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Chapter 4: Problem 24

A hydrogen discharge in a 1-T feld produces a density of \(10^{16} \mathrm{~m}^{-3}\). (a) What is the Alfv茅n speed \(v_{A}\) ? (b) Suppose o, had come out greater than \(c\). Does this mean that Alfv茅n waves travel faster than the speed of light?

Short Answer

Expert verified
(a) Alfv茅n speed \( v_A \approx 7.75 \times 10^6 \text{ m/s} \). (b) No, Alfv茅n waves cannot exceed the speed of light.

Step by step solution

01

Understand the Alfv茅n Speed Formula

The Alfv茅n speed \( v_A \) is given by the formula \( v_A = \frac{B}{\sqrt{\mu_0 \rho}} \), where \( B \) is the magnetic field strength, \( \mu_0 \) is the permeability of free space \( (4\pi \times 10^{-7} \text{ N/A}^2) \), and \( \rho \) is the mass density of the plasma.
02

Identify Given and Known Quantities

We are given the magnetic field strength, \( B = 1 \) T, and the number density of hydrogen atoms, \( n = 10^{16} \text{ m}^{-3} \). The mass of a hydrogen atom is approximately \( m_H = 1.67 \times 10^{-27} \text{ kg} \).
03

Calculate the Mass Density \( \rho \)

Calculate the mass density \( \rho \) using \( \rho = n \cdot m_H = 10^{16} \times 1.67 \times 10^{-27} \). This gives \( \rho \approx 1.67 \times 10^{-11} \text{ kg/m}^3 \).
04

Calculate Alfv茅n Speed \( v_A \)

Substitute the values into the Alfv茅n speed formula: \( v_A = \frac{1}{\sqrt{4\pi \times 10^{-7} \times 1.67 \times 10^{-11}}} \). Calculate \( v_A \approx 7.75 \times 10^6 \text{ m/s} \).
05

Interpret the Results for Alfv茅n Speed vs. Speed of Light

The speed of light \( c \) is approximately \( 3 \times 10^8 \text{ m/s} \), which is much higher than any possible Alfv茅n speed. If the calculated \( v_A \) were to exceed \( c \), it would indicate an error in calculation or assumptions, as Alfv茅n waves cannot physically exceed the speed of light.

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Key Concepts

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

Magnetic Field
A magnetic field is a fundamental concept in physics, representing a vector field around magnetic forces. It affects any charged particle within its vicinity, like those found in a plasma. In the provided exercise, the magnetic field is denoted as \( B \) and has the strength of 1 Tesla (T). This field is crucial for calculating the Alfv茅n speed, which measures how fast magnetic waves can pass through a plasma.
  • Magnetic Field Effects: It influences the movement and alignment of. charged particles.
  • Role in Alfv茅n Speed: It directly impacts the wave velocity, making it an essential factor in laboratory experiments and astrophysical studies.
The magnetic field is also important in various applications, from medical imaging (MRI) to industrial processes like welding. Understanding the influence of magnetic fields helps in controlling and optimizing these applications.
Plasma Density
Plasma density, symbolized by \( n \), refers to the number of particles in a unit volume of plasma, measured as number of particles per cubic meter \( \text{m}^{-3} \). In the given problem, the plasma density is provided as \( 10^{16} \text{ m}^{-3} \).
  • Significance: It indicates how concentrated the plasma components are, affecting the propagation of electromagnetic waves.
  • Relationship to Mass Density: Plasma density helps determine the mass density \( \rho \) when multiplied by the mass of a single particle, like a hydrogen atom.
Plasma density changes the characteristics of the Alfv茅n waves. In general astrophysical and laboratory settings, knowing the plasma density is crucial for predicting and modeling wave behavior.
Mass Density
Mass density \( \rho \) is the mass of particles per unit volume, measured in kilograms per cubic meter \( \text{kg/m}^3 \). It is calculated using the formula \( \rho = n \cdot m_H \), where \( n \) is the number density, and \( m_H \) is the mass of a hydrogen atom, approximately \( 1.67 \times 10^{-27} \text{ kg} \).
  • Calculation in Context: For the exercise, \( \rho \) was found to be \( 1.67 \times 10^{-11} \text{ kg/m}^3 \).
  • Importance: The mass density is fundamental in determining the Alfv茅n speed, with higher densities slowing down wave propagation.
This concept is essential in various scientific fields, including fluid dynamics and astrophysics. Mass density affects not only wave velocities but also the overall dynamics of any fluid or plasma in question.
Speed of Light
The speed of light \( c \) in vacuum is a constant \( 3 \times 10^8 \text{ m/s} \). This speed is the ultimate speed limit in the universe, according to Einstein's theory of relativity. In the context of the exercise, it is used to frame the Alfv茅n speed: even though Alfv茅n waves in strong magnetic fields and dense plasmas can approach high speeds, they cannot surpass the speed of light.
  • Physical Limitation: Alfv茅n waves traveling faster than light would contradict fundamental physics, indicating a need for re-evaluation of assumptions or calculations.
  • Contextual Role: It serves as a benchmark, reinforcing the concept that no information or matter can be transmitted faster than \( c \).
Understanding the role of the speed of light helps reinforce its significance in both theoretical and applied physics. It serves as a critical check-point to ensure the validity of calculations, especially in experimental and astrophysical physics.

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Most popular questions from this chapter

Hannes Alfwen, the first plasma physicist to be ararded the Nobel prize. has suggested that perhaps the primordial universe was symmetric between matter and antimatter. Suppose the universe was at one time a uniform mixtare of procons, antiprotons, electrons, and positrons, each species having a density \(s_{0}\) (a) Work out the dispersion relation for high-frequency electromagnetic waves in this plasma. You may oeglect colisions, annihilations, and thermal effects. (b) Work out the dispersion relation for ion waves, using Poisson's equation. You may aeglect \(T_{1}\) (but noc \(\left.T_{4}\right)\) and assume that all leptons foliow the Boltzmann relation.

to a remole part of the universe, there exists a plasma consisting of positrons and fully stripped antifermium nuclei of charge \(-Z e\), where \(Z=100\). From the equations of motion, continuity, and Poisson, derive a dispersion relation for plasma oscillations in this plasma, inctuding ion mocions. Define the plasma frequencies. You may assume \(K T=0, B_{0}=0\), and all other simplifying initial conditions.

For a Langmuir plasma oscillation, show that the time-averaged electron kioecic energy per \(\mathrm{m}^{3}\) is equal to the electric field energy densicy \(\frac{1}{2} \varepsilon_{0}\left(E^{2}\right) .\)

A space capsule making a reentry into the earth's atmosphere suffers a communications blackout because a plasma is generated by the shock wave in front of the capsule. If the radio operates at a frequency of \(300 \mathrm{MH}\). what is the minimum plasma density during the blackout?

Use Maxwell's equations and the electron equation of motion to derive. the dispersion relation for light waves propagating through a uniform, unmag. netized, collisionless, isotherenal plasma with density \(n\) and finite electron temperacure \(T_{e}\), (Ignore ton motions.)
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