Animated Solution for Physics - Dual Nature of Matter and Radiation: A beam of electron is used in an YDSE experiment. The slit width is $d$. When the velocity of electron is increased, then

Imagine a Young's Double Slit Experiment (YDSE), but instead of a beam of light, we are firing a beam of electrons. According to quantum mechanics, moving particles exhibit wave-like properties. This phenomenon is known as wave-particle duality. When the electron beam passes through the double slits, the matter waves interfere with each other, creating a distinct interference pattern on the screen.

To understand the interference pattern, we first need to determine the wavelength of the electron waves. According to de-Broglie's hypothesis, the wavelength $\lambda$ of a moving particle is inversely proportional to its momentum $p$.

The relationship is given by the equation:

Since momentum is the product of mass $m$ and velocity $v$, we can rewrite the de-Broglie wavelength as:

This equation tells us that if the velocity of the electrons increases, their de-Broglie wavelength must decrease.

Next, let's recall the formula for the fringe width in a standard YDSE setup. The fringe width $\beta$, which is the distance between two consecutive bright or dark fringes, depends on the wavelength $\lambda$, the distance to the screen $D$, and the slit separation $d$.

The formula is:

This shows that the fringe width is directly proportional to the wavelength of the interfering waves.

Now, let's combine our two key insights. By substituting the expression for the de-Broglie wavelength into the fringe width formula, we get a direct relationship between the electron's velocity and the fringe width:

From this combined equation, it is clear that the fringe width $\beta$ is inversely proportional to the velocity $v$ of the electrons.

Therefore, when the velocity of the electron beam is increased, the momentum increases, causing the de-Broglie wavelength to decrease. A smaller wavelength leads to a tighter interference pattern. Consequently, the fringe width decreases.

This elegant problem beautifully connects the kinetic properties of a particle with its wave-like interference behavior!