Motion of charged particles in magnetic fields Knowledge-Based Study

Study notes

1. Lorentz Magnetic Force

   When a particle carrying an electric charge $q$ moves with a velocity $\vec{v}$ through a uniform magnetic field $\vec{B}$, it experiences a magnetic deflecting force $\vec{F}_B$ given by the vector cross product:

   $$\vec{F}_B = q(\vec{v} \times \vec{B})$$

   The magnitude of this force is expressed as:

   $$F_B = |q|vB\sin\theta$$

   where $\theta$ is the angle between the velocity vector $\vec{v}$ and the magnetic field vector $\vec{B}$.

   Key Characteristics of the Force

   • Zero Force Conditions: The magnetic force is zero if the particle is at rest ($v = 0$), if the particle is electrically neutral ($q = 0$), or if the particle moves parallel or antiparallel to the field lines ($\theta = 0^\circ$ or $\theta = 180^\circ$).
   • Maximum Force: The force reaches its maximum magnitude, $F_{B,\text{max}} = |q|vB$, when the velocity is perpendicular to the magnetic field ($\theta = 90^\circ$).
   • Direction and the Right-Hand Rule: For a positive charge, point the fingers of your right hand along $\vec{v}$ and curl them toward $\vec{B}$; your extended thumb points in the direction of $\vec{F}_B$. For a negative charge, the force acts in the directly opposite direction.

   Magnetic Force Does No Work

   Because the magnetic force is defined by a cross product, $\vec{F}_B$ is always perpendicular to the instantaneous velocity $\vec{v}$ (and thus perpendicular to the displacement $d\vec{r}$):

   $$\vec{F}_B \cdot \vec{v} = 0$$

   According to the work-energy theorem, the rate of work done (power $P$) on the charged particle is:

   $$P = \frac{dW}{dt} = \vec{F}_B \cdot \vec{v} = 0$$

   Consequently:

   • A static magnetic field does zero work on a moving charged particle.
   • The kinetic energy of the particle remains constant: $\Delta K = 0$.
   • The particle's speed remains constant, even though its velocity changes continuously in direction.
   • A magnetic field can change the direction of motion, but it cannot speed up or slow down a charged particle.

   Perpendicular Injection ($\theta = 90^\circ$)

   When a charged particle enters a uniform magnetic field with its velocity strictly perpendicular to $\vec{B}$, the magnetic force acts as a centripetal force of constant magnitude. This causes the particle to undergo uniform circular motion in a plane perpendicular to $\vec{B}$.

   Equating the magnetic force to the centripetal force:

   $$|q|vB = \frac{mv^2}{r}$$

   Trajectory Parameters

   • Radius of the Orbit (Gyroradius):
     $$r = \frac{mv}{|q|B} = \frac{p}{|q|B}$$
     where $p = mv$ is the linear momentum of the particle. Particles with larger momentum trace out larger circles, while stronger fields produce tighter loops.
   • Period of Revolution:
     The time required for one complete orbit is the circumference divided by the speed:
     $$T = \frac{2\pi r}{v} = \frac{2\pi m}{|q|B}$$
   • Cyclotron Frequency:
     The frequency and angular frequency (cyclotron frequency $\omega$) are given by:
     $$f = \frac{1}{T} = \frac{|q|B}{2\pi m}, \quad \omega = 2\pi f = \frac{|q|B}{m}$$

   Crucial Observation: The orbital period $T$ and cyclotron frequency $\omega$ are completely independent of the particle's speed $v$ and radius $r$. Faster particles travel along larger circles, completing each revolution in the exact same duration as slower particles of the same charge-to-mass ratio.

   General Motion ($0^\circ < \theta < 90^\circ$)

   If the velocity vector makes an arbitrary angle $\theta$ with the magnetic field, we resolve $\vec{v}$ into components parallel and perpendicular to $\vec{B}$:

   • $v_\parallel = v\cos\theta$ (parallel to $\vec{B}$)
   • $v_\perp = v\sin\theta$ (perpendicular to $\vec{B}$)

   Because $\vec{F}_B$ has no component along $\vec{B}$, the parallel velocity $v_\parallel$ remains constant, producing uniform linear motion along the field lines. Meanwhile, $v_\perp$ undergoes uniform circular motion with radius $r = \frac{m v_\perp}{|q|B}$.

   The superposition of these two motions results in a helical path (helix). The distance moved along the field axis during one full period is called the pitch ($p$):

   $$p = v_\parallel T = (v\cos\theta)\left(\frac{2\pi m}{|q|B}\right)$$

   Practical Applications

   • Velocity Selector: Perpendicular electric and magnetic fields ($\vec{E} \perp \vec{B}$) allow only particles with speed $v = \frac{E}{B}$ to pass through undeflected.
   • Mass Spectrometer: Ions separated by a velocity selector enter a uniform magnetic field. Since $r = \frac{mv}{qB}$, measuring the orbital radius reveals the precise mass-to-charge ratio ($m/q$) of isotopes.
   • Magnetic Confinement: Non-uniform magnetic fields can trap charged particles, causing them to spiral and reflect back between high-field regions (magnetic mirrors), a principle seen in Earth's Van Allen radiation belts and fusion reactors.