For your science fair project you need to design a diffraction grating that will disperse the
visible spectrum (400 –700 nm) over 30.0
◦
in first order.
(a) How many lines per millimeter does your grating need?
(b) What is the first-order diffraction angle of light from a sodium lamp (λ = 589 nm)?
(a) To disperse the visible spectrum over 30.0° in first order, we can use the formula for angular dispersion:
D = λ / (m * d)
Where:
D is the angular dispersion
λ is the wavelength of light
m is the order of diffraction
d is the spacing between the grating lines
In this case, we want the angular dispersion to be 30.0°, the wavelength to be within the visible spectrum (400 - 700 nm), and the order of diffraction to be 1.
We can rearrange the formula to solve for d:
d = λ / (D * m)
Substituting the values given:
λ = 700 nm (taking the longest wavelength in the visible spectrum)
D = 30.0°
m = 1
d = 700 nm / (30.0° * 1)
Converting nm to mm:
d = 0.7 mm / (30.0° * 1)
d = 0.7 mm / 30.0
d ≈ 0.023 mm
Therefore, the grating needs approximately 0.023 lines per millimeter.
(b) To find the first-order diffraction angle of light from a sodium lamp (λ = 589 nm), we can again use the formula for angular dispersion:
D = λ / (m * d)
Rearranging the formula to solve for the diffraction angle:
θ = sin^(-1)(m * λ / d)
Substituting the values given:
λ = 589 nm
m = 1
d ≈ 0.023 mm
θ = sin^(-1)(1 * 589 nm / 0.023 mm)
Converting nm to mm:
θ = sin^(-1)(1 * 0.589 mm / 0.023 mm)
θ ≈ sin^(-1)(25.52)
Using a calculator or trigonometric table, the approximate value for θ is:
θ ≈ 14.9°
Therefore, the first-order diffraction angle of light from a sodium lamp with a wavelength of 589 nm is approximately 14.9°.