Final answer:
The first grating has a lower resolving power and fewer observable orders with weaker principal maxima due to fewer lines. The second grating, with many more lines spaced closer together, provides a significantly higher resolving power, a larger number of observable orders, and much higher intensities of principal maxima for a 500 nm incident wavelength.
Step-by-step explanation:
We are comparing two diffraction gratings in terms of their resolving power, observable orders, and the intensities of principal maxima for an incident wavelength of 500 nm.
Maximum Resolving Power
The resolving power (R) of a diffraction grating is given by R = nN, where n is the order number, and N is the number of lines illuminated. For the first grating with 10 lines spaced 1 cm apart, assuming only one line is illuminated, the maximum resolving power is R = n(10). For the second grating with 100,000 lines spaced 1 micron apart, the maximum resolving power is much higher: R = n(100,000).
Number of Observable Orders
The maximum number of observable orders (m) for a diffraction grating is limited by m = (d/λ) - 1, where d is the distance between the grating lines and λ is the wavelength of light. For the first grating, d = 1 cm, which means no higher orders will be observed since d > λ. For the second grating with d = 1 micron, many more orders can be observed.
Intensities of Principal Maxima
The intensity (I) of the principal maxima is proportional to the square of the number of slits (N^2), giving us I ~ N^2. Thus, the second grating with 100,000 lines will have much stronger principal maxima compared to the first grating with only 10 lines.