Motion in a uniform magnetic field

Motion in a uniform magnetic field is a fundamental concept in plasma physics and can be understood by considering the Lorentz force that acts on charged particles moving through the field.

Key Concepts:

1. Lorentz Force: The force on a charged particle moving with velocity v in a magnetic field B is given by the Lorentz force law:

   F=q(v×B)

   where q is the charge of the particle, v is its velocity, and B is the magnetic field.

2. Cyclotron Motion: In a uniform magnetic field, the Lorentz force acts perpendicular to the velocity of the particle, causing it to move in a circular path. This is known as cyclotron motion. The radius of the circular motion, called the cyclotron radius or Larmor radius, is given by:

   r=mv⊥∣q∣Br

   where m is the mass of the particle, perpv⊥​ is the component of velocity perpendicular to the magnetic field, and ∣q∣|q|∣q∣ is the magnitude of the charge.

3. Cyclotron Frequency: The frequency of the circular motion, known as the cyclotron frequency or gyrofrequency, is given by:

   ωc=∣q∣Bmω_c

   This is the rate at which the particle orbits in the magnetic field.

4. Motion Components:

   • Parallel to the Magnetic Field: If the velocity has a component parallel to the magnetic field B, this component remains unchanged because the Lorentz force acts perpendicular to B. Thus, the particle moves with a constant velocity along the direction of the field while simultaneously undergoing circular motion perpendicular to the field.

   • Perpendicular to the Magnetic Field: The component of velocity perpendicular to the field leads to circular motion with a radius and frequency as described above.

5. Magnetic Mirror Effect: In a non-uniform magnetic field, where the field strength varies with position, particles can experience a magnetic mirror effect. This occurs when the magnetic field strength increases, causing the particles to reverse their motion. This is particularly important in space plasmas and contributes to phenomena such as the trapping of charged particles in the Earth's magnetosphere.

   When a charged particle has a simple cyclotron gyration. The equation of motion of a particle of mass m and charge q in a given magnetic field is

Taking z to be the direction of B, the equation of motions are

From equation 4, we get v_z = constant

From 2 and 3

Equations 5 and 6 describe a simple harmonic oscillator having angular frequency ( defined as the cyclotron frequency)

w_c = l q l B / m ...7

By the conversion, we have choosen wc is always non negative. B is measured in tesla or wb/m², a unit equal to 10^4 Gauss. The solution of equation 5 and 6 are respectively

We obtain from 2,

Integrating 10 and 11

x - x_o = v_⟂/ iwc e^iwct

x - x_o = - i v_⟂/ wc e^iwct ....12

y - y_o = ± v_⟂/ wc e^iwct ....13

Taking real part,

x - x_o = v_⟂/ wc sin wct ....14

y - y_o = ± v_⟂/ wc cos wct ....15

Hence, ( x - x_o)² + ( y - y_o)² = ( v_⟂/ wc )²

It describes a circular path orbit around a guiding centre (xo, yo). The radius of the path is defined as the Larmor radius.

i.e r_L = v_⟂/ w_c ......17

Along with this motion, there is an orbital velocity along B (i.e. vz) that B does not affect. In general, a charged particle's space trajectory is a helix. Always, the gyration direction is such that the magnetic field produced by the charge particle is opposite to the externally imposed field. Because plasma particles have a tendency to reduce the magnetic field there, plasma is diamagnetic.

Summary

In a uniform magnetic field, a charged particle will follow a helical path due to its combined cyclotron motion (circular motion perpendicular to the field) and linear motion parallel to the field. The specifics of the motion, including the radius and frequency of the circular path, depend on the charge, mass, and velocity of the particle, as well as the strength of the magnetic field.

This note is a part of the Physics Repository