Plasma Physics
Grad-B Drift: Why Charged Particles Curve Across Magnetic Gradients
A megaelectronvolt proton trapped in Earth's inner radiation belt takes only about 30 minutes to circle the entire planet — and it never touches a wire. It drifts westward because the geomagnetic field is stronger near Earth than far away, and that stronger-here, weaker-there gradient bends its gyro-orbit into a slow, sideways march. This is the grad-B drift: the steady velocity a charged particle acquires perpendicular to both the magnetic field B and its gradient ∇B whenever the field's magnitude changes across the particle's orbit.
Grad-B drift is one of the guiding-center drifts of magnetized-plasma theory. It arises because a particle's gyroradius r_L = m·v_⊥/(qB) shrinks where the field is strong and widens where it is weak, so successive loops of the spiral don't close — the orbit "walks" across the gradient. Crucially, the drift direction depends on the sign of the charge, so ions and electrons separate and drive real currents.
- TypeGuiding-center drift (magnetic)
- RegimeMagnetized plasma, weak gradient (r_L ≪ L_B)
- Key equationv_∇B = (m·v_⊥²)/(2qB³) · (B × ∇B)
- DirectionCharge-dependent: ions and electrons drift oppositely
- Typical scaleMeV proton in inner belt: full drift orbit in ~30 min
- Observed inRadiation belts, ring current, tokamaks, mirror machines
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The Physical Setup: A Gyro-Orbit in a Non-Uniform Field
Put a charged particle in a uniform magnetic field and it does something simple: it spirals. The Lorentz force F = qv × B is always perpendicular to the velocity, so the particle circles at the cyclotron (gyro) frequency ω_c = qB/m with a gyroradius r_L = m·v_⊥/(qB). The center of that circle — the guiding center — sits still (or slides freely along B). Nothing drifts sideways.
Now make the field non-uniform: let |B| grow in some direction, say toward increasing y, so ∇B points in +y. The gyroradius is no longer constant around the loop. Where the particle swings into the strong-field region, r_L shrinks; where it swings into the weak-field region, r_L grows. The orbit is tighter on one side and looser on the other, so it no longer closes on itself.
- Requirement: the field must change gently — the fractional change over one orbit must be small, r_L·|∇B|/B ≪ 1.
- Result: the guiding center creeps in the direction perpendicular to both B and ∇B.
The Mechanism and Derivation
The cleanest way to see the drift is to average the Lorentz force over one gyro-orbit. A gyrating particle acts like a small magnetic dipole with moment μ = m·v_⊥²/(2B) (the first adiabatic invariant, conserved when the field changes slowly). A dipole in a field gradient feels a force F = −μ·∇B, pushing it from strong field toward weak field.
Any steady force F acting on a gyrating particle produces the generic force drift v = (F × B)/(qB²). Substituting F = −μ·∇B gives the grad-B drift:
- v_∇B = (μ/qB²)·(B × ∇B) = (m·v_⊥²)/(2qB³)·(B × ∇B)
- Magnitude: |v_∇B| = ½·v_⊥·r_L·(|∇B|/B).
Symbols: m = mass, q = charge (with sign), v_⊥ = speed across B, B = field magnitude, r_L = gyroradius. Because q carries its sign, positive and negative particles drift in opposite directions — the geometric origin lies in the difference between the tight and loose arcs of the non-closing orbit.
Key Quantities and a Worked Example
Take a 1 MeV proton in Earth's inner radiation belt, at roughly two Earth radii where B ≈ 1000 nT = 1×10⁻⁶ T. A 1 MeV proton has v ≈ 1.4×10⁷ m/s. Its gyroradius is r_L = m·v_⊥/(qB) ≈ (1.67×10⁻²⁷·1.4×10⁷)/(1.6×10⁻¹⁹·1×10⁻⁶) ≈ 1.5×10⁵ m ≈ 150 km — tiny next to the ~13,000 km field scale length L_B, so the drift approximation holds cleanly (r_L/L_B ≈ 0.01).
- With |∇B|/B ≈ 1/L_B, the drift speed is |v_∇B| ≈ ½·v_⊥·(r_L/L_B) ≈ ½·(1.4×10⁷)·(0.01) ≈ 7×10⁴ m/s.
- Circumference at 2 R_E ≈ 2π·1.28×10⁷ m ≈ 8×10⁷ m, so one drift orbit takes ~10³ s — of order minutes to tens of minutes, matching observed drift periods.
Two scalings matter: the drift grows with particle energy (∝ v_⊥², i.e. ∝ perpendicular kinetic energy) and it weakens in stronger fields (∝ 1/B³ through B × ∇B with |∇B|∝B). Faster, more energetic particles drift fastest.
How It's Observed, Measured, and Used
Grad-B drift is not a textbook abstraction — it is measured routinely in space and matters in the lab.
- Radiation belts: NASA's Van Allen Probes (2012–2019) tracked energetic protons and electrons circling Earth on drift timescales of minutes to hours, exactly as the ½·v_⊥·r_L·(∇B/B) scaling predicts.
- Ring current: because ions drift westward and electrons eastward, their separation is a net westward current of millions of amperes encircling Earth. During geomagnetic storms this current strengthens and depresses the surface field by tens to hundreds of nT — the Dst index directly tracks it.
