Ten narrow slits are equally spaced 0.25 mm apart and illuminated with yellow light of wavelength \(580 \mathrm{nm}\) (a) What are the angular positions of the third and fourth principal maxima? (b) What is the separation of these maxima on a screen \(2.0 \mathrm{m}\) from the slits?

Understanding the angular position of maxima is crucial when studying multiple-slit interference patterns. In physics, these angular positions refer to the specific angles where the intensity of light reaches a maximum because the light waves intersect constructively.

When light passes through multiple narrow slits, it diffracts, and each diffracted wave overlaps with others. This superimposition can either reinforce the light intensity (maxima) or cancel it out (minima). The exact angular position of these bright fringes — the maxima — is given by the formula
\[\begin{equation}θ_m = \text{{arcsin}}\left(\frac{mλ}{nd}\right)\end{equation}\]
Where \(m\) represents the order of the maximum, \(λ\) is the wavelength of light, \(n\) is the number of slits, and \(d\) is the separation between slits. By knowing these variables, one can determine where on a screen placed at a distance from the slits the bright fringes will appear. For our example, with the third and fourth maxima, the formula helps us identify their exact positions by plugging in the given data.

The interference pattern refers to the arrangement of bright and dark fringes on a screen as a result of multiple-slit interference. Each bright spot (maxima) is created where light waves from the different slits arrive in phase and reinforce each other, while dark spots (minima) occur where waves arrive out of phase, cancelling each other out.

The pattern is a direct visual evidence of the wave nature of light and is characterized by regular alternation of light and dark bands. The distance between adjacent maxima (fringe spacing) depends on the wavelength of light, slit separation, and distance from the slits to the screen, indicating how closely or widely spaced the bright fringes are. A detailed analysis of this pattern allows scientists to quantify properties of the light and the interfering system.

The wavelength of light, represented by \(λ\), is the distance over which the light wave's shape repeats. It determines the color of the light and is integral to the interference pattern produced in multiple-slit experiments.

In our example, the wavelength is given as 580 nm, representing yellow light. This value is crucial because it directly affects the position and spacing of the interference fringes. The behavior of different wavelengths through a set of slits can result in different patterns on a screen, translating to the concept of dispersion where different colors split due to their unique wavelengths.

Lastly, the slit separation (\(d\)) is the distance between adjacent slits in a multiple-slit arrangement. It's an essential factor in determining the nature of the interference pattern we observe, including the angular width of fringes and their spacing on a detection screen.

A smaller slit separation typically results in a wider spread of the maxima on the screen, all other variables held constant. Conversely, larger slit separations cause a denser fringe pattern. In the exercise, the students are provided with a slit separation of 0.25 mm, which plays a pivotal role in calculating both the angular positions of the maxima and their physical separation on the screen.