The Lorentz Force and Motion in Fields

The Lorentz Force in Plain Language

Imagine a charged particle moving through space. If there is an electric field, the particle feels a push or pull in the direction of the field (if positive) or opposite to it (if negative). This force depends only on the charge and the field strength, not on how fast the particle is moving. However, if there is also a magnetic field, the situation changes. The magnetic force is unique: it is always perpendicular to both the particle's velocity and the magnetic field direction. Because this force is always sideways, it does no work on the particle. It cannot speed the particle up or slow it down; it can only change the direction of motion. This is why a charged particle entering a uniform magnetic field travels in a circle or a helix, rather than accelerating linearly.

Equation of Motion in Uniform Fields

We begin with the total electromagnetic force acting on a point charge \( q \) of mass \( m \). According to the Lorentz force law, the force is the sum of the electric and magnetic components. In vector notation, this is expressed as:

Applying Newton's second law, \( m\dot{\vec{v}} = \vec{F} \), we obtain the equation of motion. For a uniform static magnetic field \( \vec{B} \) and zero electric field, the magnetic force provides the centripetal acceleration required for circular motion. The magnitude of the force is \( F = qv_\perp B \), where \( v_\perp \) is the component of velocity perpendicular to the field. Equating this to the centripetal force \( mv_\perp^2 / r_c \), we can solve for the radius of curvature.

The angular frequency of this circular motion, known as the cyclotron frequency, is independent of the particle's speed (for speeds well below \(c\); relativistically \(m\) is replaced by \(\gamma m\)). It is given by:

This result is crucial because it allows for the synchronization of electric fields with the particle's natural orbital period, a principle used in cyclotron accelerators.

Helical Trajectories

In general, the velocity vector \( \vec{v} \) can be decomposed into two components: one parallel to the magnetic field \( \vec{B} \) and one perpendicular to it. The component parallel to the field, \( v_\parallel \), experiences no magnetic force because the cross product \( \vec{v}_\parallel \times \vec{B} \) is zero. Consequently, the particle moves with constant velocity along the field lines. The perpendicular component, \( v_\perp \), undergoes uniform circular motion as described above. The superposition of these two motions results in a helical path.

The pitch of the helix, which is the distance traveled along the field direction during one complete revolution, is determined by the parallel velocity and the cyclotron period \( T = 2\pi/\omega_c \). This behavior explains why charged particles in the Earth's magnetosphere spiral along magnetic field lines, creating the auroras when they collide with atmospheric gases near the poles.

The \( \vec{E}\times\vec{B} \) Drift

When both electric and magnetic fields are present and uniform, the motion becomes more complex. The electric field exerts a constant force \( q\vec{E} \), which tends to accelerate the particle. However, the magnetic field deflects this acceleration. If the fields are perpendicular to each other, the particle undergoes a specific drift motion superposed on its gyration. This drift velocity is independent of the charge and mass of the particle, a remarkable result known as the \( \vec{E}\times\vec{B} \) drift.

This drift occurs in a direction perpendicular to both the electric and magnetic fields. It is a single-particle effect: during each gyration the electric field speeds the particle up on one side of its orbit and slows it on the other, so the radius changes and the orbit centre steps sideways. In plasma physics, this drift is significant because it can lead to the transport of plasma across magnetic field lines, which is a major challenge in magnetic confinement fusion devices.