Two identical pieces of rectangular plate glass are used to measure the thickness of a hair. The glass plates are in direct contact at one edge and a single hair is placed between them hear the opposite edge. When illuminated with a sodium lamp \((\lambda=589 \mathrm{nm}),\) the hair is seen between the 180 th and 181 st dark fringes. What are the lower and upper limits on the hair's diameter?

Optical interference is a phenomenon where two or more light waves superpose to form a resultant wave of greater, lower, or the same amplitude. Imagine ripples on a water surface created by two separate stones dropped at different points. Where the ripples meet, they may reinforce each other, making a bigger wave (constructive interference), or they might cancel each other out, resulting in a flat surface (destructive interference).

This principle applies to light waves when they reflect off or pass through thin films, such as oil on water or in the space between two glass plates. When a hair is inserted between the plates, as in the exercise, it creates a thin film which causes the light waves to interact. Depending on the alignment, or path difference, of these waves, fringes or bands of light and dark are observed. The dark fringes described in the problem are examples of destructive interference where the waves cancel out.

The wavelength is a core concept in understanding optical interference. It is the distance between successive crests of a wave, which determines its color and energy in the case of light. For example, the sodium lamp's wavelength in the given exercise is 589 nanometers (nm), which falls into the visible spectrum, giving the characteristic yellow glow of sodium vapor lamps.

When dealing with interference, we use the wavelength to determine where fringes of light will occur. In the exercise, the change in thickness of the hair produces a shift in the alignment of the reflected light waves, causing them to interfere constructively or destructively at different points. By counting the fringes and knowing the light's wavelength, we can accurately calculate the thickness of the hair or any thin film.

Fringes in physics, particularly in the context of interference patterns, are the bright or dark bands caused by the constructive or destructive interference of light. These patterns are not just beautiful; they are a powerful tool in precision measurement, as seen in the given problem. The position of these fringes depends on the path difference of the interfering beams of light.

In our exercise, we observed the fringes to estimate the thickness of a hair. As the light reflects off the top and bottom of the hair, it creates those alternating bands due to interference. Knowing the order of the fringe (180th, 181st) and the light's wavelength allows us to calculate the thin film's thickness with remarkable accuracy, utilizing simple mathematics to reveal microscopic details.