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

False — magnetic force hamesha velocity ke perpendicular hoti hai ($\vec{F}=q\vec{v}\times \vec{B}$), isliye woh zero work karti hai. Ions energy sirf axial electric field / potential drop se gain karte hain.

True — kyunki $r_L=m{v}_{\perp }/(qB)$ mass ke saath scale karta hai, ~$2.4\times 10^5$ guna bhaari xenon ion ka gyroradius channel se kaafi bada hota hai jabki electron ka sub-millimetre hota hai. Dekho Larmor Radius and Cyclotron Motion.

True — electrons wahan hote hain jahan ions accelerate karte hain, net charge cancel ho jaata hai, aur yehi reason hai ki Hall thruster Child–Langmuir space-charge cap se bachta hai (dekho Plasma Quasineutrality aur Child–Langmuir Law).

False — Hall current ek closed azimuthal loop hai trapped electrons ka jo kabhi bahar nahi jaata; exhaust woh stream hai jismein unmagnetized ions axially baahr nikaale jaate hain.

False — force balance $qE=q{v}_{d}B$ se milta hai $v_d=E/B$. Units confirm karte hain: $(\text{V/m})/\text{T}=\text{m/s}$, jabki $EB$ ke dimensions galat hain.

False — collisions aur anomalous (turbulent) transport electron guiding-centres ko har event mein $\vec{B}$ ke across ek gyroradius hop kara dete hain, jisse finite cross-field mobility milti hai jo discharge current ko sustain karti hai.

False — bahut zyada $B$ electron cross-field mobility choke kar deta hai (anode current starve ho jaata hai) aur eventually ions ko magnetize karna shuru kar sakta hai, us beam ko deflect karke jo seedha baahr fire karni thi.

Partly, lekin asli reason hai iska large mass (ionization energy per unit momentum low hai aur magnetization asymmetry easy hai): bhaare ions zyada momentum per charge carry karte hain, aur unka bada $r_L$ unhe unmagnetized rakhta hai jabki electrons trapped rehte hain.

Error — $\vec{E}$ (axial) $\times \vec{B}$ (radial) azimuthally (ring ke around) point karta hai, axially nahi. Yeh electrons ko ek circle mein move karta hai; yeh ion-ejection mechanism nahi hai.

Error — Hall current ko steady rehne ke liye khud par close hona padta hai. Ek tube ke ends hote hain jahan charge pile up hokar drift stall kar deta hai; annular ring azimuthal drift ko ek endless loop deta hai.

Error — thrust hai $F=\dot{m}{v}_e$ jahan $v_e$ ion exhaust speed hai ($\sqrt{2q\Delta V/m_i}$). Hall drift $v_d$ internal electron circulation hai, exhaust nahi; ise use karna dono currents ko double-confuse karna hai.

Error — magnetized ka matlab hai gyroradius channel se chota hai, yaani $r_{L,e}\ll L_{\text{channel}}$. Bada $r_L$ (ion case) unmagnetized hota hai.

Error — quasineutrality net space charge remove kar deti hai, isliye ions poore $\Delta V$ se bina kisi shielding cap ke guzarte hain. Yahi gridded Ion Thruster (Gridded) par advantage hai.

Error — bahar nikalti beam positive ions ki stream hai aur spacecraft ko negative charge kar degi, eventually ions ko waapas kheench legi. External electron source (neutralizer) exhaust aur craft charge-balance rakhne ke liye zaroori hai.

Perpendicular $B$ electrons ko accelerating $E$ ke transverse trap karta hai (virtual grid aur Hall loop create karta hai); parallel $B$ electrons ko seedha $E$ ke saath anode tak stream karne deta aur koi confinement nahi hota.

Kyunki $\vec{F}=q\vec{v}\times \vec{B}$ hamesha $\vec{v}$ ke perpendicular hai, yeh direction change karta hai speed nahi — pure centripetal force, isliye circular gyration hoti hai (dekho Larmor Radius and Cyclotron Motion).

Electrons ~$10^5$× halke hain, isliye kisi bhi given field par ek species (electrons) magnetized hoti hai aur doosri (ions) nahi — yeh single-field split hi woh cheez hai jo accelerator-plasma trap karte hue ions freely eject karne deta hai.

Co-located electrons ion space charge neutralize kar dete hain, isliye axial $E$-field quasineutral plasma ke andar strong ho sakta hai, Child–Langmuir current limit hit karne ki jagah ~5–10× zyada thrust density milti hai.

$v_d=E/B$ mein charge aur mass force balance $qE=q{v}_{d}B$ se cancel ho jaate hain; guiding-centre drift crossed fields ki ek purely geometric property hai, electrons aur ions dono ke liye same (dekho Magnetic Mirror & Guiding-Centre Drifts).

Kyunki $\Delta v=v_e\ln (m_0/m_f)$ exhaust speed ke saath badhta hai, ek high-$v_e$ Hall thruster ek given $\Delta v$ chemical rocket se kaafi kam propellant mein achieve kar leta hai (dekho Tsiolkovsky Rocket Equation).

Kam speed par bhi iska gyroradius ($r_L=m_i{v}_{\perp }/qB$) channel gap ke comparable ya zyada hota hai (~2 cm vs ~1.5 cm), isliye yeh essentially unmagnetized hai; ek baar accelerate hone par yeh aur seedha udta hai.

Us point se zero — $v_i=\sqrt{2q\Delta V/m_i}$ $\Delta V=0$ par vanish ho jaata hai; ions speed sirf anode aur exit ke beech potential drop ke across gain karte hain.

Bina field ke electrons unconfined hote hain aur seedha anode tak stream karte hain; koi Hall loop nahi aur koi virtual grid nahi, isliye yeh ek inefficient, unmagnetized plasma discharge mein degenerate ho jaata hai jismein koi directed ion beam nahi hoti.

Cross-field electron mobility zero ki taraf drop karti hai, isliye koi discharge current anode tak nahi pahunch sakta aur plasma sustain nahi ho sakta — thruster starve karke bujh jaata hai.

Nahi — ${\omega }_c\tau \to \infty$ ke saath perfectly magnetized electrons hamesha ke liye gyrate karte rehte $\vec{B}$ ke across hop kiye bina; koi colisional ya anomalous transport current anode tak le jaane ke liye zaroori hai.

Yeh tube ke ends par accumulate hota hai kyunki azimuthal drift kahin close nahi ho sakti, ek opposing field build karta hai jo drift stall kar deta hai — aur yehi reason hai ki real thrusters annular hote hain.