22: Problem 22.12

INTRO:
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 510lines/mm , and the light is observed on a screen 1.3m behind the grating.

Part A:
What is the distance between the first-order red and blue fringes?
Express your answer to two significant figures and include the appropriate units.

SOLUTION:
givens: 
λ_red = 656 nm
λ_blue = 486 nm
line density = 510 / mm
L = 1.3 m
d = 1 mm / 510 lines = 1.961⋅10^-3 mm
m = 1

the positions can be determined by using the formula θ=sin^-1(mλ/d) & Ltan(θ)=y
θ_red = sin^-1(1⋅λ_red/d) = sin^-1(656 nm / 1.961⋅10^-3 mm)
θ_red = 0.341099 rads = 19.5435°
y_red=Ltan(θ_red) = (1.3 m)tan(19.5435°) = 0.462 m
θ_blue = sin^-1(1⋅λ_blue/d) = sin^-1(486 nm / 1.961⋅10^-3 mm) 
θ_blue = 0.250443 rads = 14.349°
y_blue=Ltan(θ_blue) = (1.3 m)tan(14.349°) = 0.333 m

so the blue fringe begins at 0.333 m and the red fringe begins at 0.462 m
the distance in between the fringes is y_red-y_blue = 0.129 m = 13 cm

22: Problem 22.9

INTRO:
A 4.0-cm-wide diffraction grating has 2000 slits. It is illuminated by light of wavelength 590nm .

Part A:
What is the angle (in degrees) of the first diffraction order?
Express your answer to three significant figures and include the appropriate units.

SOLUTION:
givens: 
width = 4 cm
d = 4 cm / 2002 spaces between slits = 1.998⋅10^-3 cm
N = 2000
λ = 590 nm
m = 1

the angle can be determined by using the formula θ=m⋅λ/d
θ= 590 nm / 1.998⋅10^-3 cm
θ= 0.02953 radians = 1.69°

Part B:
What is the angle (in degrees) of the second diffraction order?
Express your answer to three significant figures and include the appropriate units.

SOLUTION:
do the same thing but with m=2
θ=2⋅590 nm / 1.998⋅10^-3 cm
θ= 0.05906 radians = 3.38°