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Angular dispersion is a measure of how the angle of diffraction varies with the wavelength of light passing through a diffraction grating. It's crucial in understanding how a spectrum is formed when white light is diffracted.

Using the diffraction formula, \( \sin \theta = m \lambda / d \), where \( m \) signifies the order, \( \lambda \) the wavelength, and \( d \) the grating line spacing, one can determine the angles at which different wavelengths emerge from the grating. A higher dispersion means a greater separation between various wavelengths, making each color in the visible spectrum more distinct. This conveys how effectively a grating can spread out light into its constituent colors, an essential aspect for spectrometry and educational demonstrations.

Linear dispersion translates the concept of angular dispersion into a measurable distance on a fixed screen. By calculating \( \tan(\theta) \) for the angles obtained through angular dispersion, and multiplying that by the grating-to-screen distance, the linear positions of each wavelength can be determined.

This aids in visually quantifying the spread of the spectrum—how broad it appears over a distance. In our example with the physics instructor's setup, the difference in linear positions for the red (\(700 nm\)) and violet (\(400 nm\)) light, projected onto a screen 2.4 meters away, would give the actual width of the spectrum for the first order (\( m=1 \) spectrum).

Spectrum separation refers to the physical distance between two sequential orders of spectra. For instance, the separation between the end of the \( m=1 \) and the beginning of the \( m=2 \) spectrum. This becomes especially significant in determining the resolving power of a grating and is critical for ensuring distinct spectral lines in spectrometry.

As per the problem's instruction, spectrum separation can be visualized as the gap on a screen between two adjacent rainbow patterns formed due to different orders of diffracted light. These separations help students understand the concept of orders in light diffraction and their practical implications in spectral analysis.

Wavelength calculation in the context of a diffraction grating involves decoding the spectrum patterns back into the component wavelengths. By applying the grating equation, \( m \lambda = d \sin \theta \) with known values of grating spacing (\( d \) from the number of lines per millimeter) and diffraction angles, the wavelengths can be extracted.

The calculation of wavelengths is essential for interpreting the recorded spectra and for further scientific analysis. Spectrums generated from white light through a diffraction grating, such as that in classroom demonstrations, can be utilized to teach students about the nature of light and its interaction with different materials at a foundational level.