Lecture 32 energy and momentum. standing waves.
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This document provides information about energy and momentum in electromagnetic waves and standing waves. It discusses how energy is carried in EM waves and is split equally between the electric and magnetic fields. It also describes how EM waves transport energy through space via the Poynting vector and introduces the concept of radiation pressure from absorbed or reflected EM waves. The document then summarizes how standing electromagnetic waves can form between conducting plates and discusses the Doppler effect for electromagnetic waves.
Lecture 32 energy and momentum. standing waves.
- 1. Lecture 32 Energy and momentum. Standing waves.
- 2. Energy in a EM wave Energy density due to an electric field: u = 1 ε 0E 2 2 Energy density due to a magnetic field: u = 1 B2 2µ 0 Energy density for an EM wave: u = but E = cB = u = B ε 0 µ0 1 1 ε 0E 2 + ε 0 µ0E 2 2 2µ 0 1 1 ε 0E 2 + B2 2 2 µ0 u = ε 0E 2 Energy density equally split between E, B fields
- 3. Energy transport How much energy goes through a surface of area A in time dt ? y propagation Energy in this “box”: dU = udV = ε 0E 2Acdt cdt x z EB 1 dU 2 = ε 0cE = Energy flow per unit time and per unit area: S = µ0 A dt r 1 r r Definition: Poynting vector S = E ×B µ0 Intensity: I = S Energy flow per unit time and per unit area
- 4. ACT: Plane harmonic wave At the time shown, the magnetic field point P (on the y axis) is: x •P Propagation A. Bmax i B. Bmax j C. 0 y r r Propagation direction is E × B , r so E is in the x direction r and B is in the y direction z x E/B are the same at all points in each yz plane! z y
- 5. Energy in the harmonic wave r ˆ E = E max cos ( kx − ωt ) j r ˆ B = Bmax cos kx − ωt k ( ) ( r 1 r 1 ˆ ˆ S = E ×B = E maxBmax cos2 ( kx − ωt ) j × k µ0 µ0 1 = E maxBmax cos2 ( kx − ωt ) iˆ µ0 I = S = ) Direction +x (as expected…) 1 1 E maxBmax cos2 ( kx − ωt ) = E maxBmax µ0 2µ 0 I = S = 1 E maxBmax 2µ0
- 6. ACT: Emax and distance An isotropic radio transmitter emits power in all directions. What is the ratio of the amplitudes of the E field at distances of 100 m and 200 m from the source E max(100)/E max(200) ? A. 1 B. 2 C. 4 Energy is uniformly distributed in a sphere of radius r (r = distance to source): I = E maxBmax 2µ0 2 E max = 2c µ0 Intensity is I = S = E max =c Bmax power power = area 4π r 2 E max µ 1 r
- 7. Emission of EM waves How does an EM wave begin? When a charge is accelerated. Oscillating dipole Moving charged infinite sheet Whenever a charge is accelerated, it loses energy due to radiation. → Bad thing when you’re trying to accelerate a particle → Good thing when you can use the radiation! • synchrotron radiation produces X-rays • detection of black holes • any emission antenna
- 8. Momentum EM waves carry energy… and momentum. (And mechanical waves, too, btw.) Basic idea of momentum: p = mv → Mass m moving with speed v (say to the right) → (Kinetic) energy flows (to the right) A very hand-waiving trick to get momentum without the mass: KE = 1 1 mv 2 = pv 2 2 p: KE v Using the proper mathematical tools (special relativity), one obtains KElight = pc
- 9. Radiation pressure If EM waves carry radiation, they can exert a force (and thus a pressure) when they hit a surface: S EB F 1 ∆p 1 KE Power I = pressure = = = = = = µ0c A A ∆t cA ∆t c c cA ∆p = KElight c if radiation is completely absorbed If radiation is completely reflected, ∆p = 2KElight c , so pressure = 2EB µ0c Light pressure, though “light”, has noticeable effects → comet’s tail pushed away from the sun*. *Note: The dust tail is pushed away by radiation; the ion tail is pushed away by the solar wind!
- 10. Standing electromagnetic waves EM wave propagating between two plates of a perfect conductor: Conducting wall ⇒ E-field must be zero there ⇒ original wave and reflected waves produce a standing wave with condition E = 0 on both ends: Original wave: E y = E max cos ( kx − ωt ) y E x Re flected wave: E y = −E max cos ( kx + ωt ) E y = E max cos ( kx − ωt ) − E max cos ( kx + ωt ) = −2E max sin ( kx ) sin ( ωt ) λ E-field nodes: λ 3λ xE −nodes = 0, , λ, ... 2 2
- 11. And the B field? Original wave: E y = E max cos ( kx − ωt ) Re flected wave: E y = −E max cos ( kx + ωt ) ⇒ ⇒ Bz = Bmax cos ( kx − ωt ) Bz = Bmax cos ( kx + ωt ) Note: No minus sign for B ! r r We need E × B = propagation z B Bz = Bmax cos ( kx − ωt ) + Bmax cos ( kx + ωt ) x = −2Bmax cos ( kx ) cos ( ωt ) B-field nodes: xB −nodes = λ 3λ 5λ , , ... 4 4 4 λ DEMO: Marshmallows and microwave
- 12. Doppler effect Just like for mechanical waves, if the source or the observer of an EM are moving, the received frequency can be different from the emitted frequency. The equations are different, though, because… Spaceship moves with speed v … nothing can go faster than light! Light from star travels at c +v relative to spaceship??? We can’t simply add velocities à la Galilean. We need relativity.
- 13. No relativity in 222, so let’s forget about the math. But here’s some examples of Doppler’s effect in EM radiation anyway: • police speed radars • weather radars (detect motion of rain droplets) • in astronomy: red shift/blue shift Most stars are made of H, so their spectrum must be the same Spectrum of the sun (optical wavelengths) Spectrum of object X Lines are at λ larger than expected (red shift) It turns out that all distant galaxies are moving away from us Object must be moving away from us The universe must be expanding!