Number of lines/cm for the grating

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The discussion revolves around calculating the maximum number of lines per centimeter for a grating that produces three bright fringes on either side of the central maximum when illuminated with light of 520 nm. The participant determines that the order of the fringe (m) is 3, leading to the equation d sin(theta) = (3 * 520 nm). After establishing that the maximum displacement occurs at theta = 90 degrees, they calculate the slit separation (d) as 1.54 E-6 meters. Finally, they conclude that the maximum number of lines per centimeter for the grating is 6410 lines/cm. The problem is resolved with the correct application of interference principles.
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The problem is:

Three, and only three, bright fringes can be seen on either side of the central maximum when a grating is illuminated with light ( = 520 nm). What is the maximum number of lines/cm for the grating?

Here is what i know / did:

since there are three fringes, m = 3

since they are bright fringes, it's constructive interference so (m lambda)

we know that d sin(theta) = (m lambda)

so, d sin(theta) = (3 * 5.20 E-7 meters)

we know that the highest displacement has to occur when theta is 90 degrees. So, sin of 90 gives you 1.

d sin(90) = (3 * 5.2 E -7); D = 1.56 E-6 meters <- This is the slit separation, I am stuck at this point.
 
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Never Mind

I figured it out

Nmax = 1 / D min

since d = 1.54 E-6 m = 1.5 E -4 cm

1 / 1.54 E -4 cm = 6410 lines/cm.
 
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