Wave optics and diffraction gratings

[Kalibasa]

Homework Statement

I'm really struggling with this one.

"A diffraction grating with 600 lines/mm is illuminated with light of wavelength 500 nm. A very wide viewing screen is 2.0 m behind the grating."

a) What is the distance between the two m=1 fringes?
b) How many bright fringes can be seen on the screen?

Homework Equations

d sinθ= mλ where d is the distance between the slits in the grating and m is the diffraction order of the fringes
y=Ltanθ where y is the distance between the fringes and L is the distance to the viewing screen

The Attempt at a Solution

I finally got the first part. I found d (d=1/N where is slits/mm) and then using d sinθ= mλ, I solved for θ for both fringes and then plugged this into y=L tanθ. I found the difference between these two y values (0.629 m each way) to get 1.3 m, which the book says is correct.

But now I have no idea how to solve the second part. How can I know how many fringes there are on the screen if I don't know the width of the screen?

 

[ehild]

What do you think, will that ray reach the screen which is diffracted at 90 degrees or more?

ehild

 

[Kalibasa]

I tried that, though. The first fringe was at 17.45 degrees, so I figured you could go out to only 90 degrees in each direction; this means you could fit five fringes in either direction, plus the one in the middle, so 11 total. Doesn't that make sense?

But the book says there are supposed to be only seven fringes...

 

[ehild]

You know that d sinθ= mλ. Plugging in the data, sinθ =0.3 m. sin(90°)=1, so 0.3 m <1. It can be 0, 0.3, 0.6, 0.9 and also -0.3, -0.6, -0.9.

ehild

 

[rwisz]

Based on the given information, we can calculate the distance between two consecutive m=1 fringes on the viewing screen using the equation y=L tanθ, where L is the distance to the screen and θ is the angle of diffraction. However, in order to determine the number of fringes on the screen, we also need to know the width of the screen.

To find the number of fringes, we can use the equation N = W/y, where N is the number of fringes, W is the width of the screen, and y is the distance between fringes calculated in the first part. If we assume that the screen is wide enough to accommodate all the fringes, then we can use the given distance of 2.0 m between the grating and the screen as the width of the screen.

Therefore, the number of fringes that can be seen on the screen is N = 2.0 m / 1.3 m = 1.54 fringes. Since we cannot have a fraction of a fringe, we can round this value down to 1 fringe. Therefore, only one bright fringe can be seen on the screen in this setup.

It is important to note that the actual number of fringes may vary depending on the exact width of the screen and the accuracy of the calculated distance between fringes. However, this calculation gives us an estimation of the number of fringes that can be seen on the screen.