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Energy and the Poynting Vector

Energy and the Poynting Vector. Let ’ s find the energy density in the wave. Now let ’ s define the Poynting vector:. It is energy density times the speed at which the wave is moving It points in the direction energy is moving It represents the flow of energy in a particular direction

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Energy and the Poynting Vector

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  1. Energy and the Poynting Vector • Let’s find the energy density in the wave • Now let’s define the Poynting vector: • It is energy density times the speed at which the wave is moving • It points in the direction energy is moving • It represents the flow of energy in a particular direction • Units:

  2. Optics Reflection and Refraction Reflection • What happens when our wave hits a conductor? • E-field vanishes in a conductor • Let’s say the conductor is at x = 0 • Add a reflected wave going other direction • In reality, all of this is occurring in three dimensions Incident WaveReflected WaveTotal Wave

  3. Waves Going at Angles • Up to now, we’ve only considered waves going in the x- or y-direction • We can easily have waves going at angles as well • What will reflected wave look like? • Assume it is reflected at x = 0 • It will have the same angular frequency • Otherwise it won’t match in time • It will have the same kyvalue • Otherwise it won’t match at boundary • kx must be negative • So it is going the other way

  4. Law of Reflection ki=kr • Since the frequency of all waves are the same, the total kfor the incident and reflected wave must be the same. • To match the wave at the boundary, kymust be the same before and after kisini=krsinr kisini krsinr sini=sinr ki kr Incident Reflected i r i=r Mirror y x

  5. Geometric Optics and the Ray Approximation • The wave calculations we have done assumethe mirror is infinitely large • If the wavelength is sufficiently tiny comparedto objects, this might be a good approximation • For the next week, we will always makethis approximation • It’s called geometric optics • Physical optics will come later • In geometric optics, light waves are represented by rays • You can think of light as if it is made of little particles • In fact, waves and particles act very similarly • First hint of quantum mechanics! i=r i r Mirror

  6. Measuring the Speed of Light ½ ½ • Take a source which produces EM waves with a known frequency • Hyperfine emission from 133Cs atom • This frequency is extremely stable • Better than any other method of measuring time • Defined to be frequency f = 9.19263177 GHz • Reflect waves off of mirror • The nodes will be separated by ½ • Then you get c from c = f • Biggest error comes frommeasuring the distance • Since this is the best way tomeasure distance, we can use this to define the meter • Speed of light is now defined as 2.99792458108 m/s 133Cs

  7. The Speed of Light in Materials • The speed of light in vacuum c is the same for all wavelengths of light, no matter the source or other nature of light • Inside materials, however, the speed of light can be different • Materials contain atoms, made of nuclei and electrons • The electric field from EM waves push on the electrons • The electrons must move in response • This generally slows the wave down • n is called the index of refraction • The amount of slowdown can dependon the frequency of the light Indices of Refraction Air (STP) 1.0003 Water 1.333 Ethyl alcohol 1.361 Glycerin 1.473 Fused Quartz 1.434 Glass 1.5 -ish Cubic zirconia 2.20 Diamond 2.419

  8. Refraction: Snell’s Law k1sin1 1 r 2 k2sin2 • The relationship between the angular frequency  and the wave number k changes inside a medium • Now imagine light moving from one medium to another • Some light will be reflected, but usually most is refracted • The reflected light again must obey the law of reflection • Once again, thefrequencies all match • Once again, the y-componentof k must match 1=r index n1 index n2 y x Snell’s Law

  9. Dispersion • The speed of light in a material can depend on frequency • Index of refraction n depends on frequency • Confusingly, its dependence is often given asa function of wavelength in vacuum • Called dispersion • This means that different types of light bendby different amounts in any given material • For most materials, the index of refractionis higher for short wavelength Red Refracts Blue Bends

  10. Prisms • Put a combination of many wavelengths (white light) into a triangular dispersive medium (like glass) • Prisms are rarely used in research • Diffraction gratings work better • Lenses are a lot like prisms • They focus colors unevenly • Blurring called chromatic dispersion • High quality cameras use a combination of lenses to cancel this effect

  11. Rainbows • A similar phenomenon occurs when light bounces off of the inside of a spherical rain drop • This causes rainbows • If it bounces twice, youcan get a double rainbow

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