Problem 25 - University Physics with Modern Physics

Question

Two rectangular pieces of plane glass are laid one upon the other on a table. A thin strip of paper is placed between them at one edge so that a very thin wedge of air is formed. The plates are illuminated at normal incidence by 546-nm light from a mercury-vapor lamp. Interference fringes are formed, with 15.0 fringes per centimeter. Find the angle of the wedge

Step-by-Step Solution

Answer

The angle of the wedge is approximately \( 4.095 \times 10^{-6} \) radians.

Step 1: Understand the Problem Setup

We have two glass plates forming a thin air wedge due to a strip of paper at one edge. The wedge is illuminated with light of wavelength \( \, \lambda = 546 \, \text{nm} \). We know the fringe density is 15.0 fringes per centimeter.

Step 2: Relation between Fringe Spacing and the Wedge Angle

The number of fringes per centimeter is the inverse of the fringe spacing. We can use the equation for the path difference between two successive bright fringes to find the angle \( \theta \) of the wedge. The path difference \( \Delta x \) for constructive interference (bright fringes) is given by \( 2t = m\, \lambda \).

Step 3: Expression for Wedge Angle

For a small angle approximation, the thickness \( t \) at a distance \( x \) from the edge where the paper is placed is \( t = x \tan \theta \approx x \theta \). The fringe spacing \( \Delta x \) is then derived from \( m \lambda = 2t \). Thus, \( x \theta = \frac{m \lambda}{2} \), giving us \( \theta = \frac{m \lambda}{2x} \).

Step 4: Calculate Fringe Spacing

We know there are 15.0 fringes per centimeter, which means each fringe is \( \frac{1}{15.0} \) cm wide or \( x = \frac{1}{15.0} \times 10^{-2} \) meters.

Step 5: Substitute Values to Find the Angle

We use \( m = 1 \) (for the first fringe), \( \lambda = 546 \times 10^{-9} \) meters, and \( x = \frac{1}{15.0} \times 10^{-2} \) meters in \( \theta = \frac{m \lambda}{2x} \) to calculate the angle. \[ \theta = \frac{1 \times 546 \times 10^{-9}}{2 \times \frac{1}{15.0} \times 10^{-2}} = \frac{546 \times 15.0}{2} \times 10^{-9} \approx 4.095 \times 10^{-6} \, \text{radians} \].

Key Concepts

Air Wedge

An air wedge is a fascinating phenomenon created when two surfaces are slightly inclined relative to each other, with air filling the gap between them. This slight inclination forms a wedge-like shape. In many experiments, placing a thin strip of material, like paper, at one edge between two glass plates can achieve such a setup. The purpose of this setup is to explore optical interference effects.

The resulting thin wedge of air can cause interference patterns to appear when monochromatic light shines on it. This happens because the path length through which light must travel differs across the wedge. Consequently, an air wedge is useful in experiments to visualize and calculate effects related to light interference, such as measuring very small angles accurately.

Wavelength

Wavelength is a fundamental property of waves, defined as the distance between successive peaks (or troughs) of a wave. For light waves, wavelength plays a crucial role in determining the behavior and color of light. In optics, the unit of measurement for wavelength is typically nanometers (nm).

In the context of the air wedge experiment, understanding the wavelength of the illuminating light is essential. In this case, a mercury-vapor lamp with a wavelength of 546 nm is used. This specific wavelength helps dictate the conditions under which interference fringes appear. Light of different wavelengths will create different fringe patterns, allowing researchers to observe how light interacts in the air wedge.

Constructive Interference

Constructive interference occurs when two waves meet in such a way that their crests and troughs align perfectly. This alignment produces a new wave with a higher amplitude resulting in bright fringes in optical experiments. The concept of constructive interference is pivotal in experiments with air wedges, where the varied path difference causes light waves to either reinforce or cancel each other out.

For two waves to constructively interfere, the condition is that the path difference must be a multiple of the wavelength, i.e., \( 2t = m \lambda \), where \( t \) is the thickness of the air wedge and \( m \) is an integer. This relationship is what results in bright fringes appearing across the wedge.

Fringe Spacing

Fringe spacing refers to the distance between consecutive bright fringes in an interference pattern. It is an important measure because it allows one to deduce properties like angles and path differences in optical setups such as the air wedge experiment. For example, in our scenario, 15 fringes per centimeter leads to specific fringe spacing.

The expression for fringe spacing \( \Delta x \) can be derived under the condition \( m \lambda = 2t \), connecting it to the wedge geometry. With \( x \approx \frac{1}{15.0} \times 10^{-2} \) meters in this problem, we find the angle of the wedge, which correlates to the physical spacing of these interference fringes.

• Fringe spacing inversely relates to the number of fringes per unit length.
• Accurate calculation of fringe spacing helps in determining small angles precisely, which is often a goal in interference experiments.