Chapter 6: Problem 42
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
Step by step solution
Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Particle Accelerators
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.
Momentum Calculation
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.
Planck Constant
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.