Class 30: Outline. Hour 1: Traveling & Standing Waves. Hour 2: Electromagnetic (EM) Waves P30-

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1 Class 30: Outline Hour 1: Traveling & Standing Waves Hour : Electromagnetic (EM) Waves P30-1

2 Last Time: Traveling Waves P30-

3 Amplitude (y 0 ) Traveling Sine Wave Now consider f(x) = y = y 0 sin(kx): π Wavelength ( λ ) = wavenumber ( k) x What is g(x,t) = f(x+vt)? Travels to left at velocity v y = y 0 sin(k(x+vt)) = y 0 sin(kx+kvt) P30-3

4 Traveling Sine Wave y y0 sin ( kx kvt) = At x=0, just a function of time: y = y sin( kvt) y sin( ωt) Amplitude (y 0 ) Period ( T ) = = 1 frequency ( f ) π angular frequency ( ω) P30-4

5 Traveling Sine Wave i i i i i i i Wavelength: λ Frequency : f π Wave Number: k = λ Angular Frequency: ω = π f 1 π Period: T = = f ω ω Speed of Propagation: v= = λ f k Direction of Propagation: + x y = y 0 sin( kx ωt) P30-5

6 This Time: Standing Waves P30-6

7 Standing Waves What happens if two waves headed in opposite directions are allowed to interfere? E = E sin( kx t) 1 0 ω E = E 0 sin( kx+ωt) Superposition: E = E + E = E sin( kx)cos( ωt) 1 0 P30-7

8 Standing Waves: Who Cares? Most commonly seen in resonating systems: Musical Instruments, Microwave Ovens E = E sin( kx)cos( ωt) 0 P30-8

9 Standing Waves: Bridge Tacoma Narrows Bridge Oscillation: P30-9

10 Group Work: Standing Waves Do Problem E = E sin( kx t) 1 0 ω E = E 0 sin( kx+ωt) Superposition: E = E + E = E sin( kx)cos( ωt) 1 0 P30-10

11 Last Time: Maxwell s Equations P30-11

12 Maxwell s Equations C C S S Qin E da= (Gauss's Law) ε0 dφ B E d s = (Faraday's Law) dt B da= 0 (Magnetic Gauss's Law) B ds = µ I + µ ε 0 enc 0 0 dφ dt E (Ampere-Maxwell Law) F= q( E+ v B) (Lorentz force Law) P30-1

13 Which Leads To EM Waves P30-13

Electromagnetic Radiation: Plane Waves http://ocw.mit.

14 Electromagnetic Radiation: Plane Waves P30-14

15 Traveling E & B Waves i i i i i i i E Wavelength: λ Frequency : f 0 π Wave Number: k = λ Angular Frequency: ω = π f 1 π Period: T = = f ω ω Speed of Propagation: v= = λ f k Direction of Propagation: + x Eˆ E sin( kx t) = ω P30-15

16 Properties of EM Waves Travel (through vacuum) with speed of light v = c= 1 = µε 0 0 E B m At every point in the wave and any instant of time, E and B are in phase with one another, with E 0 = = B 0 E and B fields perpendicular to one another, and to the direction of propagation (they are transverse): Direction of propagation = Direction of E B c s P30-16

17 PRS Questions: Direction of Propagation P30-17

18 How Do Maxwell s Equations Lead to EM Waves? Derive Wave Equation P30-18

19 Wave Equation Start with Ampere-Maxwell Eq: C d B ds = µε 0 0 E da dt P30-19

20 Apply it to red rectangle: Wave Equation Start with Ampere-Maxwell Eq: B d s = Bz( xtl, ) Bz( x+ dxtl, ) C d E y µε 0 0 d = µε 0 0 ldx dt E A t C d B ds = µε 0 0 E da dt Bz( x+ dx, t) Bz( x, t) Ey = µε 0 0 dx t So in the limit that dx is very small: B z E y = µε 0 0 x t P30-0

21 Now go to Faraday s Law Wave Equation C d E ds = B da dt P30-1

22 Wave Equation Apply it to red rectangle: E d s = Ey( x+ dxtl, ) Ey( xtl, ) C Faraday s Law: d d = ldx dt B A t B z C d E ds = B da dt Ey( x+ dx, t) Ey( x, t) Bz dx = t E y B = z x t So in the limit that dx is very small: P30-

23 1D Wave Equation for E Ey B E z B z = = µε 0 0 x t x t Take x-derivative of 1st and use the nd equation E E E x x x x t t x t y y Bz Bz y = = = = µε 0 0 y E E y = µε y 0 0 x t P30-3

24 1D Wave Equation for E E E y = µε y 0 0 x t This is an equation for a wave. Let: y ( ) E = f x vt E x E t y y = f '' x vt ( ) v f x vt = '' ( ) v = 1 µ ε 0 0 P30-4

25 1D Wave Equation for B B E z y B E z = = µε 0 0 t x x t Take x-derivative of 1st and use the nd equation B B E E 1 B = = = = t t t t x x t x z z y y z µε 0 0 y B B z = µε z 0 0 x t P30-5

26 Electromagnetic Radiation Both E & B travel like waves: E y E y B z B = µε = µε z x t x t But there are strict relations between them: B E B E z = y z = µε 0 0 t x x t y Here, E y and B z are the same, traveling along x axis P30-6

27 Amplitudes of E & B ( ) ( ) Let E = E f x vt ; B = B f x vt y 0 z 0 B z E y = t x ( ) '( ) vb f x vt = E f x vt ' 0 0 vb = E 0 0 E y and B z are the same, just different amplitudes P30-7

28 Group Problem: EM Standing Waves Consider EM Wave approaching a perfect conductor: E = xe ˆ cos( kz ωt) incident 0 If the conductor fills the XY plane at Z=0 then the wave will reflect and add to the incident wave 1. What must the total E field (E inc +E ref ) at Z=0 be?. What is E reflected for this to be the case? 3. What are the accompanying B fields? (B inc & B ref ) 4. What are E total and B total? What is B(Z=0)? 5. What current must exist at Z=0 to reflect the wave? Give magnitude and direction. Recall: cos A+ B = cos A cos B sin A sin B ( ) ( ) ( ) ( ) ( ) P30-8

Next Time: How Do We Generate Plane Waves? http://ocw.mit.edu/ans7870/8/8.

29 Next Time: How Do We Generate Plane Waves? planewaveapp30.html P30-9

Maxwell s equations and EM waves. From previous Lecture Time dependent fields and Faraday s Law

Maxwell s equations and EM waves This Lecture More on Motional EMF and Faraday s law Displacement currents Maxwell s equations EM Waves From previous Lecture Time dependent fields and Faraday s Law 1 Radar