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?
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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.