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Gyroradius
 
 
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The gyroradius (also known as radius of gyration, Larmor radius or cyclotron radius) defines the radius
RADIUS

Remote Authentication Dial In User Service is a networking protocol that provides centralized access, authorization and accounting management for people or computers to connect and use a network service....
 of the circular motion of a charged particle
Charged particle

In physics, a charged particle is a particle with an electric charge. It may be either a subatomic particle or an ion. A collection of charged particles, or even a gas containing a proportion of charged particles, is called a Plasma , which is called the fourth state of matter because its properties are quite different from solids, liq...
 in the presence of a uniform magnetic field
Magnetic field

A magnetism field is a vector field which can exert a magnetic force on moving electric charges and on magnetic dipoles . When placed in a magnetic field, magnetic dipoles tend to align their axes parallel to the magnetic field....
.

where

Similarly, the frequency
Frequency

Frequency is the number of occurrences of a repeating event per unit time. It is also referred to as temporal frequency.The period is the duration of one cycle in a repeating event, so the period is the reciprocal of the frequency....
 of this circular motion is known as the gyrofrequency or cyclotron frequency, and is given by: For electrons, this works out to be

formula for the gyroradius also holds for relativistic motion
Special relativity

Special relativity is the physical theory of measurement in inertial frames of reference proposed in 1905 by Albert Einstein in the paper "Annus Mirabilis Papers#Special relativity"....
.






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The gyroradius (also known as radius of gyration, Larmor radius or cyclotron radius) defines the radius
RADIUS

Remote Authentication Dial In User Service is a networking protocol that provides centralized access, authorization and accounting management for people or computers to connect and use a network service....
 of the circular motion of a charged particle
Charged particle

In physics, a charged particle is a particle with an electric charge. It may be either a subatomic particle or an ion. A collection of charged particles, or even a gas containing a proportion of charged particles, is called a Plasma , which is called the fourth state of matter because its properties are quite different from solids, liq...
 in the presence of a uniform magnetic field
Magnetic field

A magnetism field is a vector field which can exert a magnetic force on moving electric charges and on magnetic dipoles . When placed in a magnetic field, magnetic dipoles tend to align their axes parallel to the magnetic field....
.

where
  • is the gyroradius,
  • is the mass of the charged particle,
  • is the velocity component perpendicular to the direction of the magnetic field,
  • is the charge of the particle, and
  • is the constant magnetic field.


Similarly, the frequency
Frequency

Frequency is the number of occurrences of a repeating event per unit time. It is also referred to as temporal frequency.The period is the duration of one cycle in a repeating event, so the period is the reciprocal of the frequency....
 of this circular motion is known as the gyrofrequency or cyclotron frequency, and is given by: For electrons, this works out to be

Relativistic case

The formula for the gyroradius also holds for relativistic motion
Special relativity

Special relativity is the physical theory of measurement in inertial frames of reference proposed in 1905 by Albert Einstein in the paper "Annus Mirabilis Papers#Special relativity"....
. In that case, the velocity and mass of the moving object has to be replaced by the relativistic momentum :

For rule-of-thumb calculations in accelerator
Particle accelerator

A particle accelerator is a device that uses electric fields to propel electric charge Elementary particles to high speeds and to contain them....
 and astroparticle
Astroparticle physics

Astroparticle physics is that branch of particle physics that studies elementary particles of astronomical origin, and their relation to astrophysics and cosmology....
 physics, the physical quantities can be expressed in proper units, which results in the simple numerical formula

where
  • is the charge of the moving object in elementary units.

Derivation

If the charged particle is moving, then it will experience a Lorentz force
Lorentz force

In physics, the Hendrik Lorentz force is the force on a point charge due to electromagnetic fields. It is given by the following equation in terms of the electric field and magnetic fields:...
 given by:

where is the velocity vector, is the magnetic field vector, and is the particle's electric charge
Electric charge

Electric charge is a fundamental conserved property of some subatomic particles, which determines their electromagnetic interaction. Electrically charged matter is influenced by, and produces, electromagnetic fields....
.

Notice that the direction of the force is given by the cross product
Cross product

In mathematics, the cross product is a binary operation on two vector s in a three-dimensional Euclidean space that results in another vector which is orthogonal to the plane containing the two input vectors....
 of the velocity and magnetic field. Thus, the Lorentz force will always act perpendicular to the direction of motion, causing the particle to move in a circle (gyrate
Gyration

Gyration is another term for rotation. A center of actual rotation as well as rotational symmetry may be called gyration center, gyration point, or rotocenter....
). The radius of this circle can be determined by equating the magnitude of the Lorentz force to the centripetal force
Centripetal force

The centripetal force is the external force required to make a body follow a curved path. Hence centripetal force is a kinematic force requirement, not a particular kind of force like gravity or electromagnetism....
:

where is the particle mass
Mass

In physical science, mass refers to the degree of acceleration a body acquires when subject to a force: bodies with greater mass are accelerated less by the same force....
 (for high velocities the relativistic mass), is the velocity component perpendicular to the direction of the magnetic field, and is the strength of the field.

Solving for , the gyroradius is determined to be:

Thus, the gyroradius is directly proportional to the particle mass and velocity, and inversely proportional to the particle electric charge, and the magnetic field strength.

See also

  • Cyclotron
    Cyclotron

    A cyclotron is a type of particle accelerator. Cyclotrons accelerate charged particles using a high-frequency, alternating voltage . A perpendicular magnetic field causes the particles to spiral almost in a circle so that they re-encounter the accelerating voltage many times....
  • Magnetosphere particle motion
    Magnetosphere particle motion

    The ions and electrons of a plasma interacting with the Earth's magnetic field generally follow its magnetic field lines. These represent the force that a north magnetic pole would experience at any given point....