Maxwell s Equations and Electromagnetic Waves W13D2

Transcription

1 Maxwell s Equations and Electromagnetic Waves W13D2 1

2 Announcements Week 13 Prepset due online Friday 8:30 am Sunday Tutoring 1-5 pm in PS 10 due Week 14 Friday at 9 pm in boxes outside

3 Maxwell s Equations S E ˆndA " = 1 ρ dv ε 0 V (Gauss's Law) " B ˆndA = 0 (Magnetic Gauss's Law) S # E d s = d dt C S B d # s = µ 0 J ˆndA C S B ˆndA + µ 0 ε 0 d dt S E ˆndA (Faraday's Law) (Maxwell- Ampere's Law) 3

4 Electromagnetic Spectrum 4

5 Wavelength and frequency are related by: λ f = c 5

6 How Do Maxwell s Equations in Empty Space Lead to EM Waves? 6

7 Maxwell s Equations in Empty Space " E ˆndA = 0 (Gauss's Law) S " B ˆndA = 0 (Magnetic Gauss's Law) S # E d s = d dt C S B d d # s = +µ 0 ε 0 C Set ρ = 0 J = 0 then Maxwell s Equations in empty space are B ˆndA dt S E ˆndA (Faraday's Law) (Maxwell- Ampere's Law) 7

8 Electromagnetic Radiation: Plane Waves 8

9 Plane Waves We shall consider a class of one-dimensional waves called plane traverse electromagnetic waves in which the electric and magnetic fields are uniform on planes. That means that at each point on the plane, the vectors associated with the electric field have the same direction and magnitude. The same is true for the vectors associated with the magnetic field. 9

10 Wave Equation Start with Ampere-Maxwell Eq and closed oriented loop B d s E ˆn da C = µ 0 ε 0 d dt 10

11 Ampere s Law for Plane EM Wave Start with Ampere-Maxwell Eq: Apply it to red rectangle: B d s = B z (x,t)l B z (x + Δx,t)l C d µ 0 ε 0 dt E ˆn da = µ 0 ε 0 l Δx E (x + Δx / 2,t) y t C B d s = µ 0 ε 0 d dt E ˆn da B z (x + Δx,t) B z (x,t) Δx = µ 0 ε 0 E y (x + Δx / 2,t) t So in the limit that dx is very small: B z x = µ 0 ε 0 E y t 11

12 Group Problem: Faraday s Law for Plane EM Wave Use Faraday s Law E d s = d and apply it to dt B ˆn da C red rectangle to find the partial differential equation in order to find a relationship between E y / x and B z / t 12

13 1D Wave Equation for Electric Field E y x = B z t (1) B z x = µ 0 ε 0 E y t (2) Take x-derivative of Eq.(1) and use the Eq. (2) 2 E y x 2 = x E y x = x B z t = t B z x = µ ε E y t 2 2 E y x 2 = µ 0 ε 0 2 E y t 2 13

14 1D Wave Equation for E 2 E y x 2 2 E = µ ε y 0 0 t 2 This is an equation for a wave. Let E y = f (x vt) 2 E y x 2 2 E y t 2 = f ''( x vt) ( ) = v 2 f '' x vt v = 1 µ 0 ε 0 14

15 General Sol. to One-Dim l Wave Eq. Consider any function of a single variable, for example E y (x,t) = f (x ct) Change variables. Let u = x ct then u x = 1 and u t = c Now take partial derivatives using the chain rule E y x = E y u Similarly E y t = E y u u x = E y u f and 2 E y x 2 u t = c E y u cf and 2 E y t 2 = f x = f u = c f t = c f u u x = f u = 2 E y u 2 u t = f c2 u = 2 E y c2 u 2 Therefore 2 E y x 2 = 1 c 2 2 E y E y (x,t) satisfies the wave equation t 2 15

16 Generalization Take any function of a single variable E y (u), where then E y (x ct) or E y (x + ct) (or a linear combination) is a solution of the one-dimensional wave equation 1 2 E y (x,t) c 2 t 2 = 2 E y (x,t) x 2 u = x ± ct E y (x ct) corresponds to a wave traveling in the positive x- direction with speed c and E y (x + ct) corresponds to a wave traveling in the negative x- direction with speed c 16

17 Speed of Light Recall exact definitions of c m s 1 µ 0 4π 10 7 N s 2 C 2 The permittivity of free space ε 0 is exactly defined by ε 0 1 c 2 µ C 2 m -2 N 1 c = 1 µ 0 ε 0 = m s 1 ( in vacua) 17

18 Demo Bell Labs Wave Machine C27 18

19 Group Problem: 1D Wave Eq. for B B z t = E y x B z x = µ 0 ε 0 E y t Take appropriate derivatives of the above equations and show that 2 B z x 2 = 1 c 2 2 B z t 2 19

