Problem 98 - Master Resource Book in JEE Main Physics

Question

This question has Statement I and Statement II. Of the four choices given the statements, choose the one that describes the two statements. Statement I Davisson-Gerner experiment established the wave nature of electrons. Statement II If electrons have wave nature, they can interfere and show diffraction. (a) Statement 1 is false, Statement 11 is true. (b) Statement 1 is true, Statement \(\|\) is false. (c) Statement 1 is true, Statement 11 is true, Statement \(\|\) is the correct explanation for statement 1 (d) Statement 1 is true, statement 11 is true, statement \(\mathrm{ll}\) is not the correct explanation of statement 1 .

Step-by-Step Solution

Answer

(c) Statement I is true, Statement II is true, and it is the correct explanation.

Step 1: Understanding Statement I

Statement I claims that the Davisson-Germer experiment established the wave nature of electrons. This experiment showed electrons can produce an interference pattern when scattered from a crystal, confirming their wave-like behavior, which aligns with the de Broglie hypothesis suggesting particles have wave characteristics.

Step 2: Understanding Statement II

Statement II states that if electrons have wave nature, they can interfere and show diffraction. This is a correct description of wave behavior, as interference and diffraction are phenomena that occur when waves encounter obstacles or openings.

Step 3: Evaluating Statements I and II

Both Statement I and Statement II are individually true. Statement I describes a historical experiment that provided evidence of the wave nature of electrons, while Statement II provides a general explanation of what wave properties entail, specifically with wave interference and diffraction.

Step 4: Analyzing the Explanation Link

Statement II seems to logically support Statement I, as if electrons have wave properties (as confirmed by experiments like Davisson-Germer), they should naturally exhibit interference and diffraction patterns. This reasoning makes Statement II a valid explanation for Statement I.

Step 5: Determining the Correct Answer

Given the analysis, Statement I is true, Statement II is true, and Statement II provides the correct explanation for why Statement I is true. Thus, the best answer choice is (c).

Key Concepts

Wave Nature of Electrons

In the early 20th century, scientists began to uncover fascinating aspects of particle behavior, one of which was the wave nature of electrons. This surprising phenomenon was distinctly highlighted in the Davisson-Germer experiment.
- **What does 'wave nature' mean?** When we say electrons have wave nature, we are essentially asserting that electrons, which we typically think of as particles, can also exhibit characteristics typical of waves.
- This dual behavior suggests that under certain conditions, electrons can spread out much like water waves, rather than moving in straight paths as conventional particles do.
This concept challenges the conventional classical mechanics which described particles and waves as two distinct entities. It opened up a whole new understanding in physics, indicating that microscopic particles like electrons have a "wave-particle duality."
Thus, electrons do not always just "move" from point A to point B; instead, they have properties that can interfere and show patterns under certain conditions.

Electron Diffraction

Electron diffraction is a direct result of the wave-like behavior of electrons. It occurs when a beam of electrons passes through a crystal, or encounters an obstacle, creating a distinct pattern.
- **So, why does this happen?** Electron diffraction is akin to light bending around obstacles, which causes interference patterns in classical wave experiments.
When electrons, envisioned as waves, pass through an object like a metal crystal, they undergo diffraction because they encounter a periodic atomic structure. This structured obstacle causes the electrons to bend and spread, much like how waves might spread when meeting a barrier.
The Davisson-Germer experiment showed that, unlike a single path of travel, electrons get diffracted at angles where constructive interference takes place. This leads to maxima in an interference pattern, a phenomenon typical of wave activity, but surprising for electrons at that time.

Interference Pattern

The interference pattern is a key feature illustrating wave properties. When waves overlap, they can interfere, combining in ways that build up (constructive interference) or cancel out (destructive interference).
- **In the context of electrons:** In experiments like the Davisson-Germer, electrons create patterns that resemble ripples on water. Where peaks meet peaks, enhanced intensity occurs, creating visible patterns. This is constructive interference.
- Where troughs meet peaks, the waves cancel each other out, demonstrating destructive interference, which results in less or no visible pattern.
Such results reaffirm the unexpected choice of electrons behaving not just as particles but as entities capable of interaction typical of waves. This finding was monumental in supporting the idea of wave-particle duality.

De Broglie Hypothesis

The de Broglie hypothesis was a theoretical prediction made by physicist Louis de Broglie in 1924, proposing that all matter, including electrons, exhibit wavelike properties.
- **Essence of the hypothesis:**De Broglie suggested that every particle has an associated wave with a wavelength proportional to the Planck constant divided by the particle's momentum. The equation is given by \[ \lambda = \frac{h}{p} \] where \(\lambda\) is the wavelength, \(h\) is Planck's constant, and \(p\) is the momentum.
This was a ground-breaking prediction because it introduced the notion that tiny particles like electrons could no longer be considered just small balls but needed to be understood as having wave properties as well. The subsequent evidences, such as those from the Davisson-Germer experiment, supported this hypothesis, as electrons indeed displayed wave characteristics through patterns of diffraction and interference, confirming de Broglie's revolutionary idea.