Question bank — Hall-effect thruster — cross-field discharge, annular channel

False — the magnetic force is always perpendicular to velocity ($\vec{F}=q\vec{v}\times \vec{B}$), so it does zero work. Ions gain energy only from the axial electric field / potential drop.

True — because $r_L=m{v}_{\perp }/(qB)$ scales with mass, the ~$2.4\times 10^5$-times-heavier xenon ion has a gyroradius far larger than the channel while the electron's is sub-millimetre. See Larmor Radius and Cyclotron Motion.

True — electrons sit right where ions accelerate, cancelling net charge, which is exactly why the Hall thruster dodges the Child–Langmuir space-charge cap (see Plasma Quasineutrality and Child–Langmuir Law).

False — the Hall current is a closed azimuthal loop of trapped electrons that never leaves; the exhaust is the stream of unmagnetized ions slung out axially.

False — force balance $qE=q{v}_{d}B$ gives $v_d=E/B$. Units confirm it: $(\text{V/m})/\text{T}=\text{m/s}$, whereas $EB$ has the wrong dimensions.

False — collisions and anomalous (turbulent) transport hop electron guiding-centres one gyroradius across $\vec{B}$ per event, giving finite cross-field mobility that sustains the discharge current.

False — too strong a $B$ chokes electron cross-field mobility (starving the anode current) and can eventually start to magnetize the ions, deflecting the very beam you want to fire straight out.

Partly, but the deeper reason is its large mass (low ionization energy per unit momentum and easy magnetization asymmetry): heavy ions carry lots of momentum per charge, and their huge $r_L$ keeps them unmagnetized while electrons stay trapped.

Error — $\vec{E}$ (axial) $\times \vec{B}$ (radial) points azimuthally (around the ring), not axially. It moves electrons in a circle; it is not the ion-ejection mechanism.

Error — the Hall current must close on itself to be steady. A tube has ends where charge would pile up and stall the drift; the annular ring gives the azimuthal drift an endless loop.

Error — thrust is $F=\dot{m}{v}_e$ with $v_e$ the ion exhaust speed ($\sqrt{2q\Delta V/m_i}$). The Hall drift $v_d$ is the internal electron circulation, not the exhaust; using it double-confuses the two currents.

Error — magnetized means the gyroradius is small compared to the channel, i.e. $r_{L,e}\ll L_{\text{channel}}$. Large $r_L$ (the ion case) is unmagnetized.

Error — quasineutrality removes the net space charge, so ions fall through the full $\Delta V$ with no shielding cap. That is precisely the advantage over the gridded Ion Thruster (Gridded).

Error — the beam leaving is a stream of positive ions and would charge the spacecraft negative, eventually pulling ions back. An external electron source (neutralizer) is required to keep the exhaust and craft charge-balanced.

A perpendicular $B$ traps electrons transverse to the accelerating $E$ (creating the virtual grid and Hall loop); a parallel $B$ would let electrons stream straight down $E$ to the anode and provide no confinement.

Since $\vec{F}=q\vec{v}\times \vec{B}$ is always perpendicular to $\vec{v}$, it changes direction but not speed — pure centripetal force, hence circular gyration (see Larmor Radius and Cyclotron Motion).

Electrons are ~$10^5$× lighter, so at any given field one species (electrons) is magnetized and the other (ions) is not — this single-field split is what lets you trap the accelerator-plasma while freely ejecting ions.

The co-located electrons neutralize the ion space charge, so the axial $E$-field can be strong inside a quasineutral plasma, giving ~5–10× the thrust density instead of hitting the Child–Langmuir current limit.

In $v_d=E/B$ the charge and mass cancel out of the force balance $qE=q{v}_{d}B$; the guiding-centre drift is a purely geometric property of the crossed fields, the same for electrons and ions (see Magnetic Mirror & Guiding-Centre Drifts).

Because $\Delta v=v_e\ln (m_0/m_f)$ grows with exhaust speed, a high-$v_e$ Hall thruster achieves a given $\Delta v$ with far less propellant than a chemical rocket (see Tsiolkovsky Rocket Equation).

Even at low speed its gyroradius ($r_L=m_i{v}_{\perp }/qB$) is comparable to or larger than the channel gap (~2 cm vs ~1.5 cm), so it is essentially unmagnetized; once accelerated it flies even straighter.

Zero from that point — $v_i=\sqrt{2q\Delta V/m_i}$ vanishes at $\Delta V=0$; ions gain speed only across the potential drop between anode and exit.

With no field the electrons are unconfined and stream straight to the anode; there is no Hall loop and no virtual grid, so it degenerates into an inefficient, unmagnetized plasma discharge with no directed ion beam.

Cross-field electron mobility drops toward zero, so no discharge current can reach the anode and the plasma cannot be sustained — the thruster starves and extinguishes.

No — with ${\omega }_c\tau \to \infty$ the perfectly magnetized electrons would gyrate forever without hopping across $\vec{B}$; some collisional or anomalous transport is required to carry current to the anode.

It accumulates at the tube ends because the azimuthal drift has nowhere to close, building an opposing field that stalls the drift — which is exactly why real thrusters are annular.