Let 580 nm light be incident normally on a diffraction grating for which d=3.00D=1050nm. a) how many order (principal maxima) are present ? b) if the grating is 1.80 cm wide, what is the full angular width of each principal maximum?
I got the first part I just plugged into dsin(θ)=mλ which is 1050sin(90)=m*580 but i really don't know what to do on the second part. I tried the equation Δθ=λ/Ndcosθ
To find the full angular width of each principal maximum in a diffraction grating, you can use the formula:
Δθ = λ / N * d * cos(θ)
Where:
- Δθ represents the angular width of each principal maximum
- λ is the wavelength of light incident on the grating
- N is the number of slits (or order of the principal maximum)
- d is the spacing between adjacent slits in the grating
- θ is the angle of incidence (which is assumed to be zero for normal incidence)
In this case, you have already found that the wavelength, λ, is 580 nm and the spacing between slits, d, is 1050 nm.
a) To determine the number of orders (principal maxima) present, you can rearrange the equation d * sin(θ) = m * λ, where m is the order of the principal maximum. Since you have normal incidence (θ = 0), the equation simplifies to d = m * λ. Plugging in the values, you get:
1050 nm = m * 580 nm
Solving for m, you find that m ≈ 1.81. Since the order of the principal maximum cannot be a fraction, you can round it to the nearest whole number, which gives you m = 2. Therefore, there are 2 orders present.
b) Now, to find the full angular width of each principal maximum, you can use the formula mentioned earlier:
Δθ = λ / N * d * cos(θ)
Since you are looking for the angular width of each principal maximum, you can use N = 1 (first order) and θ = 0 (normal incidence):
Δθ = λ / 1 * d * cos(0)
The cosine of 0 is 1, so the formula simplifies to:
Δθ = λ / d
Substituting the values, you get:
Δθ = 580 nm / 1050 nm
Calculating this, you find that Δθ ≈ 0.5524 radians. To convert this to degrees, multiply by the factor 180/π:
Δθ ≈ 31.63 degrees
Therefore, the full angular width of each principal maximum is approximately 31.63 degrees.