Question

Coherent light with wavelength 450 nm falls on a pair of slits. On a screen 1.80 m away, the distance between dark fringes is 3.90 mm. What is the slit separation?

Short answer

The slit separation is approximately 2.08 x 10^{-4} m.

Step-by-step solution

Understand the Concept

The problem involves two-slit interference. The formula for the position of dark fringes in a double-slit experiment is given by: \[ d \sin \theta = (m + \frac{1}{2}) \lambda \] where \( d \) is the slit separation, \( \theta \) is the angle of the fringe, \( m \) is the order number, and \( \lambda \) is the wavelength of light. The distance between fringes on the screen can also be related to the geometry of the setup.

Identify Known Values

We are given:- Wavelength, \( \lambda = 450 \) nm = \( 450 \times 10^{-9} \) m- Distance between the screen and the slits, \( L = 1.80 \) m- Distance between dark fringes, \( \Delta y = 3.90 \) mm = \( 3.90 \times 10^{-3} \) mThese values will be used in calculations to determine the slit separation.

Relate Fringe Spacing to Given Values

For small angles, \( \sin \theta \approx \tan \theta \times \theta \approx \frac{y}{L} \). The distance between two consecutive dark fringes (fringe spacing) is given by:\[ \Delta y = \frac{\lambda L}{d} \]This formula relates the fringe spacing, wavelength, slit separation, and distance between the screen and slits.

Solve for Slit Separation

Re-arranging the equation \( \Delta y = \frac{\lambda L}{d} \) for \( d \), we have:\[ d = \frac{\lambda L}{\Delta y} \]Substitute the known values:\[ d = \frac{450 \times 10^{-9} \times 1.80}{3.90 \times 10^{-3}} \]

Calculate the Result

Carrying out the calculation:\[ d = \frac{450 \times 10^{-9} \times 1.80}{3.90 \times 10^{-3}} = \frac{810 \times 10^{-9}}{3.90 \times 10^{-3}} \approx 2.08 \times 10^{-4} \text{ m} \]Thus, the slit separation \( d \) is approximately \( 2.08 \times 10^{-4} \) meters.

Key concepts

Wavelength

In the context of the double-slit interference experiment, the wavelength is a key component that describes the distance between consecutive peaks (or troughs) of a wave. Wavelength is usually denoted by the Greek letter \( \lambda \) and is often measured in nanometers (nm) for visible light.

The wavelength in this exercise is 450 nm, equivalent to \( 450 \times 10^{-9} \) meters. This value indicates the color of the light, which in this case, is in the blue spectrum. Different wavelengths will affect how light interferes with itself, changing the pattern of light and dark bands observed on the screen. Shorter wavelengths will typically lead to closer spacing of interference fringes, while longer wavelengths result in wider spacing.

Understanding the wavelength allows us to predict the behavior of light in various interference setups, such as through different slit separations or varying distances to the screen.

Fringe Spacing

Fringe spacing refers to the distance between two consecutive bright or dark fringes on the screen in a double-slit experiment. This distance, often denoted by \( \Delta y \), is determined by the interaction of coherent light waves as they superimpose on one another after passing through the slits.

In this exercise, the fringe spacing is 3.90 mm, or \( 3.90 \times 10^{-3} \) m. It represents how the interference pattern spreads out on the screen. Larger fringe spacing occurs when the interference pattern is spread more widely, which can be the result of varying factors such as the slit separation, the wavelength of the light, or the distance to the screen.

• Formulas: The relationship between fringe spacing and other factors can be expressed as \( \Delta y = \frac{\lambda L}{d} \). This equation considers the wavelength \( \lambda \), slit separation \( d \), and the screen distance \( L \).
• Factors Influencing Fringe Spacing: Changes in wavelength or slit separation have a direct impact on \( \Delta y \), altering the interference pattern observed.

Slit Separation

Slit separation, denoted as \( d \), is an essential factor in determining the pattern created in a double-slit experiment. The distance between the two slits affects how the light waves emerge and interfere with each other.

In this exercise, solving for the slit separation involves using the formula \( d = \frac{\lambda L}{\Delta y} \). By substituting the known values of wavelength (450 nm), screen distance (1.80 m), and fringe spacing (3.90 mm), the slit separation is found to be approximately \( 2.08 \times 10^{-4} \) meters.

• As the slit separation \( d \) decreases, the interference pattern generally becomes more spread out, leading to wider fringe spacing.
• A larger slit separation causes the fringes to appear closer together on the screen.
• Understanding and calculating slit separation is crucial for predicting and manipulating the interference pattern of waves.

Screen Distance

The distance from the slits to the screen, denoted as \( L \), is a critical factor in determining the size and distribution of the interference pattern in a double-slit experiment. This parameter influences how the light waves overlap after passing through the slits.

For this problem, the screen distance is given as 1.80 meters. An increase in this distance tends to spread the fringes farther apart, leading to larger fringe spacing. Conversely, decreasing the screen distance would bring the fringes closer together, decreasing fringe spacing.

• Geometric Relationship: The angle \( \theta \) at which lights interfere is small, so \( \sin \theta \approx \tan \theta \approx \frac{y}{L} \).
• Impact on Pattern: Changes in the screen distance directly affect the interference spacing, thus influencing which order of fringes might be visible.

Understanding screen distance and its relationship with other factors such as wavelength, fringe spacing, and slit separation helps in designing experiments to observe specific interference patterns.