Question

A parallel beam of fast moving electrons is incident normally on a narrow shit. A fluorescent screen is placed at a large distance from the slit. If the speed of the electrons is increased, which of the following statements is correct?

• A. Diffraction pattern is not observed on the screen in the case of electrons
• B. The angular width of the central maximum of the diffraction pattern will increase
• C. The angular width of the central maximum will decrease
• D. The angular width of the central maximum will be unaffected

Answer 1

The Correct Option is C

To understand the behavior of the diffraction pattern in this scenario, we must consider the principles of wave-particle duality and the concept of electron diffraction. Particularly, the concept of de Broglie wavelength is crucial here. The de Broglie wavelength \(\lambda\) of a particle is given by:

\(\lambda = \frac{h}{p}\)

where:

• \(h\) is Planck's constant (\(6.626 \times 10^{-34}\) Js).
• \(p\) is the momentum of the electron, given by \(p = mv\), where \(m\) is the mass of the electron and \(v\) is its velocity.

As the speed \(v\) of the electrons increases, their momentum \(p\) also increases. Therefore, from the de Broglie equation, we see that an increase in momentum leads to a decrease in the de Broglie wavelength \(\lambda\).

The angular width (\Delta \theta) of the central maximum in a single-slit diffraction pattern for a wave of wavelength \(\lambda\) can be determined using the formula:

\(\Delta \theta = \frac{2\lambda}{a}\)

where \(a\) is the width of the slit.

From this relation, it is evident that the angular width \(\Delta \theta\) is directly proportional to the wavelength \(\lambda\). Thus, if the wavelength \(\lambda\) decreases due to an increase in the speed of the electrons, the angular width of the central maximum will decrease as well.

Therefore, the correct statement is: The angular width of the central maximum will decrease.