(a) Show that a 30,000 line per centimeter grating will not produce a maximum for visible light. (b) What is the longest wavelength for which it does produce a firstorder maximum? (c) What is the greatest number of line per centimeter a diffraction grating can have and produce a complete second-order spectrum for visible light?

Understanding visible light diffraction is key to grasping how diffraction gratings work. Diffraction occurs when a wave, such as light, encounters an obstacle or a slit that is comparable in size to its wavelength. A diffraction grating consists of many evenly spaced openings, which scatter the incoming light into several directions. The pattern of light and dark bands produced is caused by the constructive and destructive interference of the waves.

For visible light, which spans wavelengths from about 380 nm (violet) to 750 nm (red), the effects of diffraction can be clearly observed when it passes through a grating with openings or lines spaced closely together, typically a few micrometers apart. The lines act like slits, causing the light waves to spread out and overlap. When the overlapped waves are in phase, they amplify each other, leading to bright spots known as maxima. Conversely, when they are out of phase, they cancel each other, creating dark spots known as minima.

This phenomenon can help us understand why a grating with too fine a spacing, like the 30,000 lines per centimeter example in the exercise, may not produce a visible maximum—it effectively 'squeezes' the light waves too closely, leading to diffraction angles that fall outside the range detectable by human eyes or standard optical equipment.

The grating equation is the formula that relates the angles at which light rays are diffracted to the spacing between the grating lines and the wavelength of the light. The equation is given by:
\[ \sin \theta = \frac{m \lambda}{d} \]
Where \( \sin \theta \) is the sine of the angle of diffraction, \( m \) is the order of the maximum (which can be any whole number, with first-order being the primary focus), \( \lambda \) is the wavelength of the light, and \( d \) is the distance between adjacent grating lines. The grating equation is derived from the condition for constructive interference, which is when the path difference between neighboring diffracted waves is a whole number multiple of the wavelength.

Applying the grating equation to the 30,000 lines per centimeter grating, we find that this grating spacing does not allow diffraction maxima for visible light. The reason is that the calculated angles for the diffraction maxima exceed the possible range for \( \sin \theta \) (which has to be between -1 and 1). Setting \( \sin \theta = 1 \) allows us to calculate the longest wavelength that would produce a first-order maximum with this grating.

Spectral lines are specific to each element and arise due to the energy differences between electronic transitions in atoms or molecules. When light passes through a diffraction grating, the dispersed light forms a spectrum. In this spectrum, certain wavelengths are reinforced through constructive interference, resulting in bright lines called spectral lines.

In the context of the exercise, understanding the concept of spectral lines will help in determining the conditions under which the diffraction grating can produce overt spectral lines of a particular order for visible light. A spectral line corresponds to a particular wavelength where constructive interference is at a maximum. As such, to produce a complete second-order spectrum (where \( m = 2 \)), one can calculate the number of lines per centimeter necessary by using the grating equation and recognizing the requirement that the grating must be able to diffract light at wavelengths up to 750 nm into the second order. This provides an insight into the limitations for the number of lines a grating can have and still produce distinct spectral lines observable in the visible spectrum.