Problem 60 - College Physics

Question

A thin air wedge between two flat glass plates forms bright and dark interference bands when illuminated with normally incident monochromatic light. (See Fig. 24.9.) (a) Show that the thickness of the air wedge changes by \(\lambda / 2\) from one bright band to the next, where \(\lambda\) is the wavelength of the light. (b) What would be the change in the thickness of the wedge between bright bands if the space were filled with a liquid with an index of refraction \(n ?\)

Step-by-Step Solution

Answer

The thickness changes by \(\lambda/2\) in air and \(\lambda/(2n)\) in a medium.

Step 1: Understand the Bright Band Formation

The bright bands in interference patterns correspond to constructive interference, where the path difference between the reflected light beams is an integral multiple of the wavelength. When light is normally incident, the path difference that causes constructive interference must be equal to an integer multiple of the wavelength, i.e., \(m \lambda\), where \(m\) is an integer.

Step 2: Define the Condition for Bright Bands

Consider two rays of light reflecting off the top and bottom surfaces of the air wedge. The path difference is given by twice the thickness of the wedge, so we have the condition for constructive interference as \(2t = m \lambda\), where \(t\) is the thickness of the air wedge.

Step 3: Derive the Change in Thickness for Bright Bands

To find the change in thickness between successive bright bands, we calculate the difference between thickness \(t_m\) for the \(m\)-th band and \(t_{m+1}\) for the \((m+1)\)-th band. Using the formula \(2t_m = m \lambda\) and \(2t_{m+1} = (m+1) \lambda\), the change in thickness is \(t_{m+1} - t_m = \frac{\lambda}{2}\).

Step 4: Consider the Effect of a Refractive Medium

When the space is filled with a liquid of refractive index \(n\), the wavelength of light in the medium becomes \(\frac{\lambda}{n}\). The condition for constructive interference modifies to \(2t' = m \frac{\lambda}{n}\). The change in thickness between successive bright bands then becomes \(t_{m+1}' - t_m' = \frac{\lambda}{2n}\).

Key Concepts

Constructive Interference

Constructive interference occurs when two or more light waves meet and combine in such a way that their amplitudes add up. This results in a bright band or fringe in an interference pattern. Essentially, the two waves are "in phase," meaning the peaks of one line up with the peaks of the other.
This phenomenon is key to understanding interference patterns, like the ones formed in the air wedge between glass plates.For constructive interference to occur, the path difference between the waves must be an integral multiple of the wavelength, designated by the symbol \(m\lambda\), where \(m\) is an integer. This ensures that the waves reinforce each other, amplifying the resultant wave's intensity.
To find the thickness change, recognize that for each successive bright band, the thickness of the air wedge alters by \(\frac{\lambda}{2}\). This precise change maintains the condition of each pair of beams emerging in phase, thus contributing to the consistent appearance of bright bands.

Monochromatic Light

Monochromatic light refers to light of a single wavelength or color. This is a critical concept in experiments involving interference patterns because using light of only one wavelength makes it easier to predict and analyze results.
When light is monochromatic, interference patterns such as those created in an air wedge setup are uniform and distinct because each wavelength interferes in a predictable, systematic way. In the air wedge scenario, illuminating with monochromatic light ensures that the path difference needed for constructive interference remains consistent across the entire pattern. By using light with a well-defined wavelength, any variations or inconsistencies in the interference pattern can be more readily attributed to the physical setup (e.g., thickness changes in the air wedge) rather than variations in the light source itself.
Monochromatic light thus facilitates clearer observation and measurement, allowing the study of phenomena like constructive interference to be precise and accurate.

Refractive Index

The refractive index, represented by the symbol \(n\), plays a crucial role in the bending, or refraction, of light as it moves through different media. Essentially, it is a measure of how much a light wave slows down when it enters a medium.
For a liquid or glass, a higher refractive index means the light travels more slowly compared to its speed in air.In interference patterns, when the medium between the plates is altered, such as replacing air with a liquid, the effective wavelength of the light inside the medium changes. This modified wavelength is given by \(\frac{\lambda}{n}\), where \(\lambda\) is the wavelength in vacuum, and \(n\) is the refractive index of the new medium.When using a medium with a different refractive index in the wedge, the spacing of the bright bands will change. Specifically, with a liquid, the thickness change required between bright bands transitions from \(\frac{\lambda}{2}\) to \(\frac{\lambda}{2n}\).
Understanding refractive index is vital for predicting how light behaves when transitioning between different media, thereby affecting interference patterns.