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Physics 3 – Nov 4, 2016 P3 Challenge – A piece of glass of index of refraction 1.33 is coated with a thin layer of magnesium fluoride of index of refraction 1.38. It is illuminated with light of wavelength 680 nm. Determine the minimum thickness of the coating that will result in no reflection. Today’s Objective – Resolution and Doppler Effects Objectives/Agenda/Assignment Objective: 9.4 Resolution 9.5 Doppler Effects Agenda: Resolution and your eyes Doppler effects Assignment: p380 #27-35 and p389 #36-49 Resolution and our eyes Whether we see objects with our eyes or a camera, the light from the object has to pass through a circular aperture resulting in a diffraction. Resolution – our ability to distinguish between two sources. Test of visual acuity: to see the double star in the big dipper. (near arch of handle) Rayleigh criterion Condition for resolution called the Rayleigh criterion The angular separation of the objects must be greater than or equal to the angle of diffraction for the wavelength of light and the diameter of the aperture. If the aperture is circular rather than a slit, a 1.22 factor is needed for the angle of diffraction. Angle of separation, can be approximated as the separation, s, over the distance to the objects, d. 𝒔 𝒅 ≥ 𝟏. 𝟐𝟐 𝝀 𝒃 where b is the diameter of the aperture. Sample Problems A camera has an aperture of 1.33 mm. What is the minimum separation of two flashlights at a distance of 2.00 km that this camera can record as two lights rather than one light. (Assume a wavelength of 550 nm as an average for white light.) Doppler Effect The Doppler effect arises anytime there is relative motion between the source of a sound at the observer of a sound. The result is a compression of wavefronts, or a rarefaction of wavefronts relative to the wavefronts for stationary sources and observers. When source and observer are moving toward one another, the wavefronts compress…higher frequency sounds When source and observer are moving away from each other, the wavefronts refract…lower frequency sounds. Four cases Stationary observer: 1) source moving toward observer, 2) source moving away Stationary source: 3) observer moving toward source, 4) observer moving away We can ignore the cases when both observer and source are moving because we are free to choose a frame of reference with either the observer or the source at rest within the frame. Sources and observes accelerating relative to one another would result in a changing frequency and the relative velocity changes. IB only looks at the constant velocity cases. The Doppler frequency equations Stationary observer: 1) source moving toward observer, 2) source moving away 𝒇′ = 𝒇 Where f’ is the altered sound heard, f is the stationary frequency of the source, v is the velocity of sound in the medium (air = 343 m/s) and us is the speed of the source relative to the observer. Stationary case with us = 0 reduces the equations to f’ = f. Decreasing the denominator increases the fraction, giving a higher frequency. 𝒗 𝒗−𝒖𝒔 𝒇′ = 𝒇 𝒗 𝒗+𝒖𝒔 Wavefronts compress as source moves toward observer. Increasing the denominator decreases the fraction, giving a lower frequency. Wavefronts refract as source moves away from observer. The Doppler frequency equations Stationary source: 3) observer moving toward source, 4) observer moving away 𝒇′ = 𝒇 Where f’ is the altered sound heard, f is the stationary frequency of the source, v is the velocity of sound in the medium (air = 343 m/s) and uo is the speed of the observer relative to the observer. Stationary case with us = 0 reduces the equations to f’ = f. Increasing the numerator increases the fraction, giving a higher frequency. 𝒗+𝒖𝒐 𝒗 𝒇′ = 𝒇 𝒗−𝒖𝒐 𝒗 Wavefronts compress as observer moves toward source. Decreasing the numerator decreases the fraction, giving a lower frequency. Wavefronts refract as observer moves away from source. The Doppler wavelength equations Recall 𝒗 = 𝝀𝒇 for waves, so frequency and wavelength are inversely related. Therefore we can generate the four corresponding wavelength equations by taking the reciprocal of the velocity ratio fractions: Stationary observer: 1) source moving toward observer, 2) source moving away 𝝀′ = 𝝀 Stationary source: 3) observer moving toward source, 4) observer moving away 𝝀′ = 𝝀 𝒗−𝒖𝒔 𝒗 𝒗 𝒗+𝒖𝒐 𝝀′ = 𝝀 𝝀′ = 𝝀 𝒗+𝒖𝒔 𝒗 𝒗 𝒗−𝒖𝒐 Choosing correct equation Data packet identifies moving source or object for you. The part you need to think through is the proper sign to use. Think about whether wavefronts are compressing/refracting and whether you should get a higher or lower fraction value and frequency. Ex: You’re standing on the station as a train passes you by at 35.0 m/s. The sound of the train engine is a whirr at a pitch of 525 Hz. What pitch do you hear as the train approaches? After it passes you? Doppler Effect for light When the speed of the wave is much much greater than the speed of either the observer or the source, (which is true for light), all four equations and cases reduce to a single relationship: 𝚫𝒇 𝒇 = 𝒗 𝒄 Where ∆f is the change in frequency observed, f is the frequency of the light, v is the speed of source or the observer and c is the speed of light. The wavelength equation is: (notice we do not flip it) 𝚫𝝀 𝝀 = 𝒗 𝒄 Red and blue shifts When light waves are being compressed, you get higher frequencies and shorter wavelengths. Blue light has the shortest wavelengths and highest frequencies so this is a Blueshift. A blue shift means the source and observer are moving toward each other. When light waves are being refracted, you get lower frequencies and longer wavelengths. Red light has the longest wavelengths and lowest frequencies so this is a Red-shift. A red shift means the source and observer are moving away from each other. Light from all starts is red shifted. Therefore the universe is expanding. Exit slip and homework Exit Slip – A store front has a sale that they are calling attention for with a constant pitch rhythm generator with a frequency of 112 Hz. You drive past the store at 25.0 m/s. What pitch do you hear as you approach? after you pass? (Let speed of sound in air = 343 m/s) What’s due? (homework for a homework check next class) 380 #27-35 and p389 #36-49 What’s next? (What to read to prepare for the next class) Start studying for Ch 9 Wave Phenomenon Test for week of 11/14
