ISEGORIA / MATH ENCYCLOPEDIA
Plasma physics: screening, orbits, and waves
Explore Debye screening, charged-particle gyromotion, and a cold-plasma dispersion sketch.
Before you begin: Electromagnetism and differential equations
Predict, manipulate, then check your reasoning against the example and question. Graphs illustrate the mathematics; they do not replace a proof.
1. Debye screening
Put a positive test charge into a plasma. Electrons crowd towards it and ions are pushed away, so a cloud of negative charge forms around it and hides it from far away. Change the Debye length and drag the gold probe sphere: the potential falls like the Coulomb potential times \(e^{-r/\lambda_D}\), and the net charge inside the sphere shows how much of the test charge is still visible.
Worked example. At r = λD the potential is e⁻¹ ≈ 0.37 of the bare Coulomb value and 74 % of the charge is still unscreened; at r = 3λD only 5 % of the potential and 20 % of the charge remain.
Watch out. The formula comes from linearising the Boltzmann response of the electrons, which needs many particles inside a Debye sphere and fails very close to the charge. The colours show the density of the cloud on a compressed scale.
What sets the screening distance?
The Debye length \(\lambda_D\).
2. Cyclotron motion
In a uniform magnetic field the Lorentz force is always perpendicular to the velocity, so it bends the path into a circle without changing the speed. Change the signed charge-to-mass ratio and the perpendicular speed: the radius depends on |q|/m, while the sign of q decides the sense of rotation, shown against the dashed orbit of the opposite charge.
Worked example. With B = 1, q/m = 1 and v⊥ = 1, the orbit has radius 1 and period 2π. Doubling q/m halves both; flipping its sign reverses the rotation.
Watch out. Uniform field, no electric field, no collisions and speeds far below c. A velocity component along B would add a steady drift out of the page, turning the circle into a helix.
What reverses when q changes sign, and what does not?
The sense of gyration reverses; the radius, the period and the speed depend only on |q|, so they stay the same.
3. Plasma-wave dispersion
Electromagnetic waves in a cold plasma obey \(\omega^2 = \omega_p^2 + c^2k^2\): below the plasma frequency they cannot propagate at all. Drag the point along the branch. The chord from the origin gives the phase velocity, always faster than c, and the tangent gives the group velocity, always slower; the animation shows crests sliding through a wave packet whose envelope lags behind them.
Worked example. With \(\omega_p = 1\), c = 1 and k = 1.2, ω ≈ 1.562, so \(v_\varphi\) ≈ 1.302 and \(v_g\) ≈ 0.768, and their product is exactly c² = 1.
Watch out. Cold, collisionless and unmagnetised: temperature adds the Langmuir branch and a magnetic field adds more. The wave packet is drawn to first order, moving at the group velocity without spreading.
What happens at \(k=0\)?
The branch begins at the plasma frequency \(\omega_p\).