Double slit interference – problems and solutions

1. Yellow light passes through two slits and an interference pattern is observed on a screen.

(1) The bright fringes will increase in width if the yellow light is replaced blue

(2) The bright fringes will increase in width if the distance between slits minimized

(3) The intensity of light decreases if it is far from the central fringe

(4) The intensity of light is constant if it is far from the central fringe

Which is the correct statement?

Solution

*The distance between slits is smaller than the distance between the slit and screen so that angle is very small. Then,*

**The equation of double-slit interference (constructive interference****)**

*d = **distance between slits**, y =**Distance between bright line and the central fringe**, l = **distance between screen and slit**, n = orde**r**, **λ = **wavelength*

**(1) ****T****he bright ****fringes ****will increase in width if the yellow light is replaced blue**

Based on the above equation, the number of bright lines (n) is inversely proportional to the wavelength (λ). If the wavelength decreases, the number of bright lines (n) increases. Yellow light has a larger wavelength (smaller frequency) than blue light. If the yellow light is changed blue, the wavelength decreases, so the number of bright lines (n) increases.

*This statement is correct.*

**(2) ****The bright fringes will increase in width if the distance between slits minimized**

Based on the above formula, the distance between slits (d) is directly proportional to the number of bright lines (n). If the distance between the slits is minimized, the number of bright lines (n) decreases.

*This statement is incorrect.*

**(3) **T**he intensity of light decreases if it is far from the cent****ral fringe**

Intensity relates to light level. Intensity is inversely proportional to the distance if the distance the greater the intensity the smaller (the light dimmer).

*This statement is correct.*

**(4) **The intensity of light is constant if it is far from the central fringe

*This statement is incorrect.*

2. A light falls on two slits 2-mm apart and produces on a screen 1 m away from the fourth-order bright line 1-mm from the center of the pattern. What is the wavelength of the light used?

__Known :__

Distance between slits (d) = 2 mm = 2 x 10^{-3 }m

Order (n) = 4

Distance between screen and slit (l) = 1 meter

Distance between the fourth-order bright line and the center of the pattern (y) = 1 mm = 1 x 10^{-3 }m = 10^{-3 }m

__Wanted :__ Wavelength (λ)

__Solution :__

The equation of the double slit interference :

d sin θ = n λ

The wavelength of the light (λ) :

3. Two slits 3-mm apart, 1 meter from the screen. If produced the sixth-order bright line 1-mm from the center of the pattern, what is the wavelength of the light used?

__Known :__

Distance between slits (d) = 3 mm = 3 x 10^{-3 }m

Order (n) = 6

Distance between screen and slit (l) = 1 meter

Distance between the sixth-order bright line and the center of the pattern (y) = 1 mm = 1 x 10^{-3 }m = 10^{-3 }meter

__Wanted :__ The wavelength of the light (λ)

__Solution :__

The wavelength of the light (λ)

**1. Question:** What is the fundamental principle behind double-slit interference?

**Answer:** Double-slit interference arises from the superposition of waves emanating from two closely spaced slits, leading to regions of constructive and destructive interference.

**2. Question:** How are bright and dark fringes formed on the screen?

**Answer:** Bright fringes (maxima) result from constructive interference when wave crests overlap, while dark fringes (minima) result from destructive interference when a crest from one slit overlaps with a trough from the other.

**3. Question:** Why is monochromatic light typically used in the double-slit experiment?

**Answer:** Monochromatic light ensures a consistent wavelength, producing a clear and stable interference pattern.

**4. Question:** How does changing the wavelength of light affect the interference pattern?

**Answer:** Increasing the wavelength increases fringe spacing, whereas decreasing the wavelength decreases it.

**5. Question:** How does the spacing between the two slits influence the interference pattern?

**Answer:** The greater the spacing between the slits, the closer together the interference fringes become on the screen.

**6. Question:** Can double-slit interference be observed with particles like electrons?

**Answer:** Yes, particles such as electrons demonstrate wave-particle duality, showing interference patterns similar to light when passed through double slits.

**7. Question:** What is the path difference between waves arriving at a bright fringe?

**Answer:** The path difference at a bright fringe is an integral multiple of the wavelength, such as 0, λ, 2λ, etc.

**8. Question:** How does the distance between the double slits and the screen influence the interference pattern?

**Answer:** As the distance between the slits and the screen increases, the separation between the fringes becomes larger.

**9. Question:** What does the central maximum represent in the interference pattern?

**Answer:** The central maximum is the brightest spot directly opposite the double slits, where waves from both slits travel the same distance to the screen, resulting in constructive interference.

**10. Question:** Can the interference pattern be seen with the naked eye?

**Answer:** Typically, the interference fringes are too close together to be resolved with the naked eye, so a viewing screen or detector is used.

**12. Question:** What happens to the interference pattern when the source of light is replaced with a white light source?

**Answer:** White light produces a central white fringe flanked by colored fringes because different wavelengths interfere at slightly different locations.

**13. Question:** How is coherence related to double-slit interference?

**Answer:** For a clear interference pattern, the light sources from the two slits must be coherent, meaning they maintain a constant phase relationship.

**14. Question:** Why are single-photon experiments significant in understanding double-slit interference?

**Answer:** Single-photon experiments demonstrate that individual photons can interfere with themselves, further supporting the wave-particle duality concept.

**15. Question:** How does the width of each slit influence the interference pattern?

**Answer:** A wider slit will cause broader diffraction patterns, which will overlap and modify the interference pattern’s visibility and sharpness.

**16. Question:** What role does the principle of superposition play in interference?

**Answer:** The principle of superposition states that the total displacement of overlapping waves is the sum of their individual displacements. This leads to constructive and destructive interference in double-slit experiments.

**17. Question:** Can double-slit interference occur with other types of waves besides light?

**Answer:** Yes, interference is a fundamental wave phenomenon and can be observed with sound waves, water waves, and even matter waves like electrons.

**18. Question:** What happens to the interference pattern when one of the slits is covered?

**Answer:** Covering one slit will eliminate the interference pattern, and only a diffraction pattern from the single open slit will be observed.

**19. Question:** How is the double-slit experiment related to the wave-particle duality concept?

**Answer:** The double-slit experiment shows that particles such as electrons can exhibit wave-like behavior (interference) and particle-like behavior (detectable as individual particles), underscoring wave-particle duality.

**20. Question:** What happens to the interference pattern when particles are observed through which slit they pass?

**Answer:** The act of observation collapses the wave function, and the interference pattern disappears, illustrating the principle of quantum mechanics that the observer affects the observed.

Understanding the double-slit interference is foundational in both classical wave optics and quantum mechanics, revealing the intricate nature of light and matter.

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