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$\bullet$$\bullet$ You need a diffraction grating that will disperse the visiblespectrum $(400$ nm to 700 $\mathrm{nm})$ through $30.0^{\circ}$ in the first-orderpattern. What must be the line density of this grating? (Hint:Use the fact that $\sin (A+B)=\sin A \cos B+\cos A \sin B . )$

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1230 lines/mm

Physics 102 Electricity and Magnetism

Physics 103

Chapter 26

Interference and Diffraction

Electromagnetic Waves

Reflection and Refraction of Light

University of Michigan - Ann Arbor

University of Washington

Hope College

University of Sheffield

Lectures

02:30

In optics, ray optics is a geometric optics method that uses ray tracing to model the propagation of light through an optical system. As in all geometric optics methods, the ray optics model assumes that light travels in straight lines and that the index of refraction of the optical material remains constant throughout the system.

10:00

In optics, reflection is the change in direction of a wavefront at an interface between two different media so that the wavefront returns into the medium from which it originated. Common examples include the reflection of light, sound and water waves. The law of reflection says that for specular reflection the angle at which the wave is incident on the surface equals the angle at which it is reflected. Reflection may also be referred to as "mirror image" or "specular reflection". Refraction is the change in direction of a wave due to a change in its speed. The refractive index of a material is a measure of its ability to change the direction of a wave. A material with a higher refractive index will change the direction of a wave to a greater degree than a material with a lower refractive index. When a wave crosses the boundary between two materials with different refractive indices, part of the wave is refracted; that is, it changes direction. The ratio of the speeds of propagation of the two waves determines the angle of refraction, which is the angle between the direction of the incident and the refractive rays.

05:27

You need a diffraction gra…

04:34

Problem 26.57You need …

04:04

A diffraction grating is m…

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A diffraction grating is …

03:58

06:15

Assume light of wavelength…

03:39

You have a diffraction gra…

04:08

A diffraction grating illu…

03:59

A 5.000-cm-wide diffractio…

All right, folks. This one is algebraic Lee Intense. So buckle up. It's gonna take a minute. Okay, so we have the diffraction grating that wants to disperse the visible spectrum 400 to 780 meters through 30 degrees in the first order pattern. We want to find the density of the line. They're the line density of the grating. So what that means is that first wave length 490 meters, our second wave length this 700 Nana meters. And when we shine, though, through the diffraction grating, it's gonna fan them out so that whatever angle we get our first maximum here, over here on our bigger one, it's gonna be 30 degrees larger. Cool, Cool. Because if we look at our condition for constructive interference, that's d signed data equals M lambda. So whatever the bigger wavelength ISS is gonna have the bigger angle. So that's why on my 700 nanometers, it gets the plus 30. So whatever angle my 480 meters air at my 700 leaders is gonna be the same angle plus 30. Okay, that means do you sign? Call this data one is equal and Lambda. We're looking at first order. So it's gonna be one table, the Landau one, Similarly here. Do you sign say that too is equal to a Lambda too. But we have the condition that stated to is equal a state A one plus 30 degrees. Okay, so I do a little rearranging. You know, a sign of Saito one as he will lambda one over deep on that sign made a one plus 30. I was able to land it to over D. All right, they also give us this handy dandy trigger identity. That sign of a plus B is equal to sign a coast be plus sign. Be coast, eh? So we can use that to rewrite this one. So sign of data one lost 30 is equal to sign. Say that one coast 30 plus sign 30 Coast Data one. Okay, We can also use the trigger identity sine squared. Seita plus Co sign squared data, missy. For the one so co sign Seita ISS one minus. Sign squared, Data square rated. And we're gonna use that over here. So co sign of state A one is the square root. Well, one minus sign squared. So It's Lambda One squared over D squared. Okay, now I'm gonna put all this together, so I left off with signs. Data lawn, his land on over D Coast data one square of one, minus Landau. One squared over. Do you squared? Sure. And from my other one sign Zeta one co sign 30 plus sign. 30 Coast 81 was equal to land it too. Over D. So I'm gonna take these and stick him in there. So science data one Lambda one over D Coast 30. Signed 30 Coast. They won just the square root of one minus named Owen squared over D squared is equal to number two already. All right, I personally don't like fractions, so I'm gonna multiply both sides. I d. So that gives me an do you hear to cancel out that one? Give me a D here and d here to cancel that one. So I land the one goes 30 plus sign 30 Do you? Very one minus lambda. One squared over D squared. Zeke will delay into two. So now I'm gonna take this d and distribute it inside of my square root sign. So if I'm gonna distribute it inside. That means it's gonna be squared inside. So Lamda one coast 30 plus sign. 30 square root D squared times one minus lambda one squared d squared over D squared. Yeah, And I did that to get rid of the last of my fractions cause I really don't like fractions when we're dealing with algebra cause and it gets all confusing and gross. Yeah. Oh, okay. It's now murdered. Willy Boxley had Lambda. One was coast 30. Kuala signed 30 Swear a D squared minus lambda. One squared was equal to landed too. Okay, so I want to do the tedious process of solving for D squared. First, let's subtract my Lambda one coz 30 snow I have signed 30 square you d squared minus lambda Long squared sequel The Lambda too. Minus land one, co sign three. Next, we're gonna divide both sides by sign 30. It's getting the license, girls very satisfying to scratch. All right, so now I have square root of d squared minus lambda. One squared is equal to landed two minus lambda one coz I'm 30 all over, so I'm dirty. Okay, so I can get rid of my square root by squaring both sides. Well, next page. So I have d squared minus Lambda. One squared is equal to I am too minus Lambo. One those 30. Yeah, all over. Sign 30. All of that squared. Now I'm gonna add my lambda One squared to both sides. Some d squared. Was he able to land too minus lambda One coast. I'm 30 over. Sign 30 squared plus Lambda one squared. Who? Okay, now finally square. Root both sides to get d by itself. Finally! Well, actually not. Finally D is equal to the square root of I am too. Ones lambda want coz 30 blood all over. Sign 30 You squared plus Lambda one squared. But then the lines basing. There's one over D Who are so we have one over looks. Yeah, make this real big. Okay, well, I am too too. Was our bigger wavelengths about 700 nano meters. Linus Lam, the 1 400 Nana leaders Times Coast 30 a loader sign 30 squared plus Lambda One which was 400. You know, neither Sze squared. It's still inside the square roots on So we finally get an answer. Uh, 1230 wines. Her Millie Meter hoo boy

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