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Protons can be accelerated to speeds near that of light in particle accelerators. Estimate the wavelength (in \(\mathrm{nm})\) of such a proton moving at \(2.90 \times 10^{8} \mathrm{~m} / \mathrm{s}\) (mass of a proton \(\left.=1.673 \times 10^{-27} \mathrm{~kg}\right)\)

Short Answer

Expert verified
The wavelength of the proton is approximately \(1.37 \times 10^{-6}\) nm.

Step by step solution

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01

Identify de Broglie Wavelength Formula

The de Broglie wavelength, \( \lambda \), of a particle is given by the formula: \( \lambda = \frac{h}{p} \) where \( h \) is the Planck constant \( (6.626 \times 10^{-34} \text{ J s}) \) and \( p \) is the momentum of the particle.
02

Calculate the Momentum

The momentum \( p \) of the proton can be calculated using the formula: \( p = mv \), where \( m = 1.673 \times 10^{-27} \text{ kg} \) is the mass of the proton and \( v = 2.90 \times 10^{8} \text{ m/s} \) is its velocity. Substituting these values gives \( p = 1.673 \times 10^{-27} \times 2.90 \times 10^{8} = 4.8517 \times 10^{-19} \text{ kg m/s} \).
03

Apply the de Broglie Formula

Substitute \( h = 6.626 \times 10^{-34} \text{ J s} \) and \( p = 4.8517 \times 10^{-19} \text{ kg m/s} \) into the de Broglie equation to find \( \lambda \):\[\lambda = \frac{6.626 \times 10^{-34}}{4.8517 \times 10^{-19}} \approx 1.37 \times 10^{-15} \text{ m}\]
04

Convert Wavelength to Nanometers

To convert the wavelength from meters to nanometers, use the conversion factor: \( 1 \text{ m} = 10^{9} \text{ nm} \). Thus, \( \lambda \approx 1.37 \times 10^{-15} \times 10^{9} = 1.37 \times 10^{-6} \text{ nm} \).

Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Particle Accelerators
Particle accelerators are fascinating devices used to speed up subatomic particles to incredibly high speeds, often to a fraction of the speed of light. These machines are integral to modern physics research as they allow scientists to probe the fundamental constituents of matter.
When a proton is accelerated in such a device, it gains significant kinetic energy. This energy allows particles to be smashed into targets or among themselves, revealing new particles and enabling detailed study of the forces and components at play.
Particle accelerators come in different types, with linear accelerators (linacs) and circular accelerators (synchrotrons or cyclotrons) among the most common. Each type has a specific application depending on the energy required and the experiments being conducted. In these machines, protons, and other particles, are accelerated by electric fields and kept in paths by magnetic fields. During acceleration, relativistic effects take over, especially as speeds approach the speed of light.
  • Used in medical therapies for cancer treatment.
  • Vital in discovering fundamental particles like the Higgs boson.
  • Contribute to advancements in material sciences and chemistry.
Understanding these accelerators provides insight into the journey of particles like protons that help unravel the mysteries of the universe.
Momentum Calculation
Calculating momentum is crucial in understanding the behavior of particles in accelerators. Momentum, denoted as \( p \), is the product of a particle's mass and velocity. It underpins the calculations needed to determine a particle's de Broglie wavelength.
The formula for momentum is simple yet powerful: \( p = mv \). Here, \( m \) represents mass, and \( v \) is velocity. For a proton moving at near-light speeds, this velocity significantly affects the momentum.
To get accurate results, units must be consistent. For example, mass should be in kilograms (kg), and velocity in meters per second (m/s). This consistency ensures that the momentum is calculated in units of kg m/s.
  • In particle physics, precise momentum calculations are essential for predictions and experiments.
  • The velocity of protons in accelerators can reach astounding levels, amplifying their momentum tremendously.
  • This momentum plays a critical role in determining the interactions and behaviors of particles during collisions.
For students, mastering momentum calculations is a stepping stone to understanding various physical phenomena.
Planck Constant
The Planck constant is a fundamental quantity in physics, especially pivotal in quantum mechanics. It acts as a bridge between the macroscopic observations of classical physics and the microscopic behaviors examined in quantum physics.
Represented by \( h \), it has a value of approximately \( 6.626 \times 10^{-34} \text{ J s} \). This constant appears in Heisenberg's uncertainty principle and is a crucial component of the de Broglie wavelength formula, \( \lambda = \frac{h}{p} \).
The influence of the Planck constant stretches far beyond von de Broglie's hypothesis; it sets the scale at which quantum effects become significant. It defines the quantization of physical quantities and introduces the concept that energy levels are discrete.
  • Central to the theory outlining the particle-wave duality of quantum objects.
  • Helps calculate energy in photons through the relation \( E = hf \), where \( f \) is frequency.
  • Sets the groundwork for technologies like Quantum computers and GPS.
Understanding the Planck constant aids in grasping larger quantum theories and the dimensionality of our universe.

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