From last time Waves Interference Thursday, Sep. 4
From last time Waves Interference Thursday, Sep. 4 Please pick up pack of color sheets Phy208 Lecture 2 1 Interference of 2 speakers constructiv destructive e interferenc e,loud tone interference
quit tone crest troug h Thursday, Sep. 4 Phy208 Lecture 2 2 A little more detail Path-length difference = (Path 2) - (Path 1) = Path 2) - (Path 2) - (Path 1) = Path 1) = 0 Same distance, same phase Path 1 Path 2 Path 1 Path 2
d Thursday, Sep. 3 Phy208 Lecture 2 3 Other angles? Path-length difference 0 Interference: m Constructive: = =0, , 2 , 3 K 2m + 1 Destructive: 2 = 1 3 5 Path 2
= , , K 2 2 2 Path 1 = Extra path length / d = sin = d sin d Thursday, Sep. 3 Phy208 Lecture 2 4
Question Suppose that the frequency of the sound wave increases by a factor of two. The adjacent maxima are A) Farther apart B) Closer together C) Same Higher frequency, -> shorter wavelength Less path length diff required to make 1 wavelength diff. D) Need to know speaker spacing Thursday, Sep. 4 Phy208 Lecture 2 5 Question In your room you have two speakers in different corners. At your desk you are exactly 1 meter from each, so that there is no interference. Your
roommate moved one of your speakers 0.25 m further away from your desk. At what frequency will you hear destructive interference? A. B. C. D. E. 170 Hz 340 Hz 680 Hz 1000 Hz 1350 Hz Thursday, Sep. 3 Phy208 Lecture 2 6 Math Analysis
Constructive interference: =d sin = 0 =0 =d sin = 1 sin = / d = Path 2 340m / s v = = 0.34 fd (1000 / s)(1m) = 0.347 rad = 19.9 =d sin = 2 v sin = 2 /d = 2
fd Path 1 = Extra path length / d = sin = d sin d Thursday, Sep. 4 Phy208 Lecture 2 7 For small angles sin = /d for constructive interference Small angle approximation For small , sin with in radians e.g. sin = /d = 0.34
Exact value: 0.34 radians =0.347 radians Approx breaks down for large Thursday, Sep. 4 Phy208 Lecture 2 8 Light waves Light is a wave just like sound. But can propagate in vacuum at speed c = 3x108 m/s Has same wave properties, e.g.
c vac = f Can also propagate in glass, water, but slower Speed in medium = v = c/n c vac med = = nf n Has same superposition / interference properties n = Index of refraction Thursday, Sep. 4 Phy208 Lecture 2 9
Interference of light Thursday, Sep. 4 Phy208 Lecture 2 10 Projection on screen Light wavelength much shorter than sound Interference fringes much closer together y /L = tan y Light beam Recordin g plate Foil with two
narrow slits Thursday, Sep. 4 L Phy208 Lecture 2 11 Small-angle approximation =d sin = Extra path length y = position on screen = L tan Thursday, Sep. 4 sin tan
y Extra path length = ~ d L for 0 Phy208 Lecture 2 12 Fringe separation & wavelength For bright fringes: extra path length = m y bright L = m m = 0, 1, 2 K d For dark fringes: extra path length = (Path 2) - (Path 1) = m+1/2) L 1 y dark =
m + m = 0, 1, 2 K d 2 Bright Fringe separation = Thursday, Sep. 4 Phy208 Lecture 2 L d 13 Question A two-slit interference pattern is observed in air (Path 2) - (Path 1) = n=1). Then the entire system is immersed in water (Path 2) - (Path 1) = n=1.33). The interference fringes are A) 1.33 times closer together B) 1.33 times farther apart C) 2.66 times closer together D) 2.66 times farther apart E) Spacing unchanged
Thursday, Sep. 4 Phy208 Lecture 2 14 Interference of multiple sources Evenly-spaced slits have maxima at same location as two-slits. Maxima become sharper, more intense Interference of multiple sources often called diffraction Thursday, Sep. 4 Phy208 Lecture 2 15 Diffraction gratings Diffraction grating is pattern of multiple slits. Very narrow, very closely spaced. Expect very narrow interference peaks.
(Path 2) - (Path 1) = 15,000 lines/inch)x(Path 2) - (Path 1) = 1 inch/2.54 cm)x(Path 2) - (Path 1) = 100 cm/m)=5.9x105/m So a spacing of 1/(Path 2) - (Path 1) = 5.9x105/m) = 1.7x10-6 m = 1700 nm. 1700 nm Thursday, Sep. 4 Phy208 Lecture 2 16 Question Suppose white light is shined through a diffraction grating. What would the pattern on the screen look like? A) L y bright = m d m = 0, 1, 2 K
B) C) D) Thursday, Sep. 4 Phy208 Lecture 2 17 Diffraction: only one slit, but wide Huygens principle: each portion of the slit acts as a source of waves These sources interfere according to path-length difference. Thursday, Sep. 4 Phy208 Lecture 2
18 Diffraction minima Interference-like pattern from a single slit. How wide is wide? Thursday, Sep. 4 Phy208 Lecture 2 19 Phase difference Waves start in phase Travel different distances (Path 2) - (Path 1) = extra path length = ) No longer in phase when combined (Path 2) - (Path 1) = Phase diff ) =/2 = = =2 General reln: =2
Constructive: =2mm Thursday, Sep. 4 Phy208 Lecture 2 Destuctive: =2m(m+1/2) 20 Thin film interference Black Colors changing with thickness Thursday, Sep. 4 Phy208 Lecture 2 21 Thin film interference
no phase shift air: n=1 n>1 180 (Path 2) - (Path 1) = radians) phase shift /n t If film is much thinner than wavelength, ~ no phase shift from extra path length But top reflection has 180 phase shift, bottom not Destructive interference for all wavelengths, film appears black Thursday, Sep. 4 Phy208 Lecture 2 22 Thicker parts
180 phase shift from reflection Phase difference comes from air: n=1 t n>1 vac/n Extra path length=2 t air: n=1 Phase difference = + 2 Reflection phase shift
Thursday, Sep. 4 Phase shift from reflection (Path 2) - (Path 1) = top) Phase shift from extra path length Extra path length for normal incidence = 2t 2t = ( vac / n ) 2m constructive (2m 1) destructive
# wavelengths in extra path length Phy208 Lecture 2 23 Constructive interference 1 2t = m + 2 n 2t = m n (m = 0,1,2K ) constructive interference (m = 0,1,2K ) Thursday, Sep. 4
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