties Polaroids and linear and circular polarization Last time 􀁇􀁇⎧∇iE = 4πρ 􀂄 Completed Maxwell’s equations ⎪ 􀁇􀁇 ⎪∇iB = 0 􀂄 Displacement currents ⎪ 􀁇 􀁇 1 ∂B 􀁇 􀂄 Kirchoff’s laws are legitimate! ⎨∇× E = − ∂⎪ ct⎪ 􀁇 􀁇 4π 􀁇 1 ∂E 􀁇 ⎪∇× B = J + 􀂄 Solved Maxwell’s equations in vacuum ⎩ cc ∂t 􀂄 Derived wave equation for EM waves 􀁇2 􀂄 They travel at speed of light: light is EM wave! 􀁇2 􀁇 1 ∂ E∇ E = 2c ∂t 2 􀂄 Started studing properties of the general solution f(x-vt) 􀂄 Today we will complete the study of these properties… G. Sciolla –MIT 8.022 – Lecture 20 2 1 Plane waves 􀂄 Fourier Theorem: 􀂄 Any periodic function can be expressed as a linear combination of sin and cos functions 􀃆 sin and cos are the building blocks of all waves! 􀂄 Plane waves in the most general form: 􀁇􀁇􀁇􀁇 􀁇 􀂄 E = E sin(kr −ω t) = E sin(k x + ky + kz −ω t)0 i 0 xyz􀁇􀁇􀁇􀁇 􀁇􀂄 B = B sin(kr −ω t) = B sin(k x + k y + kz −ω t)0 i 0 xyz where: 􀁇 􀁇 ˆk = wavevector; k = wavenumber; k = propagation direction G. Sciolla –MIT 8.022 – Lecture 20 3 4 Plane waves vs f(x-ct) 􀂄 satisfies the wave equation 􀂄 ? 􀂄 􀂄 ω and c: ( )kr tω± 􀁇􀁇i ( )xct± ( )fx ct± ˆ( ) ( )fx ct ckt± ⇒ ± 􀁇 ˆ ˆ( ) ( )kr t kt ckt ck k ωω ω ± = ± = = ± 􀁇 􀁇 􀁇􀁇 􀁇 􀁇i i i G. Sciolla – MIT 8.022 – Lecture 20 We proved thatHow to connect to the the argument of plane waves From 1D to 3D: Relation between k, f r k r k r 2 G. Sciolla – MIT 8.022 – Lecture 20 􀂄 􀃆 5 More on k and ω 􀂄 􀂄 􀂄 λ 􀂄 􀂄 0 )xE E tω= − 0 )xE E= 0 )E E tω= |E(x)| x λ t Τ 2 2 ck cT πω ω= = = = Choose a system of coordinates so that our wave vector k is oriented //to x axis: plane wave solution for E is Let’s consider only the spatial variation of the wave (e.g. t=0): = wavelength Let’s now consider the time variation of the wave (e.g. x=0): Relations between variables: sin( k x 􀁇 􀁇 sin( k x 􀁇 􀁇 sin( 􀁇 􀁇 |E(t)| πν λν 6 equations? 􀂄 􀂄 􀂄 0 0 0 0 0 ) ( cos( ) x y z x x y y z z x y z E E t Ek Ek t t ω ω ω ⎡ ⎤∇ = ∇ + + − = ⎣ ⎦ + + + + − − 􀁇 􀁇 i i 􀁇 􀁇􀁇 􀁇 􀁇 􀁇 􀁇i i 2 2 2 2 0 2 2 0 2 2 1 ( ) 1 ( ) ω ω ⎧ ∂∇ =⎪ ⎧ = −⎪ ⎪∂ ⇒⎨ ⎨∂ = −⎪⎪ ⎩∇ = ⎪ ∂⎩ 􀁇􀁇􀁇 􀁇􀁇 􀁇 􀁇i 􀁇􀁇 􀁇􀁇􀁇 i EE E E tc t B B B tB c t 0∇ = 􀁇􀁇 iE 0 0igE⇒∇ = = ⇒ ⊥ 􀁇􀁇 􀁇 􀁇 i i G. Sciolla – MIT 8.022 – Lecture 20 Do plane wave satisfy Maxwell’s EM waves are a consequence of Maxwell’s equations in the sense that we used the 4 Maxwell’s Equations to derive the wave equations for E and B: Does the solution of the EM wave equation satisfy all Maxwell’s Equations? Not necessarily! Let’s start with Gauss’s law: sin( ) cos( )=k x k y kz E k k x k y kz kE kr sinsin􀁇 􀁇 k r k r 0 when E is to wave's drection of propaation kE 3G. Sciolla – MIT 8.022 – Lecture 20 7 More constraints on plane waves 􀂄 Constraints following from 􀂄 Constraints following from 􀂄 Time derivative