AP Physics - More Dadgum Interference

Transcription

1 AP Phsics - More Dagu Interference Single Slit Diffraction: A single slit will also for an interference pattern when light passes through it. Each part of the slit acts as source of waves. This is escribe in Hugen s principle. Hugen s principle Ever point on a wave front acts as a source of tin wavelets that ove forwar with the sae spee as the wave. The wave front at a later instant of tie is the surface that is tangent to the wavelets. You can iagine that across the with of the slit, little wavelets originate an travel through the slit. These waves pass through the slit an for a bright central fringe on the screen, which is at a far istance fro the slit. This istance is so far that all the waves are essentiall parallel to one other. All the wavelets travel the sae istance an arrive at the screen in phase with each other an we get constructive interference. This creates a bright central fringe at the center of the screen irectl opposite the slit. The wavelets that originate in the slit can also interfere estructivel. Here s a rawing that shows the ver thing. incient wave Central Maxia Ver istant screen ipoint of center axia First ark fringe ipoint of center axia incient wave Wavelet froation for ark fringe How interference takes place 508

2 ight fro one part of slit interferes with light fro another part of slit, foring the patterns. Again the cause of the interference is the path ifference for the waves (wavelets in this case). The patterns that for can be escribe in this wa: There will be a bright central fringe surroune b two ark fringes, then a set of weaker bright fringes, a set of ark fringes, a set of weaker (than the central fringe) bright fringes, an so on. et s look at the geoetr of the thing. l The sae euation is use with single slit iffraction as with the ouble slit iffraction, except that the angle we get,, is the angle fro the center of the slit to the center of the ark fringe. sin This euation escribes estructive interference. is the with of the slit, is the angle to the center of the ark fringe, is the integer orer nuber, an is the wavelength of the light. We can analze it the sae wa we i the ouble slit eal to fin the spacings between the central fringe an the ark fringes. sin sin tan 509

3 This general case is: x While this is the sae euation as for a ouble slit eal, for a single slit it gives ou the istance fro the center of the bright central fringe to the esire ark fringe Sin = 2l Sin = l Sin = Sin = Sin = l 2l 575 n light passes through a slit of with An observing screen is set up 3.00 awa. (a) Fin the position of the first ark fringe. (b) What is the with of the central axia? 1 (a) This is the first inia, so = 1. The spacing is given b: x 1575 x x x10 Diffraction Grating: 6 1 Diffraction gratings are a recent invention (well, a lot ore recent than the ol ouble slit eal). Basicall, a iffraction grating is a piece of transparent aterial that has parallel cuts scribe in it. The scribings are so sall that ou can t reall see the. At an rate, the grating has a ver 510

4 large nuber of euall space parallel slits cut into it. This woul be on the orer of hunres to several thousan lines per centieter. The grating acts like a ouble slit setup. It prouces a large nuber of ver bright, sharp fringes separate fro one another b fairl wie inia. Maxia are given b the sae euation as we have seen before: sin is the orer nuber, is the spacing between the slits, is the wavelength of the light, an is the angle fore b a noral to the grating to a line at the center of the fringe. ight fro a istant star enters a telescope an then passes through a iffraction grating onto a screen. A first orer re line appears on the screen at an angle of The lines of the grating are separate b 1.50 x What is the wavelength of the light? sin sin x10 sin x x n n laser bea passes through a iffraction grating that has lines per centieter. An observing screen is set up 3.00 awa. Fin separation of the axia. The slit separation is the inverse of slit ensit c x10 c x sin sin The angle is not sall an we cannot ake the assuption that the sine of is eual to the tangent. We can fin fro the euation an then use the tangent to fin. Y x10 sin x x10 3 o o tan tan tan

5 The pattern of axia fro a iffraction grating looks like this: M l l 0 l 2l The axia are ver bright an sharp, the are also wiel separate fro one another. For this reason, the iffraction grating is preferre to ouble slits. So iffraction gratings are better because: Get ver sharp axia Get ver wie ark areas With a few eas to ake easureents, one can easil calculate the wavelength of the light. Newton s Rings: The last thing we want to talk about in our lovel iscussion of light an optics are Newton s rings. These are not the ugl lines he ha uner his eeballs, nor are the the fabulous finger wear he sporte. This is an optic effect thingee. 2 1 To the right ou can see a rawing of the thing. A rectangular glass plate lies on a table top. ing atop the plate is a circular lens that is flat on top an has a constant raius curvature on its lower surface. It s our basic half of a lens. P r O 512

