Light waves through grating. Min and Max measurements of wavelenghts in nm

[Mugen112]

Homework Statement

Light passes through a 220 lines/mm grating and is observed on a 1.4m wide screen located 1.2m behind the grating. Three bright fringes are seen on both sides of the central maximum.

What are the minimum and maximum possible values of the wavelength?
Express your answers in nm.

Homework Equations

y=LtanØ
dsinØ=m*wavelength

The Attempt at a Solution

I started by looking at the first equation that is listed. y=LtanØ
To find y, I just looked at the 220lines/mm and that is .00022lines/nm. So then I just converted L into nm and plugged in the numbers.

700000000 = 1200000000tanØ
then
tan^-1(700000000/1200000000) = about 30 degrees? ... I then put that into the second equation and get something wayyyy off for the wavelength. like .00037nm?
I've been working on this problem for over an hour and can't seem to find what I'm doing wrong or how to approach this the correct way. Any help is appriciated. Thanks!

 

[Carid]

d is the distance between two lines in the grating (hint: it is not a density)

 

[thomate1]

I would suggest approaching this problem using the second equation listed: dsinØ=m*wavelength. This equation relates the distance between the lines on the grating (d), the angle of diffraction (Ø), the order of the fringe (m), and the wavelength of light passing through the grating.

In this case, we know that there are three bright fringes on either side of the central maximum, so m=3. We also know that the distance between the grating lines is 1/220 mm, or 0.004545 nm. Plugging in these values, we get the following equation:

0.004545*sinØ = 3*wavelength

Solving for wavelength, we get:

wavelength = 0.004545*sinØ/3

To find the minimum and maximum possible values of the wavelength, we need to consider the smallest and largest possible values for the angle of diffraction. Since the light is passing through the grating and being observed on a screen, the angle of diffraction cannot be larger than 90 degrees. Therefore, the minimum and maximum values of the angle of diffraction are 0 degrees and 90 degrees, respectively.

Plugging these values into the equation for wavelength, we get:

Minimum wavelength = 0.004545*sin(0)/3 = 0 nm

Maximum wavelength = 0.004545*sin(90)/3 = 0.004545 nm

So the minimum and maximum possible values of the wavelength are 0 nm and 0.004545 nm, respectively. This makes sense because the grating can only diffract light within a certain range of wavelengths, and in this case, that range is from 0 nm to 0.004545 nm.