Question

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

Short answer

Question: Based on the three scenarios discussed above, describe how the interference pattern changes when (a) the wavelength of light is increased, (b) the slit separation is increased, and (c) the experimental apparatus is placed in water. Answer: (a) When the wavelength of light is increased, the interference pattern becomes more spread out. (b) When the slit separation is increased, the interference pattern becomes more compact. (c) When the experimental apparatus is placed in water, the interference pattern also becomes more compact.

Step-by-step solution

Recap of double slit interference formula

For the double-slit interference pattern, the formula relating the distance between the central bright fringe and the adjacent bright fringe is given by: \(y = \frac{m\lambda L}{d}\) where: - \(y\) is the distance between the central bright fringe and the adjacent bright fringe. - \(m\) is the order of the bright fringe (integer values). - \(\lambda\) is the wavelength of the light. - \(L\) is the distance between the double-slit and the screen. - \(d\) is the separation between the slits.

Part (a) - Increasing the wavelength

If the wavelength (\(\lambda\)) is increased, the distance between the bright fringes (\(y\)) would also increase, according to the formula: \(y = \frac{m\lambda L}{d}\) As a result, the interference pattern would become more spread out.

Part (b) - Increasing the slit separation

If the separation between the slits (\(d\)) is increased, the distance between the bright fringes (\(y\)) would decrease, according to the formula: \(y = \frac{m\lambda L}{d}\) As a result, the interference pattern would become more compact.

Part (c) - Placing the experimental apparatus in water

If the experimental apparatus is placed in water, the light's wavelength (\(\lambda\)) would decrease as it enters the medium with a higher refractive index. As water has a refractive index greater than 1, the light's speed becomes slower and its wavelength decreases. The new wavelength \(\lambda_n\) is given by: \(\lambda_n = \frac{\lambda}{n}\) where: - \(\lambda_n\) is the new wavelength in water. - \(n\) is the refractive index of water. Using the formula for the interference pattern distance, \(y = \frac{m\lambda L}{d}\), we can deduce that the distance between the bright fringes (\(y\)) would also decrease. As a result, the interference pattern would become more compact.

Key concepts

Wave Interference

Wave interference is a fundamental phenomenon occurring when two or more waves meet while traveling through the same medium. Imagine ripples on a pond colliding and momentarily forming larger, smaller, or even flat areas in the water. This is similar to what happens in double-slit interference.

When light waves pass through two close slits, they overlap and can constructively interfere, creating bright fringes, or destructively interfere, resulting in dark areas. Constructive interference amplifies the wave's effect, while destructive interference can cancel waves out. This pattern of dark and light bands is a hallmark of wave interference and demonstrates the wave nature of light.

This interaction is beautifully guided by the principle of superposition, which dictates that the resulting wave's amplitude at any point is the sum of the individual waves' amplitudes. For students visualizing this concept, it can be helpful to think of it as adding the heights and depths of waves together to see what new wave emerges.

Wavelength Influence on Diffraction

Wavelength is a key player in the symphony of diffraction. It is the distance between consecutive crests or troughs in a wave and is crucial in determining how light bends around obstacles or passes through slits, such as in the double-slit experiment.

Different wavelengths bend differently, a phenomenon which explains why shorter (blue light) and longer (red light) waves create distinctive patterns on a screen. As the wavelength increases, diffraction causes the waves to spread out more after they pass through the slits. This means that for a given slit separation, increasing the wavelength results in a more stretched interference pattern. Shorter wavelengths bend less, while longer wavelengths bend more, a concept that can be likened to how different musical notes travel through an instrument and emerge with distinct pitches.

Exercise Improvement Advice

In exercises, visual aids can be incredibly helpful; consider using diagrams or simulations to show wavefronts and interference patterns for waves of different wavelengths. This can greatly enhance students' understanding of wavelength's impact on diffraction and interference patterns.

Refractive Index Effect on Light

The refractive index of a material is a measure of how much it reduces the speed of light and, consequently, affects the light's wavelength within the material. When light enters a higher refractive index medium like water, it slows down. This slowdown shortens the wavelength of the light while the frequency remains constant, as frequency is determined by the source of the light and does not change during transmission.

This shortened wavelength in a material like water, compared to air, changes the conditions for interference in the double-slit experiment. According to the equation \(y = \frac{m\lambda L}{d}\), a smaller wavelength reduces the spacing \(y\) between the bright fringes on the screen, making the pattern more compact.

Exercise Improvement Advice

When explaining the refractive index effect on light, analogies, such as thinking of light as a car slowing down as the road becomes rougher, can make the concept more accessible. Also, emphasizing that the frequency remains the same while the speed and wavelength change can be crucial for grasping the underlying physics.