does not change direction of B 􀃆 􀂄 Same conclusion follows from: 0∇ = 􀁇􀁇 iB 0 cos( ) 0 0 B is to direction of propagation B kB kr t kBω∇ = − ⊥ =⇒ = ⇒ 􀁇 􀁇 􀁇􀁇 􀁇 􀁇􀁇i i i 􀁇 i Conclusion: E Bk k⊥ ⊥ ⊥ 􀁇 􀁇􀁇 􀁇 0∇=iB1 ∂∇× = ∂ 􀁇􀁇􀁇 EB ct ⊥ 􀁇 􀁇 E B 1 ∂∇× = − ∂ 􀁇􀁇􀁇 BE ct 􀁇 E 􀁇 B 􀁇 k G. Sciolla – MIT 8.022 – Lecture 20 8 More constraints on plane waves 􀂄 Let’s now calculate: 􀂄 Using and the fact that B0 =const 0 0 0 1 sin( ) sin( ) cos( ) ω ω ω ω ∂ = − = − ∂ ∇× =∇× + + − 􀁇 􀁇 􀁇􀁇 􀁇􀁇 􀁇i i 􀁇􀁇􀁇􀁇 x y z E E k r t kE k r t ct c B B k x k y k z t 1 ∂∇× = ∂ 􀁇􀁇􀁇 EB ct ()∇× =∇× +∇ × 􀁇 􀁇 􀁇􀁇 􀁇 􀁇 vs v s v ( ) ( ) ( ) 0 0 0 0 0 cos( ) cos( ) cos( ) ˆ ˆ ˆ sin( ) sin( ) ω ω ω ω ω ⎡ ⎤∇× =∇× + + − =⎣ ⎦ =∇× − +∇ + + − × = = − + + + + − × = =− × − 􀁇􀁇􀁇 􀁇 􀁇􀁇􀁇 􀁇 􀁇􀁇i 􀁇 􀁇 􀁇􀁇 􀁇i x y z x y z x y z x y z B B k x k y k z t B k r t k x k y k z t B k x ky k z k x ky k z t B kB k r t 4G. Sciolla – MIT – 9 More constraints on plane waves 􀂄 From 􀂄 Important consequences: 􀂄 􀂄 i􀂄 􀂄 1 ∂∇× = ∂ 􀁇 EB ct ( )0 0 0 0 ˆ( ) sin( ) rs ω ω− × − ⇒ = − 􀁇 􀁇 􀁇􀁇 􀁇􀁇 􀁇 􀁇 􀁇􀁇 􀁇 􀁇 i ikB t t E k B 0 00 0 ˆ ⇒ = 􀁇 􀁇 EkB BE 2 0 0 0 0 0 ˆ ˆ⇒ × = 􀁇 􀁇 􀁇 E k B E B E k 0 0×E B 􀁇 E 􀁇 B 􀁇 k G. Sciolla – MIT 8.022 – Lecture 20 it follows that In cgs, E and B have the same magnitude s parallel to the propagation of wave NB: E x B has an important physical meaning that we will soon see 􀁇􀁇 or k, E and B are right handed ort sinogonal vecto =− × k r kE k r =− × =− × 􀁇􀁇 􀁇􀁇 8.022 10 Polarization of EM waves 􀂄 􀂄 􀂄 􀂄 􀂄 􀂄 0 and B0 “linearly polarized” ˆ ˆ////􀁇 􀁇 E y B z 0 0 ˆ( ˆ ) ( )ω ω = − =⎪⎨ ⎪⎩ −⎧ 􀁇 􀁇 E B B E kx = direction of electric field Lecture 20 Did we use all of our freedom in choosing the waves? No, we can still choose the so called “polarization state” Linear polarization: Consider a plane wave propagating in the x direction Choose the coordinate system so that at t=0 If the directions of Eare constant in time, the wave is and coscos kx t z t y NB: direction of polarization 5G. Sciolla – MIT 8.022 – Lect e 20 11 When receiving antenna is //to broadcasting one, good reception Linear Polarization of EM waves 􀂄 􀂄 􀂄 􀂄 􀂄 􀂄 How do we receive the signal? 􀂄 oscillate Demo K1 ~ E ur How to produce linearly polarized waves? Oscillating charge distribution in a conductor Broadcasting antenna How to produce such charge? Long conductor driven by oscillating current Receiving antenna When receiver is perpendicular to broadcasting antenna: no reception because there is not enough room for charges to 12 Demo K1: E of microwaves 􀂄 polarized EM waves 􀂄 􀂄 􀂄 􀁇 E 􀁇 B􀁇 k Differential G. Sciolla – MIT 8.022 – Lecture 20 A microwave generator produces a signal of 10.5 GHz How should we orient antenna to detect a signal on the scope? Antenna //E will detect signal