- Fusion devices: in tokamaks and magnetic mirrors, the 1/R fall-off of the toroidal field drives grad-B (plus curvature) drift that separates ions from electrons vertically, creating a vertical electric field. Left uncorrected, the resulting E × B drift flings the whole plasma outward — the reason a pure toroidal field cannot confine plasma and why the rotational transform (twisted field lines) was invented.
Instruments infer the drift indirectly from particle flux anisotropies, drift-echo signatures after injections, and the induced magnetic perturbations.
Grad-B vs Its Close Cousins
Grad-B drift is one member of a family of guiding-center drifts, and confusing them is the classic beginner error.
- Curvature drift is its constant companion. Where field lines bend with radius R_c, a particle's parallel motion feels a centrifugal force m·v_∥²/R_c, giving v_R = (m·v_∥²/qB)·(R_c × B)/(R_c²·B). In a current-free field (∇×B = 0) the two combine neatly: v_R+∇B = (m/qB²)·(v_∥² + ½v_⊥²)·(R_c × B)/R_c². Both scale as 1/q, so both separate charges the same way.
- E × B drift is fundamentally different: it has no charge or mass dependence, so all particles move together and it drives no current.
- Polarization drift responds to a changing E field and scales as m/q.
A useful distinction: grad-B and curvature drifts are magnetic drifts driven by field geometry and grow with particle energy; E × B is an electric drift set purely by the fields. Only the charge-dependent drifts (grad-B, curvature, polarization, gravitational) produce net currents.
Significance, Famous Cases, and Open Questions
Grad-B drift is a cornerstone of the guiding-center theory formalized by Hannes Alfvén in the 1940s, work that helped earn him the 1970 Nobel Prize in Physics for magnetohydrodynamics. James Van Allen's 1958 discovery of the radiation belts, using Explorer 1's Geiger counter, gave the drift its most spectacular showcase: trapped particles that bounce between magnetic mirror points and drift in longitude, mapping out toroidal shells around the planet.
- Confinement lesson: the vertical charge separation from grad-B/curvature drift is precisely why a simple magnetic torus leaks — a foundational result that shaped tokamak and stellarator design.
- Space weather: the ring current's grad-B-driven intensification is a primary driver of magnetic storms and satellite hazards.
Open frontiers involve the breakdown of the guiding-center picture: when r_L becomes comparable to L_B (very energetic particles, sharp gradients, or reconnection sites), the adiabatic invariant μ is no longer conserved, drifts become non-adiabatic, and full-orbit or kinetic treatments are required. Accurately modeling these transitions remains active work in magnetospheric and fusion plasma physics.
| Drift | Driving cause | Velocity (magnitude) | Charge/mass dependence |
|---|---|---|---|
| Grad-B (gradient) | Gradient in |B| across the orbit | ½·v_⊥·r_L·(∇B/B) | Charge sign → direction; ∝ 1/q |
| Curvature | Centrifugal force along curved field lines | (m·v_∥²/qB)·(R_c/R_c²) | Charge sign → direction; ∝ 1/q |
| E × B | Perpendicular electric field E | (E × B)/B² | None — same for all species |
| Polarization | Time-changing E field (dE/dt) | (m/qB²)·dE⊥/dt | Charge sign → direction; ∝ m/q |
| Gravitational (F × B) | Any steady force F (e.g. gravity) | (F × B)/(qB²) | Charge sign → direction; ∝ 1/q |
Frequently asked questions
What is grad-B drift in simple terms?
It is the slow sideways motion of a charged particle spiraling in a magnetic field whose strength changes across the orbit. Because the gyroradius is smaller where the field is strong and larger where it is weak, the spiral fails to close and the guiding center creeps in the direction perpendicular to both the field B and its gradient ∇B.
What is the formula for grad-B drift velocity?
v_∇B = (m·v_⊥²)/(2qB³) · (B × ∇B), where m is mass, q is the signed charge, v_⊥ is the speed perpendicular to B, and B is the field magnitude. Its magnitude equals ½·v_⊥·r_L·(∇B/B), with r_L the gyroradius. The drift grows with perpendicular kinetic energy and is perpendicular to both B and ∇B.
Why do ions and electrons drift in opposite directions?
The drift velocity contains the charge q with its sign, so reversing the charge reverses the direction of B × ∇B/q. In Earth's field this sends energetic protons westward and electrons eastward. Their separation constitutes the westward ring current, which depresses the surface magnetic field during geomagnetic storms.
How is grad-B drift different from E × B drift?
Grad-B drift is caused by a gradient in the magnetic field magnitude and depends on the sign of the charge and on particle energy (∝ v_⊥²), so it separates species and drives currents. E × B drift is caused by an electric field, is identical for all charges and masses, moves the whole plasma together, and drives no net current.
What is the difference between grad-B drift and curvature drift?
Grad-B drift comes from the change in |B| across the orbit and depends on perpendicular energy (v_⊥²). Curvature drift comes from the centrifugal force as particles follow bent field lines and depends on parallel energy (v_∥²). In a current-free field they combine into one expression proportional to (v_∥² + ½v_⊥²), and both scale as 1/q.
Where does grad-B drift actually matter?
It governs the longitudinal drift of trapped particles in the Van Allen belts, builds the magnetospheric ring current behind magnetic storms and the Dst index, and — in fusion devices — causes the vertical charge separation that makes a pure toroidal magnetic field unable to confine plasma, motivating the twisted field lines used in tokamaks and stellarators.