Question

The two most prominent wavelengths in the light emitted by a hydrogen discharge lamp are 656 nm(red) and 486 nm (blue). Light from a hydrogen lamp illuminates a diffraction grating with 500 lines per mm, and the light is observed on a screen 1.50 m behind the grating.

What is the distance between the first-order red and blue fringes?

Answer 1

?_red = 656 nm
?_blue = 486 nm
line density = 500 / mm
L = 1.5 m
d = 1 mm / 500 lines = 2?10^-3 mm
m = 1

the positions can be determined by using the formula ?=sin^-1(m?/d) & Ltan(?)=y
?_red = sin^-1(1??_red/d) = sin^-1(656 nm / 2?10^-3 mm)
?_red = 0.334186 rads = 19.14743°
y_red=Ltan(?_red) = (1.5 m)tan(19.14743°) = 0.521 m
?_blue = sin^-1(1??_blue/d) = sin^-1(486 nm / 2?10^-3 mm)
?_blue = 0.245457 rads = 14.06367°
y_blue=Ltan(?_blue) = (1.5 m)tan(14.06367°) = 0.376 m

so the blue fringe begins at 0.376 m and the red fringe begins at 0.521 m
the distance in between the fringes is y_red-y_blue = 0.145 m = 14.5 cm