Question

What would happen to a double-slit interference pattern if a) the wavelength is increased? b) the separation distance between the slits is increased? c) the apparatus is placed in water?

Short answer

Answer: (a) When the wavelength of light is increased, the interference pattern will have more widely spaced bright fringes. (b) When the distance between the slits is increased, the interference pattern will have more closely spaced bright fringes. (c) When the apparatus is placed in water, the interference pattern will also have more closely spaced bright fringes due to the decreased wavelength in the medium.

Step-by-step solution

Relate wavelength and interference pattern

The interference pattern is determined by the constructive and destructive interference of the light waves that pass through the two slits. The condition for constructive interference is given by: nλ = d sin(θ) where n is the order of interference, λ is the wavelength of the light, d is the separation between the slits, and θ is the angle of the interference.

Analyze the effect of increasing the wavelength

When the wavelength of the light is increased, according to the constructive interference condition, either the angle θ will increase, or the order of interference (n) will decrease for the given separation distance (d). In other words, the bright fringes in the interference pattern will be more widely separated when the wavelength is increased. b) The separation distance between the slits is increased

Relate separation distance and interference pattern

Again, considering the same constructive interference condition mentioned above, we can analyze the effect of increasing the separation distance (d) between the slits on the interference pattern.

Analyze the effect of increasing the separation distance

When the separation distance (d) is increased, the angle θ for constructive interference will decrease for a given wavelength and order of interference. This means the bright fringes in the interference pattern will be more closely spaced when the separation distance is increased. c) The apparatus is placed in water

Determine the effect of the medium on the wavelength

When light travels through a medium other than vacuum, its speed and wavelength change. The wavelength in a medium is related to the wavelength in a vacuum (λ₀) as follows: λ = λ₀ / n_m where λ is the wavelength in the medium, and n_m is the refractive index of the medium.

Determine the refractive index of water

The refractive index of water is approximately 1.33.

Analyze the effect of placing the apparatus in water

When the apparatus is placed in water, the wavelength of the light in the medium will decrease. As the refractive index of water is greater than 1, the wavelength in water will be smaller than the wavelength in a vacuum. From the analysis in part a), we know that decreasing the wavelength will result in a narrower interference pattern with closely spaced bright fringes. In summary, if the wavelength is increased or the separation distance is increased, the interference pattern will have more widely spaced bright fringes. If the apparatus is placed in water, the interference pattern will have more closely spaced bright fringes.

Key concepts

Constructive and Destructive Interference

Understanding the phenomena of constructive and destructive interference is crucial to interpreting the workings of a double-slit experiment. When two light waves cross paths, they superimpose, creating a resultant wave whose amplitude can be greater or smaller, depending on the alignment of the two waves.

Constructive Interference

Occurs when the crests and troughs of two overlapping waves coincide. At this point, their amplitudes add up, resulting in a brighter or more intense light spot, known as a bright fringe in the interference pattern. This amplification is governed by the formula: λ = d sin(θ), where each variable has a distinct role in the interference pattern.

Destructive Interference

On the other hand, when the crest of one wave aligns with the trough of another, they cancel each other out. This leads to darkness or a less intense spot, known as a dark fringe. For complete cancellation, the path difference between the two waves must equal an odd multiple of half the wavelength (e.g., (2n + 1)λ/2).

• Constructive interference results in increased light intensity (bright fringes).
• Destructive interference results in decreased light intensity (dark fringes).
• The angles at which these occur are crucial in determining the spacing of the fringes in the interference pattern.

Learning to apply these principles will enable students to predict alterations in the interference pattern when experimental conditions, such as the wavelength of light or separation distance between the slits, are changed.

Wavelength of Light

Light can be characterized by its wavelength, which is the distance between two consecutive peaks (crests) or troughs of a light wave. The wavelength of light not only determines its color but also influences its behavior in phenomena like diffraction and interference.

Impact of Light's Wavelength on Interference

When you alter the wavelength in a double-slit experiment, you're directly affecting the interference pattern produced. As mentioned in the exercise's solution, increasing the wavelength of light leads to a larger separation between the bright fringes. This is because the increased wavelength affects the constructive interference conditions, requiring a larger angle to satisfy the equation λ = d sin(θ), effectively spreading out the bright fringes of the pattern.

• An increase in wavelength results in wider spacing between bright fringes.
• A decrease in wavelength leads to a more compact interference pattern.
• The wavelength is a fundamental aspect when predicting how interference patterns will shift under various conditions.

Refractive Index

The refractive index of a medium quantifies how much the speed of light is reduced compared to its velocity in a vacuum. It is a dimensionless number indicating the degree of bending, or refraction, light undergoes when passing from one medium to another.

Refractive Index's Effect on Wavelength

The equation λ = λ₀ / n_m explains how the wavelength of light changes when it enters a medium with a refractive index, n_m. Since the refractive index of water is around 1.33, the actual wavelength λ in water will be shorter than the wavelength in a vacuum λ₀. This means that when the double-slit experiment is immersed in water, the interference pattern's bright fringes move closer together, resulting in a more concentrated pattern.

• Higher refractive index implies a slower speed of light in the medium.
• As refractive index increases, the wavelength in the medium decreases.
• Changes in the medium can lead to significant adjustments in the interference pattern's appearance.

Understanding how the refractive index influences the wavelength of light is essential for comprehending the subsequent effects on double-slit interference patterns.