# Diffraction grating – problems and solutions

1. A grating containing 4000 slits per centimeter is illuminated with a monochromatic light and produces the second-order bright line at a 30° angle. What is the wavelength of the light used? (1 Å = 10^{-10} m)

__Known :__

The distance between slits (d) = 1 / (4000 slits / cm) = 0.00025 cm = 2.5 x 10^{-4} cm = 2.5 x 10^{-6} meters

Order (n) = 2

Sin 30^{o} = 0.5

1 Å = 10^{-10} m

__Wanted:__ Wavelength of the light (λ)

__Solution :__

2. A monochromatic light with a wavelength of 2.5x10^{-7 }m strikes a grating containing 10,000 slits/cm. Determine the angular positions of the second-order bright line.

__Known :__

The distance between slits (d) = 1 / (10,000 slits / cm) = 0.0001 cm = 1 x 10^{-4} cm = 1 x 10^{-6} m

Order (n) = 2

Wavelength (λ) = 2.5 x 10^{-7 }m

__Wanted :__ Angle (θ)

__Solution :__

3. A monochromatic light with wavelength of 5.10^{-7} m strikes a grating. Distance between slit and the viewing screen is 2 m, distance between the third-order fringe and the central fringe

__Known :__

Wavelength (λ) = 5 x 10^{-7} m

l = 2 m

n = 3

y = 150 cm = 1.5 m

__Wanted :__ distance between slits

__Solution :__

sin θ ≈ tan θ = y / l = 1.5 / 2 = 0.75 = 75 x 10^{-2}

Distance between slits :

d sin θ = n λ

d (75.10^{-2}) = (3)(5.10^{-7})

d (75.10^{-2}) = 15.10^{-7}

d = (15.10^{-7}) / (75.10^{-2})

d = (15/75).10^{-5}

d = (1/5)(10^{-5}) m

d = (0.2)(10^{-5}) m

d = 2.10^{-6 }m

4. A monochromatic light with wavelength of 500 nm (1 nano = 10^{-9}) strikes a grating and produces the fourth-order bright line at an 30° angle. Determine the number of slits per centimeter.

__Known :__

Wavelength (λ) = 500.10^{-9} m = 5.10^{-7} m

θ = 30^{o}

n = 4

__Wanted :__ number of slits per centimeter

__Solution :__

Distance between slits :

d sin θ = n λ

d (sin 30^{o}) = (4)(5.10^{-7})

d (0.5) = 20.10^{-7}

d = (20.10^{-7}) / 0.5

d = 40.10^{-7}

d = 4.10^{-6 }m

Number of slits per centimeter :

x = 1 / 4.10^{-6 }m

x = 0.25.10^{6} / m

x = 0.25.10^{6} / 10^{2} cm

x = 0.25.10^{4} / cm

x = 25.10^{2} / cm

x = 2500 / cm

5. A monochromatic light with wavelength of 500 nm (1 nm = 10^{-9} m) strikes a grating and produces the second-order bright line at an 30° angle. Determine the number of slits per centimeter.

__Known :__

Wavelength (λ) = 500.10^{-9} m = 5.10^{-7} m

θ = 30^{o}

n = 2

__Wanted :__ number of slits per centimeter

__Solution :__

Distance between slits :

d sin θ = n λ

d (sin 30^{o}) = (2)(5.10^{-7})

d (0.5) = 10.10^{-7}

d = (10.10^{-7}) / 0.5

d = 20.10^{-7}

d = 2.10^{-6 }m

Number of slits per centimeter :

x = 1 / d

x = 1 / 2.10^{-6 }m

x = 0.5.10^{6} / 1 m

x = 0.5.10^{6} / 10^{2} cm

x = 0.5.10^{4} / cm

x = 5.10^{3} / cm

x = 5000 / cm