Magnetized Plasmas

Dynamics of plasmas constrained by external magnetic fields. Theory, formulas, worked examples and interactive graphs.

Conceptual context

A magnetized plasma is an ionized gas whose charged particles are not simply free projectiles. The $B$ bends the perpendicular part of their motion and makes them gyrate around magnetic-field lines. The central question is what microscopic length appears when inertia competes with magnetic bending. That length is the $r_L$. If the $B$ increases while $m$, $q$, and $v_{\perp}$ stay fixed, the orbit becomes tighter. The plasma is then more strongly tied to the field, with particle motion threaded along invisible magnetic rails.

🟢 Essential level

In a magnetized plasma, a charged particle does not simply cross the $B$ in a straight path.

The part of its motion perpendicular to the field, measured by $v_{\perp}$, is continuously bent and forms a local orbit. The size of that orbit is the $r_L$. When the $B$ increases, the same particle is deflected more strongly at each instant, so it needs less distance to turn. Therefore the $r_L$ decreases. The $v_parallel$ does not create the gyration; it carries the particle along the field line and turns the circular orbit into a helix. A plasma is well magnetized when the $r_L$ is small compared with the $L_B$, because the particle then experiences an almost constant field during each magnetic turn.

In the Van Allen belts, protons trapped by Earth's $B$ gyrate with a $r_L$ of only a few hundred meters, tiny compared with the belt's size of thousands of kilometers. That huge separation of scales is exactly what allows treating these particles' motion as orbits guided by field lines rather than chaotic trajectories.

🔵 Formal level

Basic formulas

The microscopic motion is described by splitting the velocity into $v_{\perp}$ and $v_parallel$ relative to the $B$. The perpendicular rotation is controlled by the $\omega_{c}$, which increases with the magnitude of $q$ and with $B$, and decreases with $m$.

Cyclotron frequency

The orbit size is given by the $r_L$, which compares the particle’s perpendicular inertia with the magnetic bending strength.

Larmor radius

Real student questions

Does the $B$ do work on the particle? Not in the pure magnetic model: it changes the direction of $v_{\perp}$, not its magnitude.

Why does a helix appear? Because $v_{\perp}$ produces gyration and $v_parallel$ produces simultaneous advance.

Do an ion and an electron have the same $r_L$? Not necessarily. The $m$ and $q$ of each species change the orbit scale.

Does small $r_L$ mean the plasma is motionless? No. It means transverse motion is strongly curved; the plasma can still move through drifts, waves, or parallel transport.

What should I read in the graph? The decrease of $r_L$ as $B$ increases, and whether that scale lies below $L_B$.

Final synthesis

In a magnetized plasma, increasing $B$ reduces $r_L$ if species and $v_{\perp}$ stay fixed. The particle gyrates faster, follows a smaller orbit, and becomes more strongly guided by the field.