Chapter 39: Problem 63
(a) What is the energy of a photon that has wavelength 0.10 \(\mu\)m ? (b) Through approximately what potential difference must electrons be accelerated so that they will exhibit wave nature in passing through a pinhole 0.10 \(\mu\)m in diameter? What is the speed of these electrons? (c) If protons rather than electrons were used, through what potential difference would protons have to be accelerated so they would exhibit wave nature in passing through this pinhole? What would be the speed of these protons?
Short Answer
Step by step solution
Energy of Photon
Electron Potential Difference
Speed of the Electrons
Proton Potential Difference
Speed of the Protons
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
De Broglie Wavelength
- \( \lambda \) is the de Broglie wavelength,
- \( h \) is Planck's constant \( 6.63 \times 10^{-34} \text{ Js} \),
- \( m \) is the mass of the particle, and
- \( v \) is the velocity of the particle.
For instance, to reveal the wave nature of electrons passing through a pinhole with a size of 0.10 \( \mu \)m, one must use their de Broglie wavelength. This governs the behavior and properties when their size becomes comparable to their de Broglie wavelength.
Electron Acceleration
By applying a potential difference \( V \), electrons obtain kinetic energy, which influences their de Broglie wavelength. The crucial relationship for energy acquired by accelerated electrons is given by \( eV = \frac{1}{2}mv^2 \), where:
- \( e \) is the electronic charge \( 1.6 \times 10^{-19} \text{ C} \),
- \( V \) is the potential difference in volts,
- \( m \) is the electron mass \( 9.11 \times 10^{-31} \text{ kg} \), and
- \( v \) is the speed of the electron.
In practice, to observe electrons' wave behavior, you accelerate them through a calculated potential difference.This pushes their de Broglie wavelength to align with the scale of the experimental constraints, such as the size of a diffraction grating or a pinhole.
Proton Acceleration
- For protons, the equation \( eV = \frac{1}{2}mv^2 \) still applies, albeit modified for their increased mass.
This means subatomic calculations require tailoring, as increasing mass affects acceleration and thus changes the kinetic energy equation.In the research example, a much higher potential difference of \( 8.4 \times 10^4 \text{ V} \) is required to match the de Broglie wavelength needed for protons to pass through a 0.10 \( \mu \)m pinhole.
Planck's Constant
As part of the de Broglie relationship \( \lambda = \frac{h}{mv} \), it showcases how wave-like behavior is possible in particles. In photon energy calculation, it is used with the speed of light \( c \) and wavelength \( \lambda \) as \( E = \frac{hc}{\lambda} \).
Ultimately, Planck's constant reflects the limitations of classical mechanics, paving the way to the advent of quantum theory.
Potential Difference
- This is vital for inducing wave behavior in particles, where their de Broglie wavelengths become prominent.
- The potential difference specifies the energy a particle gains or loses, impacting both speed and wavelength.
Speed of Particles
- Here, \( e \) represents the charge of the particle, and \( m \) its mass.
As the velocity of a particle increases, its wavelength (based on the de Broglie concept) decreases.
This allows high speeds to be achievied due to significant potential differences, which results in high-energy electron beams useful in many practical applications like electron microscopy or proton therapy in medicine.