Gyroradius

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The gyroradius (also known as radius of gyration or cyclotron radius) defines the radius of the circular motion of a charged particle in the presence of a magnetic field.

Derivation

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

[\vec = q(\vec \times \vec)]

Where [\vec] is the velocity vector, [\vec] is the magnetic field vector, and [q] is the particle's electric charge. Notice that the direction of the force is given by the cross product 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). The radius of this circle [r_g] can be determined by equating the magnitude of the Lorentz force to the centripetal force:

[\frac = qvB]

Here [m] is the particle mass. Solving for [r_g], the gyroradius is determined to be:

[r_g = \frac]

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

Similarly, the frequency of this circular motion is known as the gyrofrequency, and is given by:

[\nu = \frac]

See also

• Cyclotron
• Magnetosphere particle motion

References & further reading

 

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