Antenna perpendicular to E: no signal Scope Antenna amplifier 613 Polaroids 􀂄 direction 􀂄 􀂄 􀂄 􀂄 Charges move 􀃆 􀃆 plastic heats: light stopped 􀂄 ): 􀂄 Conclusion: 􀂄 G. Sciolla – MIT 8.022 – Lecture 20 Sheet of plastic embedded with organic molecules extended in one They can carry current in that particular direction: behave like antennas! When linearly polarized light hits the polaroid: If E is aligned with orientation of molecules: current is generated If E is perpendicular to orientation of molecules (“preferred direction”Charges will not be able to move in that direction: light goes through Polaroids are transparent to light polarized //to their preferred direction and opaque to light polarized in the direction perpendicular to their preferred direction Polaroids and polarization direction 􀂄 What happens when the light is polarized in a direction in between the preferred direction and its perpendicular? 􀂄 Example: light polarized along x axis; polaroid oriented at θ angle􀁇 E = E0 cos( kz −ω )ˆt x pˆ = xˆ cos θ+ yˆ sin θ 􀂄 Light will go through partially 􀂄 Since E has a component //to preferred direction of polaroid 􀂄 E coming out is overlap between incoming E and polaroid’s orientation 􀁇 􀁇 = E p E cos( kz −ωt) ( xˆ cos θ+ yˆ sin θ)i xˆ = E0 cos θcos(kz −ωt)i ˆ = 0Eout 􀁇 􀁇 = i ˆˆp (parallel to polaroid's orientation)E p Eout Conclusion: Polaroids reduce the amplitude of linearly polarized light by cosθ (angle between E and polaroid’s orientation) and rotates the orientation of E by θ G. Sciolla –MIT 8.022 – Lecture 20 14 7 Unpolarized light 15 Polarization of random light 􀂄 􀂄 􀂄 􀂄 􀂄 Conclusion: 􀂄 􀂄 0 0ˆ ˆ( ) ( ) ( )θ ω ω θ= − = −∑ ∑􀁇 out i i i i E E x t E x t 0 ˆ ˆ( ) )θ θ ω= + −∑􀁇 i i i E E x y t G. Sciolla – MIT 8.022 – Lecture 20 Light from a bulb, sunlight, etc is not polarized Superposition of many plane waves, each with its own polarization When light passes through a polaroid becomes linearly polarized If polaroid is oriented //x axis: Polaroids can be used to produce linearly polarized light The intensity of the light will be reduced cos coscoscos kz kz cos sin cos( random kz 16 Demo: 3 vs 2 polaroids 􀂄 lock light 􀂄 i􀂄 􀂄 􀂄 ize light //x 􀂄 0’=E0 o 􀂄 0’=E0 (cos45o)2= E0/2 DEMO T1 G. Sciolla – MIT 8.022 – Lecture 20 2 polaroids with orthogonal preferred direction will bFirst polaroid (P1) polarizes lght in the direction x (for example) Second polaroid (P2)oriented in the y direction, but E is now just //x Now place a third polaroid P3 in between p1 and p2 (at 45 degrees) P1 will polarP3 will select only component //to its preferred direction and rotate direction of polarization by 45 deg. Ecos45P2 will select component y direction that now is not 0 anymore. Intensity further reduced, but not 0! E 8ure 20 18 17 Polarization of microwaves (K3) 􀂄 􀂄 l􀂄 􀂄 i i􀂄 􀃆 light! 