How does the de Broglie wavelength of an electron change if its velocity is doubled?

The wavelength is halved. According to $\lambda = h/mv$, wavelength is inversely proportional to velocity, so doubling the speed results in a wavelength that is $1/2$ the original value.

What is the difference between the evidence for light as a particle and electrons as a wave?

Light's particle nature is evidenced by the photoelectric effect (discrete energy packets), while the electron's wave nature is evidenced by diffraction patterns (interference). One shows quantization of energy, the other shows spatial spreading and superposition.

Why is a thin layer of graphite used in electron diffraction experiments instead of a standard slit?

The de Broglie wavelength of an electron is extremely small (roughly $10^{-10}$ m). Graphite has interatomic spacings of a similar magnitude, acting as a natural diffraction grating that is small enough to cause observable diffraction.

What common error is made when calculating the speed of an electron from its kinetic energy?

Students often forget to use the mass of the electron ($9.11 \times 10^{-31}$ kg) or fail to convert energy from eV to Joules. Additionally, some mistakenly use the speed of light $c$ instead of calculating the actual velocity $v$.

How does the accelerating voltage $V$ affect the diameter of the rings on a diffraction screen?

Increasing $V$ increases the electron's kinetic energy and momentum, which decreases the de Broglie wavelength. A shorter wavelength results in less diffraction, causing the rings to have a smaller diameter.

What is the formula for the de Broglie wavelength in terms of Planck's constant and momentum?

$\lambda = \frac{h}{p}$. Here, $\lambda$ is the wavelength in meters, $h$ is Planck's constant ($6.63 \times 10^{-34}$ J s), and $p$ is the momentum in kg m/s.

Why do we not observe the wave nature of a moving car?

A car has a very large mass compared to an electron. Since $\lambda = h/mv$ and $h$ is incredibly small, the resulting wavelength is far smaller than any possible aperture, making diffraction effects impossible to detect.

What happens to the diffraction pattern if the electrons are replaced by slower-moving neutrons?

Slower neutrons have lower momentum, which results in a longer de Broglie wavelength. This would cause the diffraction rings to spread out further, increasing the diameter of the pattern on the screen.

What is the 'work done' on an electron in an accelerating tube, and how is it calculated?

The work done is the electrical potential energy converted into kinetic energy, calculated as $W = eV$. $e$ is the elementary charge ($1.6 \times 10^{-19}$ C) and $V$ is the accelerating potential difference in Volts.

When calculating wavelength, what is a common mistake regarding the square root in the energy-momentum relation?

Students often forget that momentum $p = \sqrt{2mE_k}$. This leads to the incorrect assumption that wavelength is inversely proportional to energy, when it is actually inversely proportional to the square root of energy.

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5. Waves & Particle Nature of Light

The Wave Nature of Electrons

Summary

The wave nature of electrons describes the phenomenon where subatomic particles, traditionally viewed as discrete matter, exhibit wave-like behaviors such as interference and diffraction. This duality is quantified by the de Broglie wavelength, which establishes that any moving particle possesses an associated wavelength inversely proportional to its momentum.

1. Definition & Core Concepts

Wave-Particle Duality: This principle states that electrons are not merely point-like particles but also behave as waves under specific conditions. While they have mass and charge, they also possess a wavelength that allows them to undergo diffraction when passing through apertures of appropriate size.
Matter Waves: Also known as de Broglie waves, these represent the probability waves associated with any moving particle. In the context of electrons, these waves describe the likelihood of finding an electron at a particular point in space, rather than a physical oscillation of a medium.
Electron Diffraction: This is the primary experimental evidence for the wave nature of matter. When a beam of electrons passes through a crystalline structure, they spread out and interfere, creating a pattern of maxima and minima similar to light passing through a diffraction grating.

Isometric diagram of an electron diffraction experiment showing an electron gun, a thin graphite target, and a fluorescent screen displaying concentric diffraction rings.

2. Underlying Principles

3. Methods & Techniques

4. Key Distinctions

5. Exam Strategy & Tips

Voltage-Wavelength Relationship: Always remember that increasing the accelerating voltage decreases the wavelength. In exams, if a question describes rings getting closer together, you should immediately associate this with a higher voltage or higher electron speed.
Unit Consistency: Ensure all values are in SI units before calculating. Mass must be in kg, velocity in m/s, and energy in Joules; often, energy is given in electronvolts (eV) and must be converted ( $1 \text{ eV} = 1.6 \times 10^{-19} \text{ J}$ ).
Significant Figures: When using Planck's constant or the mass of an electron, maintain the precision provided in the data sheet. Rounding too early in multi-step calculations (like finding velocity then wavelength) can lead to significant errors in the final answer.

6. Common Pitfalls & Misconceptions