In the context of the Young's double-slit experiment using electrons, the behavior of the interference pattern is influenced by the velocity of the electrons. When you increase the velocity of the electrons, the fringe width actually decreases. So, the correct answer to your question is option c) fringe width decreases.
Understanding Fringe Width in the Double-Slit Experiment
The Young's double-slit experiment demonstrates the wave nature of particles, such as electrons. When electrons pass through two closely spaced slits, they create an interference pattern on a screen behind the slits. This pattern consists of alternating bright and dark fringes, which are a result of constructive and destructive interference of the electron waves.
What is Fringe Width?
The fringe width (β) is defined as the distance between two consecutive bright or dark fringes in the interference pattern. It can be calculated using the formula:
Where:
- λ is the wavelength of the electrons,
- L is the distance from the slits to the screen, and
- d is the distance between the slits.
Relationship Between Velocity and Wavelength
Now, when we talk about the velocity of electrons, we need to consider how it affects their wavelength. According to de Broglie's hypothesis, the wavelength (λ) of a particle is inversely proportional to its momentum (p), which is the product of mass (m) and velocity (v):
Here, h is Planck's constant. As the velocity of the electrons increases, their momentum increases, leading to a decrease in wavelength. This is crucial because the fringe width is directly proportional to the wavelength.
Impact of Increased Velocity on Fringe Width
When you increase the velocity of the electrons:
- The wavelength (λ) decreases.
- Since fringe width (β) is directly proportional to λ, a decrease in λ results in a decrease in β.
Thus, as the velocity of the electrons increases, the fringe width decreases, leading to a more closely spaced interference pattern on the screen. This relationship highlights the wave-particle duality of electrons and how their wave properties can be manipulated by changing their velocity.
Summary of Key Points
- A decrease in wavelength results in a decrease in fringe width.
- The correct answer to the question is option c) fringe width decreases.
Understanding these concepts not only deepens your grasp of quantum mechanics but also illustrates the fascinating interplay between wave and particle behavior in the realm of subatomic particles.
