Question

Coherent light with wavelength 400 nm passes through two very narrow slits that are separated by 0.200 mm, and the interference pattern is observed on a screen 4.00 m from the slits. (a) What is the width (in mm) of the central interference maximum? (b) What is the width of the first-order bright fringe?

Short answer

The width of the central maximum is 16 mm, and the width of the first-order bright fringe is 8 mm.

Step-by-step solution

Understanding the Problem

We have light with a wavelength of 400 nm passing through two slits separated by 0.200 mm. We need to find the width of the central interference maximum and the first-order bright fringe when observed on a screen 4.00 m away.

Convert Wavelength to Meters

Convert the wavelength from nanometers to meters:Given: \( \lambda = 400 \text{ nm} = 400 \times 10^{-9} \text{ m} \).

Apply the Double-Slit Interference Formula

For the central maximum, the formula for fringe separation is:\[ y = \frac{\lambda L}{d} \]where: - \( y \) is the distance from the central maximum to the next bright fringe. - \( \lambda = 400 \times 10^{-9} \text{ m} \) (wavelength of light) - \( L = 4.00 \text{ m} \) (distance to screen) - \( d = 0.200 \times 10^{-3} \text{ m} \) (separation between slits)

Calculate the Distance between the Central Maximum and First Bright Fringes

Insert the values into the formula:\[ y = \frac{(400 \times 10^{-9} \text{ m}) \times 4.00 \text{ m}}{0.200 \times 10^{-3} \text{ m}} = 0.008 \text{ m} = 8 \text{ mm} \]Therefore, the distance from the central maximum to the first bright fringe is 8 mm.

Determine the Width of the Central Maximum

The width of the central maximum is twice the distance from the central maximum to the first bright fringe:\[ \text{Width of central maximum} = 2 \times 8 \text{ mm} = 16 \text{ mm} \].

Determine the Width of the First-order Bright Fringe

The first-order bright fringe is the same as the distance from the center line (central maximum) to one first-order maximum, which we calculated as 8 mm.

Key concepts

Wavelength of Light

The wavelength of light is a key characteristic that determines how light behaves when it encounters obstacles like slits. In this exercise, we use a wavelength of 400 nanometers (nm). To grasp this, remember that a nanometer is one billionth of a meter. Wavelengths in the visible spectrum range from about 380 nm to 750 nm. This is a region of the electromagnetic spectrum that human eyes can detect, giving rise to different colors.
The wavelength impacts the interference pattern we see. If you change the wavelength, you can see shifts in the pattern, as it affects the precise locations of the bright and dark bands formed on the screen.
To convert wavelengths from nanometers to meters, a simple calculation is needed: Multiply by \(10^{-9}\). So: \(400 \text{ nm} = 400 \times 10^{-9} \text{ meters}\). Understanding how to convert and use the wavelength is vital when exploring how light behaves in interference set-ups.

Interference Pattern

The interference pattern is a fascinating concept resulting from the phenomenon of two or more waves overlapping. In the context of double-slit interference, this happens when coherent light waves pass through two narrow slits. Coherent light means the light waves are in phase, which means they peak and trough simultaneously, vital to creating a stable pattern.
The pattern itself consists of alternating bright and dark bands or fringes on a screen. This is produced by the constructive and destructive interference of light waves:

• Constructive interference happens when peaks of waves align, forming bright fringes.
• Destructive interference occurs when peaks align with troughs, creating dark fringes.

The distance between these fringes, sometimes referred to as fringe width, is determined by the separation between the slits, the wavelength of the light, and the distance to the screen. This creates a predictable pattern that provides valuable insight into the properties of light waves.

Central Interference Maximum

The central interference maximum is the very heart of the interference pattern, located directly in the center. It’s the brightest spot on the interference pattern because this is where light from both slits travels the same distance to meet on the screen, resulting in perfect constructive interference.
To determine its width, we consider the distance between this central bright fringe to the first-order bright fringes on either side. Formulas used in the exercise show us how to calculate these using the variables of wavelength, slit separation, and distance to the screen:
\[ y = \frac{\lambda L}{d} \]
Where \( y \) is the distance between the central maximum and the first bright fringe. Doubling this distance gives us the width of the central maximum. In our exercise, this is calculated to be 16 mm. The understanding of the central maximum helps in determining the scale and structure of the entire pattern, shedding light on how continuous waves interact in a confined space.