Problem 71 - University Physics with Modern Physics

Question

When the light is passed through the bottom of the sample container, the interference maximum is observed to be at 41\(^\circ\); when it is passed through the top, the corresponding maximum is at 37\(^\circ\). What is the best explanation for this observation? (a) The microspheres are more tightly packed at the bottom, because they tend to settle in the suspension. (b) The microspheres aremore tightly packed at the top, because they tend to float to the top of the suspension. (c) The increased pressure at the bottom makes the microspheres smaller there. (d) The maximum at the bottom corresponds to \(m\) = 2, whereas the maximum at the top corresponds to \(m\) = 1.

Step-by-Step Solution

Answer

Option (a): Microspheres settle and are more tightly packed at the bottom.

Step 1: Analyze Interference Pattern

Interference patterns are formed due to constructive and destructive interference of light waves. The maxima or bright spots occur at specific angles based on the path difference created by the different material properties or structures they pass through.

Step 2: Understand Impact of Packing Density

If microspheres are more packed, this affects the refractive index and consequently, the angle at which the interference maximum occurs. A higher density of microspheres means the light waves travel through more material, increasing the path length.

Step 3: Relate Maximum Angle to Packing Density

When interference maximum shifts to a larger angle (41° vs 37°), this indicates a higher effective path difference. Therefore, a greater angle suggests more densely packed microspheres resulting in increased index of refraction.

Step 4: Consider Answer Options

(a) If microspheres are denser at the bottom due to settling, the light path increases, explaining why the maximum shifts to the larger angle at the bottom (41°). (b) It's unlikely for microspheres to be denser at the top (floating). (c) Smaller microspheres at the bottom would reduce the angle, not increase it. (d) Different m-values usually indicate different wavelengths or layer sequences rather than angle differences caused by medium density.

Step 5: Determine Correct Explanation

Given the significant angle difference, the most plausible explanation is option (a): the microspheres settle, becoming more densely packed at the bottom of the container, which results in a larger angle for the interference maximum.

Key Concepts

Interference Pattern

An interference pattern arises when waves overlap as they travel through different paths. This phenomenon is most commonly associated with light waves, but it applies to any situation where wave patterns interact. When light waves meet, they can interfere with each other in two ways: constructively or destructively. Constructive interference leads to the formation of bright spots or maxima in the interference pattern. These occur when the path difference between two waves is an integer multiple of their wavelength, resulting in waves that are in phase. Destructive interference, on the other hand, happens when the waves are out of phase, canceling each other out and forming dark spots. The angles at which these maxima occur depend on the physical properties of the material the light waves pass through, such as the refractive index and the density of particles or structures present.

Refractive Index

The refractive index of a material is a measure of how much it bends or refracts light. When light enters a medium, its speed changes and so does its direction. The refractive index, denoted by the symbol \( n \), is calculated as the ratio of the speed of light in a vacuum to the speed of light in the medium. When microspheres are packed closely together, their collective refractive index increases. Such a change in the refractive index will affect the angle of the light wave’s interference pattern, causing maxima to shift. A larger change in angle suggests a higher refractive index, indicating denser packing of the microspheres which increases the optical path length of the waves traversing the medium.

Microsphere Packing Density

Microsphere packing density refers to how tightly microspheres are packed within a given volume. In a suspension, microspheres can settle over time or be distributed unevenly due to various factors such as gravity or pressure. The density affects how light moves through the suspension; densely packed areas will cause light to travel through a longer path due to the increased number of microspheres. The result is a shift in the interference maximum to a greater angle due to the higher effective refractive index. In our scenario, the microspheres being denser at the bottom of the container would lead to a larger angle for the interference peak (41° at the bottom compared to 37° at the top), supporting the idea that packing density impacts the observed interference pattern.

Constructive Interference

Constructive interference occurs when two wave peaks meet, reinforcing each other and creating a resultant wave with greater amplitude. For light, this results in brighter spots in the interference pattern called maxima. These occur precisely when the path difference is an integer multiple of the wavelength: \( (m \times \lambda) \) where \( m \) is an integer, and \( \lambda \) is the wavelength. The concept is fundamental to understanding why certain angles produce bright and others dark spots in interference experiments. The observed change in angles of maxima (from 37° to 41°) in the exercise suggests a different path length due to a more densely packed medium, creating a specific condition where constructive interference leads to a maxima at a larger angle. This illustrates the relationship between packing density, refractive index, and the constructive interference that defines the resulting patterns.