20 For the plane wave Plane Waves Summary E(x,t) = E y (x,t)ĵ B(x,t) = B z (x,t) ˆk Both electric & magnetic fields travel like waves with speed 2 E y x 2 = 1 2 E y c 2 t 2 2 B z x 2 = 1 c 2 2 B z t 2 c = 1/ µ 0 ε 0 and the components of the fields satisfy B z t = E y x B z x = µ 0 ε 0 E y t 20

21 Sinusoidal Plane Waves In particular we shall study a special class of one-dimensional waves called plane traverse linearly polarized sinusoidal electromagnetic waves. The following is an example of such a plane wave propagating in the positive x-direction at the speed of light with a non-zero component in the y-direction transverse to the direction of propagation. E(x,t) = E y (x,t) ĵ = E sin 2π y,0 λ B(x,t) = B z (x,t) ˆk = B z,0 sin 2π λ (x ct) ĵ (x ct) ˆk 21

22 Group Problem: Traveling Sine Wave E(x,t) = E y,0 sin 2π (x ct) Let λ ĵ B(x,t) = B z,0 sin 2π (x ct) λ ˆk a) Use the differential relations B z = E y t x to show that b) Show that B z,0 = E y,0 / c E y,0 sin 2π λ satisfies the one-dimensional wave equation 1 2 E y (x,t) c 2 t 2 c) draw an arrow for the vector direction of the (x ct) ĵ electric and magnetic fields for each of the each of the ten points pictured = for t = 0 2 E y (x,t) x 2 22

23 Electromagnetic Waves: Plane Sinusoidal Waves Watch 2 Ways: 1) Sine wave traveling to right (+x) 2) Collection of out of phase oscillators (watch one position) Don t confuse vectors with heights they are magnitudes of electric field (gold) and magnetic field (blue) 23

24 Wavelength and Wave Number E y (x,0) E 0 x x = Consider Set t = 0. E y (x,t) = E 0 sin 2π (x ct) λ = E 0 E y (x,0) = E 0 sin 2π λ x = E sin(kx) 0 sin( k(x ct) ) When x = λ sin(2π ) = sin(kλ) = 0 kλ = 2π k = 2π / λ λ is called the wavelength, SI units [m] k is called the wave number, SI units [rad m -1 ] 24

25 CQ: Wave Number The graph shows a plot of the function E y (x,0) = E 0 sin(kx) The value of k is 1. k = 2π / (2 m) 2. k = 2π / (1 m) 3. k = 2π / (0.5 m) 4. k = 2π / (4 m) E y (x,0)

26 Period and Angular Frequency E y (0,t) E 0 t Consider Set x = 0. When t = T E y (x,t) = E 0 sin 2π (x ct) λ = E 0 sin( kx ωt) ) E y (0,t) = E 0 sin 2π λ ct = E sin(ωt) 0 t = T sin 2π λ ct = sin(ωt ) = sin(2π ) = 0 2πcT / λ = 2π T = λ / c; ωt = 2π ω = 2π / T T is called the period, SI units [s] ω is called the angular frequency, SI units [rad s 1 ] 26

27 Demo Ripple Tank C31 27

28 1-Dim l Sinusoidal EM Waves Because c = λ / T ω = 2π / T k = 2π / λ We can write plane waves two different ways E = E y (x,t) ĵ = E sin 2π 0 λ (x ct)) ĵ, B = B z (x,t) ˆk = E 0 c sin 2π λ (x ct)) ˆk E = E y (x,t) ĵ = E sin(kx ωt) ĵ, 0 B = B z (x,t) ˆk = B 0 sin(kx ωt) ˆk 28

29 Properties of 1 Dim l EM Waves 1. Travel (through vacuum) with speed of light c = 1 m = µ 0 ε s 0 2. At every point in the wave and any instant of time, electric and magnetic fields are in phase with one another, amplitudes obey E 0 / B 0 = c 3. Electric and magnetic fields are perpendicular to one another, and to the direction of propagation (they are transverse): 4. Direction of propagation = Direction of E B. 29

30 Direction of Propagation E = E 0 sin(kx ωt)ĵ; B = B0 sin(kx ωt) ˆk dir( E B) = î Special case generalizes as follows: at any instant in time if the direction of E and B are known then direction of propagation is direction of E x B, (use right hand rule) dir E dir B dir E B î ĵ ˆk ĵ ˆk î ˆk î ĵ ĵ î ˆk ˆk ĵ î î ˆk ĵ 30

31 1. +x direction 2. x direction 3. +z direction 4. z direction CQ: Direction of Propagation The figure shows the E (yellow) and B (blue) fields of a plane wave. This wave is propagating in the 31

32 CQ: EM Wave The electric field of a sinusoidal plane wave is: E(z,t) = E 0 sin(kz + ωt)ĵ The magnetic field of this wave is given by: B(z,t) = B 0 sin(kz + ωt)î B(z,t) = B 0 sin(kz + ωt)( î) B(z,t) = B 0 sin(kz + ωt) ˆk B(z,t) = B 0 sin(kz + ωt)( ˆk) 32

