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Motion Of A Charged Particle In A Magnetic Field01:22

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A charged particle experiences a force when moving through a magnetic field. Consider the field to be uniform and the charged particle to move perpendicular to it. If the field is in a vacuum, the magnetic field is the dominant factor determining the motion. Since the magnetic force is perpendicular to the direction of motion, a charged particle follows a curved path. The particle continues to follow this curved path until it forms a complete circle. Another way to look at this is that the...
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Electric Field of a Non Uniformly Charged Sphere01:22

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Gauss's law states that the electric flux through any closed surface equals the net charge enclosed within the surface. This law is beneficial for determining the expressions for the electric field for a particular charge distribution if the electric flux is known.
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Magnetic dipoles in magnetic materials are aligned when placed under an external magnetic field. For paramagnets and ferromagnets, dipole alignment occurs in the direction of the magnetic field. However, the dipoles align opposite to the field in the case of diamagnets. This state of magnetic polarization due to the external field is called magnetization. Magnetization is defined as the dipole moment per unit volume. It plays a similar role to polarization in electrostatics.
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Magnetic Field due to Moving Charges01:23

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A stationary charge creates and interacts with the electric field, while a moving charge creates a magnetic field.
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Atomic Nuclei: Nuclear Magnetic Moment00:59

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All atomic nuclei are positively charged. When they have a nonzero spin, they behave like rotating charges. As a consequence of their charge and spin, these nuclei generate a magnetic field (B). This, in turn, gives rise to a magnetic moment (μ), which is randomly oriented in the absence of an external magnetic field. When an external magnetic field (B0) is applied, the magnetic moment vectors can align with the field or against it in 2 + 1 orientations. A hydrogen nucleus, which is just a...
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In the absence of an external magnetic field, nuclear spin states are degenerate and randomly oriented. When a magnetic field is applied, the spins begin to precess and orient themselves along (lower energy) or against (higher energy) the direction of the field. At equilibrium, a slight excess population of spins exists in the lower energy state. Because the direction of the magnetic field is fixed as the z-axis,  the precessing magnetic moments are randomly oriented around the z-axis.
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Periodic bouncing modes for two uniformly magnetized spheres. I. Trajectories.

Boyd F Edwards1, Bo A Johnson1, John M Edwards2

  • 1Department of Physics, Utah State University, Logan, Utah 84322, USA.

Chaos (Woodbury, N.Y.)
|February 5, 2020
PubMed
Summary

Researchers studied the motion of two magnetized spheres, finding 1243 unique periodic collision paths. These paths exhibit complex behaviors and symmetric trajectories, offering insights into nonlinear dynamics.

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Area of Science:

  • Physics
  • Nonlinear Dynamics
  • Magnetohydrodynamics

Background:

  • Investigates the dynamics of interacting magnetized spheres.
  • Focuses on elastic collisions and periodic motion in a frictionless environment.

Purpose of the Study:

  • To find and characterize periodic solutions for the nonlinear equations of motion governing a free sphere interacting with a fixed sphere.
  • To explore the behavior of large-amplitude modes and their relationship to small-amplitude solutions.

Main Methods:

  • Employs Runge-Kutta integration for numerical analysis of large-amplitude modes.
  • Derives closed-form mathematical solutions for small-amplitude modes to validate numerical findings.

Main Results:

  • Identified 1243 distinct periodic modes originating from the stable equilibrium position.
  • Characterized bifurcations from radial bouncing modes to states with significant angular motion.
  • Observed complex trajectories with numerous collisions and angular oscillations per period.

Conclusions:

  • The system exhibits a rich variety of behaviors and symmetric trajectories.
  • Periodic solutions are abundant and demonstrate complex dynamics beyond simple radial motion.
  • The study validates numerical findings with analytical solutions for small-amplitude oscillations.