White light strikes a grating with 1300 lines/cm at normal incidence. How many complete visible spectra will be formed on either side of the central maximum? (Take the range of visible wavelengths to be 400 - 700 nm.)
I used the equation
d sintheta = m wavelength: (1/1300cm)(1000)= [0.769m x sin90]/400nm = 1 (since you are supposed to round down). I tried with 700nm and 700-400=300nm. So, my answers were 0,1,2, or 3; all are wrong! Any help?
To solve this problem, we can use the formula for the number of maxima produced by a diffraction grating:
mλ = d(sinθ + sinα),
where:
m is the order of the maximum,
λ is the wavelength of the light,
d is the spacing between the slits in the grating,
α is the angle of incidence (which is 0 degrees in this case since it is normal incidence), and
θ is the angle at which the maximum occurs.
First, let's rearrange the equation to solve for m:
m = (d(sinθ + sinα)) / λ.
Given:
d = 1/1300 cm (0.769 μm),
λ = 400 - 700 nm (0.4 - 0.7 μm).
We need to find the range of m for which there are visible spectra formed. The central maximum occurs at m = 0.
Now, let's calculate m for the upper limit of the visible wavelength (700 nm):
m = (0.769 μm × (sinθ + sin0)) / 0.7 μm.
For the lower limit of the visible wavelength (400 nm), we calculate m as follows:
m = (0.769 μm × (sinθ + sin0)) / 0.4 μm.
Now, let's find the maximum value of m for which the visible spectra are formed on either side of the central maximum.
For the upper limit (700 nm):
m = (0.769 μm × (sinθ + sin0)) / 0.7 μm
⇒ (sinθ + sin0) / 0.7 = m / 0.769
⇒ sinθ + 0 = (m × 0.7) / 0.769
⇒ sinθ = (m × 0.7) / 0.769
Now, calculate the corresponding value of θ using the inverse sine function:
θ = arcsin((m × 0.7) / 0.769)
Using this equation, calculate the value of θ for m = 1. We will consider the value of θ in radians.
θ = arcsin((1 × 0.7) / 0.769)
Repeat the same process for the lower limit (400 nm):
θ = arcsin((m × 0.7) / 0.769)
Calculate the value of θ for m = 1.
θ = arcsin((1 × 0.7) / 0.769)
Now, find the range of values of m for which the visible spectra are formed. Since we have calculated the value of m for the upper and lower limits, we can determine the number of complete visible spectra by considering the difference between the two values of m.
The number of complete visible spectra on either side of the central maximum is equal to the difference between the two values of m.
I hope this helps!