33 CQ: Traveling Wave The B field of a plane EM wave is B( y,t) = B 0 sin(ky ωt) ˆk The electric field of this wave is given by 1. E( y,t) = E 0 sin(ky ωt)ĵ 2. E( y,t) = E 0 sin(ky ωt)( ĵ) 3. E( y,t) = E 0 sin(ky ωt)î 4. E( y,t) = E 0 sin(ky ωt)( î) 33

34 Appendix Derivation of Three Dimensional Wave Equations 34

35 Consider Traveling Wave E(x,t) = E y (x,t)ĵ = E y,0 2π sin (x ct) λ ĵ The variables x and t appear together as x - ct E y (x,t) = E y,0 sin ((2π / λ)(x ct) ) At t = 0: E y (x,0) = E y,0 sin((2π / λ)x) At ct 1 = 1 m: E y (x,t 1 ) = E y,0 sin((2π / λ)(x 1m)) At ct 2 = 2 m: E y (x,t 2 ) = E y,0 sin((2π / λ)(x 2 m)) E y (x,t) 0 E y,0 sin((2 / )x) E y,0 sin((2 / )(x 1m)) E y,0 sin((2 / )(x 2 m)) c x E y (x ct) is traveling in the positive x-direction with speed c 35

36 Differential Version Maxwell s Equations E = ρ ε 0 B = 0 E = B t B E = µ 0 J + µ0 ε 0 t (Gauss's Law) (Magnetic Gauss's Law) (Faraday's Law) (Maxwell- Ampere's Law) 36

37 Maxwell s Equations in Free Space In free space where ρ = 0, J = 0 Maxwell s Equations simplify (notice the symmetry) E = 0 (Gauss's Law) B = 0 (Magnetic Gauss's Law) E = B (Faraday's Law) t B E = µ 0 ε 0 (Maxwell- Ampere's Law) t 37

38 CQ: Two of Maxwell s Equation for Plane Waves For a linear polarized plane wave with electric and magnetic fields given by E(x,t) = E y (x,t)ĵ B(x,t) = B z (x,t) ˆk The two Maxwell equations E = B imply that t ; E B = µ0 ε 0 t 1. E y x = B z t ; B z x = µ 0 ε 0 E y t 3. E y x = B z t ; B z x = µ 0 ε 0 E y t 2. E y x = B z t ; B z x = µ 0 ε 0 E y t 4. E y x = B z t ; B z x = µ 0 ε 0 E y t 38

39 Wave Equation for Electric Field Vector Identity (try to prove this in Cartesian Coordinates): ( E) = 2 E + ( E) Derivation of Wave Equation for Electric Field in Vacuum: Start with Faraday's Law: E = B t take curl of both sides: ( E) = t ( B) apply vector identity: 2 E + ( E) = t ( B) apply E = 0 : 2 E = t ( B) apply generalized Ampere'e Law B = µ 0 ε 0 E t : 2 E = µ 0 ε 0 2 E t 2 use ε 0 µ 0 = 1/ c 2 : 2 E = 1 c 2 2 E t 2 39

40 Group Prob.: Wave Equation for Magnetic Field Use B E = µ 0 ε 0 t and the vector identity ( B) = 2 B + ( B) along with B = 0, E = B t, and ε µ = 1/ c2 0 0 to derive the wave equation for the magnetic field in vacuum 2 B = 1 c 2 2 B t 2 = 2 B 40

41 Summary: Maxwell s Equations and Three Dimensional Wave Equations Maxwell s Equations can be rewritten as three dimensional wave equations for the electric and magnetic fields 2 E = 1 c 2 2 E t 2 2 B = 1 2 B c 2 t 2 where (in Cartesian coordinates) 2 = 2 x y z 2 41

42 Special Case: Plane Wave Consider a an electric field, E(x,t), in vacuum that only varies with respect to x and t, but is independent of y and z. Then the zero divergence of the electric field requires that E = E x x + E y y + E z z = 0 Because the components of the fields are independent of y and z E y y = 0 and E z z = 0 E x x = 0 There are two solutions to this equation: E x = 0 or E x is uniform in space. We have ruled out uniform fields and therefore the electric field only has non-zero y- and z- components which are each independent of y and z. E(x,t) = E y (x,t)ĵ+ E z (x,t) ˆk We shall consider the special case (linear polarization) in which there is only a non-zero y-component: E(x,t) = E y (x,t)ĵ 42

43 Electromagnetism Review E fields are associated with: (1) electric charges (Gauss s Law ) (2) time changing B fields (Faraday s Law) B fields are associated with (3a) moving electric charges (Ampere-Maxwell Law) (3b) time changing E fields (Maxwell s Addition (Ampere- Maxwell Law) Conservation of magnetic flux (4) No magnetic charge (Gauss s Law for Magnetism) 43

44 Electromagnetism Review Conservation of charge: closed surface J d A = d dt volume enclosed ρ dv E and B fields exert forces on (moving) electric charges: F q = q( E + v B) Energy stored in electric and magnetic fields U E = u E dv = all space all space ε 0 2 E 2 dv U B = u B dv = all space 1 B 2 dv 2µ all space 0 44