6 Ra 1 is reflecte fro the surface of the plate. It unergoes a 180 phase change. Ra 2 is reflecte fro botto of lens. Because air has a saller inex of refraction than glass there is no phase change on the reflecte ra. Because of this, there exists a path ifference between the two ras. This will give us interference patterns inia an axia. These will appear in the lens when ou look own into it. You will see ark an bright concentric circles the look like rings, hence the nae. Newton was the ue who iscovere the. Dispersion: The inex of refraction, n, is a function of wavelength. The angle of refraction is varies for ifferent wavelengths of light. The inex of refraction generall ecreases with increasing wavelength. The eans that re light ( = 600 n) bens less than oes blue light ( = 470 n). This phenoenon is calle ispersion. It was iscovere b Isaac Newton. He built a pris an use it to separate out or isperse all the visible colors fro white light. Priss are especiall valuable in evices calle spectroeters. The pris separates light into a spectru of colors which can then be focuse onto a screen with a sste of lenses. ight fro istant stars can be analze in this wa to eterine the star s cheical coposition. re ispersion in a pris blue white light pris increasing wavelength Dispersion is responsible for rainbows. ittle roplets of water act like priss hanging in the sk an isperse the sun s light, foring a spectru, which we call a rainbow. Here s an interesting thing how o rainbows work? Ever woner about that? 513

7 Oka, here s the eal with rainbows. You see rainbow right before, after, uring rain, right? You also see the aroun sprinklers, waterfalls. &tc. The basic iea is that the are aroun water spras. This is where ou get water rops floating in the air. The rainbow is the result of light unergoing ispersion in these tin rops of water suspene in the atosphere. The water rops act like priss. ight enters near the top of the rop on one sie, is reflecte (total internal reflection), which eans it unergoes two refractions. (Refraction as it enters the water rop an refraction when it leaves the water rop.) The colors are therefore isperse. Violet light is refracte the ost (since it goes through the water at a slower spee, recall the higher the freuenc of light, the greater the refraction) an re light is bent the least. The angle between the sunlight entering the rop an the re light leaving the rop is fort two egrees (abe that's the real uestion for the ultiate answer). The angle for the violet light is 40. The other colors are soewhere in between. This ispersion oes not happen with just one rainrop. It happens with gazillions of the little evils. Therefore one sees colors everwhere along a arc. We call this effect a rainbow. To see the rainbow, ou ust have our back to the sun. It appears as a circle, but roughl half the circle is in the groun so ou on't notice the lower half of the circle. You onl see the upper half. If ou see a rainbow fro the air, like when ou are fling in a plane, ou will see a raincircle instea of a rainbow. You can see onl one color fro each rainrop. So if violet light enters our ee fro a rop, the re light travels in a path fro a rop that is actuall a bit higher than the rop that provie ou with the re color. So when ou see a rainbow, re light is the color that is at the top of the rainbow an violet light is below. All the other colors fall in between - re, orange, ellow, green, blue, an violet. 514

8 You see the colors irectl overhea (at 42 ), but ou also see the in an arc that is 42 on either sie of ou. Soeties ou not onl see one rainbow but a bonus, extra one as well. The first rainbow is a regular one - re light on top; violet light on botto. The secon rainbow is above this one, but the colors are reverse. Violet is on top an re is on the botto. This secon rainbow is cause b light being reflecte twice in the water rops. It enters the botto of a rop, is reflecte upwar, reflecte et again, an then leaves the rop. The secon reflection of the light is not at the critical angle, so soe of the light escapes, so the secon rainbow is ier an not as bright. Rainbows appears as cones that extens fro our ees. Your view of the rainbow is uniue - no one else can see it in the sae wa. It's our own personal, private rainbow! As ou ove, the rainbow oves with ou. This is wh chasing after the en of a rainbow is a no win eal - ou can never get there because as ou ove towars it, it oves with ou. So chasing after the gol pot at the en of a rainbow is a foolish task. Sort of the ultiate wil goose chase. Our ancestors knew this, hence the stories of gol at the en of a rainbow - the were cruel jokers on occasion. Another relate effect that one can see is the classic "ring aroun the oon". This is cause b the refraction of light through ice crstals. There is no internal reflection within the ice crstals, so there is no ispersion, so no rainbow. 515