The diffraction grating
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State the formula that relates the diffraction grating spacing (d), the angle to the nth order maximum (θ), the order number (n), and the wavelength of light (λ).
The formula is d sin θ = nλ. This equation relates the grating spacing, angle to the maxima, the order number of the maximum, and the wavelength of the incident light.
What does 'd' represent in the formula d sin θ = nλ and what are its common units?
'd' represents the grating spacing, which is the distance between adjacent slits on the diffraction grating. It is commonly measured in meters (m), but can also be given in mm or μm. Ensure you convert to meters before using it in the formula.
Explain how a diffraction grating produces a series of bright fringes (maxima) when illuminated by monochromatic light.
A diffraction grating produces bright fringes due to the interference of light waves that have diffracted through the slits. Constructive interference occurs when the path difference between waves from adjacent slits is equal to an integer multiple of the wavelength (nλ), leading to bright fringes at specific angles.
A diffraction grating has 500 lines per mm. Calculate the grating spacing 'd'.
The grating spacing, d, is the inverse of the number of lines per unit length. d = 1 / (lines per unit length). Here, d = 1 / (500 lines/mm) = 1/(500 x 10^3 lines/m) = 2 x 10⁻⁶ m.
Describe how you would use a diffraction grating to determine the wavelength of monochromatic light.
Shine the light through the grating and measure the angle θ to a specific order maximum (n). Knowing the grating spacing (d) and order number (n), use the formula λ = d sin θ / n to calculate the wavelength λ.
Monochromatic light of wavelength 600 nm is incident on a grating with a spacing of 2.0 x 10⁻⁶ m. Calculate the angle of the first-order maximum.
Using d sin θ = nλ, rearrange to get sin θ = nλ / d. For the first order (n=1), sin θ = (1 * 600 x 10⁻⁹ m) / (2.0 x 10⁻⁶ m) = 0.3. Therefore, θ = arcsin(0.3) = 17.46 degrees.
What is the maximum number of orders of diffraction that can be observed for a given wavelength λ and grating spacing d?
The maximum number of orders is determined by the condition that sin θ ≤ 1. Therefore, n_max = d / λ, rounded down to the nearest whole number. If d/λ = 3.5, then a maximum of three orders can be observed either side of the zero order.
Why are diffraction gratings preferred over double slits for measuring the wavelength of light?
Diffraction gratings produce sharper and brighter fringes than double slits, making it easier to measure the angle θ accurately. The greater number of slits leads to increased intensity at the maxima and a more precise determination of wavelength.
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