Final answer:
The maximum number of lines per centimeter a diffraction grating can have to produce a complete first-order spectrum for visible light is 10,000 lines/cm. Gratings with a higher density than this will not resolve the entire visible spectrum, with 30,000 lines/cm being too dense to display the longest wavelengths in the first order.
Step-by-step explanation:
The maximum number of lines per centimeter a diffraction grating can have to produce a complete first-order spectrum for visible light depends on the equation θ = dsinθ = mλ, where d is the distance between slits (“grating constant”), θ is the diffraction angle, m is the order of the spectrum, and λ is the wavelength of light. The grating needs to resolve the entire spectrum of visible light, typically described as ranging from 380 nm to 780 nm. To produce a first-order spectrum (m=1) for the longest wavelength of visible light (780 nm), the sine of the diffraction angle must be at most 1, as this is the maximum possible value. From this, it can be found that the maximum grating density d is given by 1/d = 1/780nm, which equals approximately 1282 lines/mm or 12820 lines/cm. Among the options given, the closest under this value would be 10,000 lines/cm.
Therefore, the maximum number of lines per centimeter a diffraction grating can have and still produce a complete first-order spectrum for visible light is 10,000 lines/cm. A 30,000 lines/cm grating would not produce a first-order diffraction maximum for visible light because its grating constant is too small to satisfy the condition for the longest wavelength of visible light. Moreover, a higher density grating would be more useful for ultraviolet spectra, while a grating with fewer lines per centimeter would be beneficial for infrared spectra, given that UV light has shorter wavelengths and IR longer wavelengths than visible light.