11.3 Motion of a Charged Particle in a Magnetic Field
3 min read•june 24, 2024
Charged particles in magnetic fields move in fascinating ways, creating circular or helical paths depending on their initial velocity. The magnetic force acts perpendicular to both the particle's motion and the field, causing without changing the particle's speed.
Understanding these motions is crucial for many applications, from particle accelerators to plasma confinement in fusion reactors. The radius and period of the circular path depend on the particle's properties and the , while helical trajectories occur when particles enter at an angle.
Motion of a Charged Particle in a Uniform Magnetic Field
Motion of charged particles in magnetic fields
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Charged particle experiences magnetic force when moving through magnetic field
Magnetic force is perpendicular to both particle's velocity and magnetic field ()
Force acts as centripetal force, causing particle to move in circular path (cyclotron motion)
Direction of depends on sign of particle's charge
Positive charges () circle counterclockwise
Negative charges () circle clockwise
Particle's speed remains constant as magnetic force does no work on particle
Magnetic force only changes direction of particle's velocity, not magnitude ( conserved)
Circular path calculations for charged particles
Radius of circular path depends on particle's mass (m), charge (q), velocity (v), and magnetic field strength (B)
Radius (r) given by: r=qBmv (also known as )
Larger mass (heavy ions) or velocity results in larger radius
Larger charge () or magnetic field strength results in smaller radius
Period (T) of circular motion is time for particle to complete one revolution
Period given by: T=qB2πm
Period depends on particle's mass, charge, and magnetic field strength
Period independent of particle's velocity (frequency of revolution constant)
The inverse of the period is known as the
Helical trajectories in uniform magnetic fields
When charged particle enters magnetic field with velocity not perpendicular to field, path becomes helix
Particle's velocity decomposed into components parallel and perpendicular to magnetic field
Perpendicular component causes particle to move in circular path (as described above)
Parallel component unaffected by magnetic field, causing particle to move along field lines (uniform motion)
Resulting motion combines circular motion in plane perpendicular to magnetic field and constant velocity along field lines
Combination creates (spiral motion)
Pitch of helix depends on ratio of parallel velocity component to perpendicular velocity component
Larger parallel component results in longer pitch (stretched helix)
Larger perpendicular component results in shorter pitch (compressed helix)
The angle between the particle's velocity vector and the magnetic field lines is called the
Particle behavior in non-uniform magnetic fields
In non-uniform magnetic fields, particles can experience magnetic mirroring
of a particle is conserved in slowly varying magnetic fields
As a particle moves into a stronger magnetic field region, its pitch angle increases
If the magnetic field strength increases sufficiently, the particle can be reflected back towards weaker field regions