电场与非均匀磁场中的Speed sorting及带电粒子轨道特性问询

Great question—this ties together some fundamental drift dynamics in plasma physics, so let’s unpack each part clearly:

1. Does "Speed Sorting" exist in this environment? #

Yes, speed sorting (or velocity-dependent separation) absolutely occurs here when collisions are negligible. Here’s why:

• The grad-B drift velocity for a charged particle follows:
  v_gradB = (v⊥² / (2B)) * (B × ∇B) / B²
  Notice this drift scales with the square of the particle’s perpendicular velocity (v⊥²). Faster particles have a much stronger upward grad-B drift than slower ones.
• You noted the electric field creates a drift that opposes this upward motion (the E×B drift: v_E×B = E × B / B²). But since the grad-B drift’s magnitude depends on speed, the net drift differs drastically between fast and slow particles:
  • Fast particles: Their upward grad-B drift dominates the opposing E×B drift, so they net-drift upward over time.
  • Slow particles: Their weak grad-B drift is overwhelmed by the downward E×B drift, so they net-drift downward.

Over time, this velocity-dependent net drift causes fast particles to accumulate at the top of the gradient, and slow ones at the bottom—exactly the "speed sorting" effect you’re asking about.

2. What’s the overall effect of adding the electric field? #

The electric field doesn’t eliminate the grad-B drift, but it introduces a competing drift that creates a velocity threshold:

• Particles with v⊥ above a certain value will drift upward (grad-B drift wins out)
• Particles below that threshold will drift downward (E×B drift dominates)

This acts like a "filter" that separates particles by their perpendicular speed, concentrating them in distinct regions along the magnetic field gradient. Without the electric field, all particles would drift upward (just at different rates), but the field adds a directional split that amplifies the sorting effect.

3. What shape is the positive charge’s orbit (ignoring out-of-plane motion)? #

It’s not a circle. Here’s the breakdown:

• The particle still undergoes cyclotron motion (tight circular orbits around its guiding center) due to the magnetic field.
• But the guiding center itself is slowly drifting upward (from the combined grad-B + E×B drift).

The combination of these two motions creates a cycloidal (or trochoidal) orbit—think of a circle rolling along a straight line, tracing a curved path that loops forward as it drifts. The exact shape depends on the relative magnitudes of the cyclotron speed and the drift speed, but it’s never a perfect circle because the guiding center is in constant motion.

内容的提问来源于stack exchange，提问作者Ludvig Mörtberg