􀁇 E 􀁇 B 􀁇 k Ch2: receiver Transmitter Receiver G. Sciolla – MIT 8.022 – Lecture 20 10.5 GHz polarized microwaves Rotate the receiver to find the direction of poarization of signal Now introduce a conductive “comb” in between transmitter and receiver When teeth of comb are //E: sgnals blocked When they are perpendicular to E: signal can go through Exactly the same behavior of Polaroid for Scope Ch1: transmitter Circular polarization 􀂄 􀂄 􀂄 Easier to understand if we look at z=0 􀂄 ω 􀂄 E and B vectors describe circles over time 0 0 0 0 ˆ ˆ( ) ( ) ˆ ˆ) ( ) ω ω ω ω = − + − = − − − 􀁇 􀁇 E Ex t t B By t B x t 0 0 0 0 ˆ ˆ) ( ) ˆ ˆ) ( ) ω ω ω ω =− + =− − 􀁇 􀁇 E t t B t B x t G. Sciolla – MIT 8.022 – Lect Consider a wave with the following form: What is it? Electric and magnetic fields rotate at frequency Circular polarization because sincossin( coskz E y kz kz kz sin( cossin( cosE x E y B y 919 Circular polarization (2) 􀂄 􀂄 Rotating dipole 􀂄 􀂄 􀂄 i-+ E 􀁇-+ -+ -+-+ ~ ~ G. Sciolla – MIT 8.022 – Lecture 20 How to produce it? 2 antennas at 90 deg driven by currents off by 90 deg NB: circular polarization does exist in nature Example: circular polarzation filters used in photography 20 Elliptical Polarization 􀂄 For a given kplane waves, e.g. 2 possible directions of E 􀂄 combinations of these 􀂄 φ1=φ2: linear polarization 􀂄 φ1=φ2+90ο􀂄 All the rest: elliptical polarization E B E B 1 0 1 2 0 2 ˆ ) ˆ ) ω φ ω φ = − + = − + 􀁇 􀁇 E t E t G. Sciolla – MIT 8.022 – Lecture 20 , there are 2 independent solutions for the All other solutions are just linear : linear polarization cos( cos( E x kz E y kz 1022 21 Summary and outlook 􀂄 􀂄 􀂄 􀂄 􀂄 􀂄 􀂄 􀂄 Next Tuesday: 􀂄 􀂄 􀂄 G. Sciolla – MIT 8.022 – Lecture 20 Today: Electromagnetic plane waves Constraints on E, B and k following from Maxwell’s equations E, B and k are always perpendicular to each other Amplitude of E and B are the same in cgs Polarization of EM waves Polaroids and linear and circular polarization Energy and momentum carried by EM waves Poynting vector Transmission lines 8.022 subject evaluation 􀂄 Fast: 􀂄 􀂄 Important: 􀂄 􀂄 You can be honest! 􀂄 􀂄 A volunteer will􀂄 NB: 􀂄 􀂄 the evaluation for ONE tutor: 􀂄 iiG. Sciolla – MIT 8.022 – Lecture 20 5-10 minutes of your time Your chance to make comments about the class We will not look at the forms until after grades are registered collect the results and will bring them to the PEO(?) Fill in both sides of the form; side 1 will be read by computer Staff: in addition to lecturer and recitation instructor you can fill in Michael Shaw OR Min Liang Zhao 22 Reward: how to “hack” a rado staton 11