CH32: Electromagnetic Waves • Maxwell’s equations and electromagnetic waves • sinusoidal electromagnetic waves • Passage of electromagnetic waves through matter • Energy and momentum of electromagnetic waves • *(Not required)Wave addition and the formation of a standing electromagnetic wave Introduction • If an electric field vector propagates, it generates a magnetic field vector. Or, is it the other way? • What do you think? Or is it just another “chicken and egg” debate • Electromagnetic waves are everywhere: visible light, radio waves, microwave, gamma rays, X rays! Even in vaccum! Maxwell’s equations • After Ampere and Faraday came James Clark Maxwell. He penned a set of four equations that draw Gauss, Ampere, and Faraday’s laws together in a comprehensive description of the behavior of electromagnetic waves. • Don’t hate Maxwell for these equations. Aren’t they elegant? Qencl E • d A = ∫ ε0 ∫ B • dA = 0 dΦ ∫ E • dl = 0 − dtB dΦ ∫ B • dl = µ0 (iC + ε 0 dtE ) € Creating electromagnetic waves • A point charge can generate electromagnetic wave if is being accelerated. • The point charge below is oscillating in simple harmonic motion (what does it mean for acceleration?) • How would the magnetic field lines look like? (Imagine the moving charge as current) Maximum +y Maximum -y Electromagnetic waves occur over a wide range • Where wavelength is large, frequency is small. • The range extends from low energy and frequency (radio and television) to high energy and small wavelength (gamma rays). • Shorter wavelength means higher frequency, thus higher energy The visible spectrum • The visible spectrum is a very small range compared to the entire electromagnetic spectrum. • Visible light extends from red light at 700 nm to violet light at 400 nm. The propagation of electromagnetic waves and speed of light c= € 1 ε 0 µ0 Experimentally, people know this in the 1800’s. Is it a coincidence? Qencl E • d A = ∫ ε0 ∫ B • dA = 0 In order to satisfy Gauss’s laws, can E and B have components in the direction of wave propagation? E € Plane wave: E, B are uniform behind The wave front, and 0 in front of it. (This is an approximation) B (There is no excess charge, or currents!) Propagation of electromagnetic waves II • How would a propagation of EM wave satisfy Faraday’s law, and Ampere’s law? Assuming E and B are perpendicular to each other, what can we find out? Farady’s law dΦB = −Ea dt dΦB BdA B(a(cdt)) = = = Bac dt dt dt ∴ E = cB ∫ E • dl = − Isn’t this strange? € Ampere’s law ∫ B • dl = µ ε 0 0 dΦE = Ba dt dΦE = Eac dt ∴Ba = µ0ε 0 Eac Ba = µ0ε 0 (cB)ac 1 ⇒ c2 = µ0ε 0 € So strange! Care to verify it? Key properties of Propagation of electromagnetic waves • The wave is transverse, moving at unchanging c in a vacuum, with electric and magnetic fields in a definite ratio, and requiring no medium (like water or air). • E=cB E⊥B • Propagation direction: E × B • Question: What if E and B change directions during the €propagation. For example, they rotate. And they do! € Q32.1 In a vacuum, red light has a wavelength of 700 nm and violet light has a wavelength of 400 nm. This means that in a vacuum, red light A. has higher frequency and moves faster than violet light. B. has higher frequency and moves slower than violet light. C. has lower frequency and moves faster than violet light. D. has lower frequency and moves slower than violet light. E. none of the above Q32.2 At a certain point in space, the electric and magnetic fields of an electromagnetic wave at a certain instant are given by This wave is propagating in the A. positive x-direction. B. negative x-direction. C. positive y-direction. D. negative y-direction. E. none of the above Q32.3 A sinusoidal electromagnetic wave in a vacuum is propagating in the positive z-direction. At a certain point in the wave at a certain instant in time, the electric field points in the negative x-direction. At the same point and at the same instant, the magnetic field points in the A. positive y-direction . B. negative y-direction. C. positive z-direction. D. negative z-direction. E. none of the above Electromagnetic waves may be treated as plane waves • Far enough from the source and considering one polarization of the vector planes only, the representations of electric and magnetic fields may be treated as orthogonal and sinusoidal waves. E y (x,t) = E max cos(kx − ωt) Bz (x,t) = Bmax cos(kx − ωt) 2π λ c λ = Why? f ω = 2πf = ck k= € Example: Fields of a laser beam • A carbon dioxide laser emits a sinusoidal em wave in –x direction. Λ=10.6µm. Emax=1.5MV/m What is E(x,t) and B(x,t) By (x,t) = Bmax cos(kx + ωt) = 2π λ c λ= f ω = 2πf = ck E max cos(kx + ωt) c k= Why? € If the graph shows E and B switched positions,would the wave still Propagate in –x direction? Q32.4 In a sinusoidal electromagnetic wave in a vacuum, the electric field has only an x-component. This component is given by Ex = Emax cos (ky + ωt) This wave propagates in the A. positive z-direction. B. negative z-direction. C. positive y-direction. D. negative y-direction. E. none of the above Q32.7 The drawing shows a sinusoidal electromagnetic wave in a vacuum at one instant of time at points between x = 0 and x = λ. At this instant, at which values of x does the instantaneous Poynting vector have its maximum magnitude? A. x = 0 and x = λ only B. x = λ/4 and x = 3λ/4 only C. x = λ/2 only D. x = 0, x = λ/2, and x = λ Energy in an EM wave, the Poynting vector 1 1 u = ε0 E 2 + B2 = ε0 E 2 2 2µ0 This should be no surprise, E and B contribute equally to the EM energy density dU = udV = (ε 0 E 2 )(Acdt) 1 dU S≡ ⇒ S: Energy flow per unite area per unit time A dt EB 2 S = ε 0cE = µ0 1 S= E×B µ0 Poynting vector: (not pointing) € Plane wave example: EM wave in matter EM wave travels slower in matter. Instead of We have 1 in vacuum µ0ε 0 1 v2 = for the wave µε c2 = speed in matter. In the next chapter, we will learn n=c/v is the refraction index for Light € (or any EM wave) “Solar wind” and solar sail • Proposals have also been made to use solar wind to gain acceleration. It is very small, but after a long time, one can reach extreme high speed. Standing EM waves (Not required) • Microwave use the principle of standing waves. • An EM wave can be reflected so waves can add constructively and E y (x,t) = E max [cos(kx − ωt) + cos(kx + ωt)] resonate in a cavity. cos(A + B) = cos Acos B − sin Asin B ⇒ E y (x,t) = −2E max sin kx sin ωt E is 0 at the conducting ends! Bz (x,t) = −2Bmax cos kx cosωt And 0 at the nodes. Why does microwave have a rotating plate? €