A diffraction grating gives a second-order maximum at as angle of 31�‹ for violet light (ƒÉ = 4.0 �~ 102 nm). If the diffraction grating is 1.0 cm in width, how many lines are on this diffraction grating?
To find the number of lines on the diffraction grating, we need to use the formula for angular position of diffraction maxima:
d * sin(θ) = m * λ
Where:
d is the spacing between the lines on the grating,
θ is the angular position of the diffraction maximum,
m is the order of the maximum, and
λ is the wavelength of light.
In this case, we are given:
θ = 31°,
m = 2 (second order maximum),
λ = 4.0 × 10^(-7) meters (converted from nm to meters).
We need to find the value of d.
Rearranging the formula, we have:
d = (m * λ) / sin(θ)
Substituting the given values, we get:
d = (2 * 4.0 × 10^(-7)) / sin(31°)
Calculating this, we find the value of d to be approximately 7.75 × 10^(-7) meters.
Now, the width of the grating (w) is given as 1.0 cm. We can convert this to meters:
w = 0.01 meters.
The number of lines (N) on the grating can be calculated using the formula:
N = w / d
Substituting the values, we get:
N = 0.01 / 7.75 × 10^(-7)
Calculating this, we find the number of lines on the diffraction grating to be approximately 12,903.
Therefore, there are approximately 12,903 lines on this diffraction grating.
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