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 approximately 14,300 lines/cm. This value ensures that the entire range of visible light is diffracted within a 90-degree angle.
Step-by-step explanation:
The maximum number of lines per centimeter that a diffraction grating can have to produce a complete first-order spectrum for visible light depends on the wavelength range of the visible spectrum. Visible light typically ranges from about 400 nanometers (nm) to 700 nm. Using the formula θ = sin⁻¹(mλ/d), where θ is the diffraction angle, λ is the wavelength of light, d is the spacing between adjacent lines on the grating, and m represents the order of the spectrum, we can calculate the required grating density.
The grating must fulfill the condition that the entire first-order spectrum (m=1) is visible, which means that sin⁻¹(700 nm / d) ≤ 90 degrees, since 90 degrees is the maximum possible diffraction angle for light to be visible. A complete first-order spectrum occurs at the grating line separation that corresponds to the longest wavelength of visible light (700 nm) diffracted at an angle of 90 degrees.
Therefore, d must be at least 700 nm for the first-order maximum to be at 90 degrees. Converting 700 nm to centimeters (1 nm = 1 x 10⁻⁷ cm), we get d = 700 nm = 7 x 10⁻⁵ cm. Hence, the maximum number of lines per centimeter is 1 / d, which equals approximately 1.43 x 10⁵ lines/cm, or 14,300 lines/cm. This would represent the grating with the highest number of lines per centimeter that can still display the full range of visible light in the first-order spectrum.