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

3.3.42 · Physics › Rocket Propulsion

YE design KYUN exist karta hai? (vo problem jo ye solve karta hai)

Poori cleverness ek mass/mobility asymmetry hai:

Electrons ( $m_{e}$ ) xenon ions ( $m_{i}$ ) se ~ $1 0^{5}$ × halke hain.
$B$ aise choose karo ki electrons magnetized hon lekin ions nahi.

"Cross-field" aur "magnetized" ka MATLAB

First principles se gyroradius derive karo

$B$ mein ek charged particle Lorentz force feel karta hai jo circular-motion centripetal force supply karta hai:

$q v_{⊥} B = \frac{m v _{⊥}^{2}}{r _{L}} \Rightarrow r_{L} = \frac{m v _{⊥}}{q B} .$

Ye step kyun? Magnetic force koi kaam nahi karta (ye $⊥$ to $v$ hai), isliye ye sirf path ko bend karta hai — ek circle. $q v B$ ko $m v^{2} / r$ se balance karne par seedha radius milta hai.

Kyunki $r_{L} \propto m$ hai, same speed par ion radius $\sim m_{i} / m_{e} \approx 2.4 \times 1 0^{5}$ (Xe) zyada bada hota hai — ek hi $B$ electrons ko magnetize karta hai jabki ions essentially seedha udte rehte hain.

Hall current — electrons KYUN ek circle mein drift karte hain

Crossed $E$ aur $B$ ek steady drift velocity produce karte hain dono ke perpendicular.

$E \times B$ drift derive karo

Steady drift ka matlab hai average acceleration zero hai. Guiding-centre force balance lo:

$q (E + v_{d} \times B) = 0.$

Ye step kyun? Drift frame mein net force vanish ho jaati hai; electron ek centre ke around circle karta hai jo uniformly translate karta hai. $B$ se cross karo:

$q E \times B + q (v_{d} \times B) \times B = 0.$

$(v_{d} \times B) \times B = (v_{d} \cdot B) B - B^{2} v_{d}$ use karke aur $v_{d} ⊥ B$ lete hue:

$E \times B - B^{2} v_{d} = 0.$

Kyunki channel annular (ek ring) hai, $E$ (axial) $\times B$ (radial) azimuthally point karta hai — drift ring ke around ek loop ki tarah khud par close ho jaata hai. Ye closed-loop electron current hi Hall current hai, aur isiliye annular geometry zaroori hai: ye drift ko ek complete circuit form karne deti hai.

Ion acceleration & thrust

Ions unmagnetized hain, isliye ye simply axial potential drop $Δ V$ (anode se exit tak) neeche gir jaate hain:

$q Δ V = \frac{1}{2} m_{i} v_{i}^{2} \Rightarrow v_{i} = \frac{2 q Δ V}{m _{i}} .$

Ye step kyun? Ek charge ke liye energy conservation jo ek potential difference cross karta hai; plasma quasineutral hai isliye ions space-charge cap ke bina pura $Δ V$ "dekhte" hain.

Figure — Hall-effect thruster — cross-field discharge, annular channel

Worked examples

Common mistakes

Active recall

Recall Khud ko test karo (answers hide hain)

Kaun si species magnetized hai, aur kyun? → Electrons; $r_{L} \propto m$ hai isliye halke electrons ka tiny gyroradius hota hai jabki bhaare ions seedha udte hain.
Hall current ki direction? → Azimuthal (ring ke around), $E_{axial} \times B_{radial}$ se.
Ions ko actually kya accelerate karta hai? → Axial electric field (ek potential drop $Δ V$ ), $B$ nahi.
Annular kyun? → Taaki $E \times B$ drift ek loop mein close ho sake.
Gridded ion engines par kya advantage hai? → Quasineutral plasma space-charge (Child–Langmuir) current limit se bachata hai → zyada thrust density.

Recall Feynman: ek 12-saal ke bachche ko samjhao

Ek round racetrack imagine karo. Tum bhaare marbles (ions) ko ek taraf se bahut tezi se bahar phenk'na chahte ho. Unhe push karne ke liye ek strong "hawa" banane ke liye, tumhe choti tez bees (electrons) ki ek bheed chahiye. Lekin bees halki hain aur ud jaayengi — isliye tum ek invisible magnetic fence lagate ho jo bees ko track ke around circles mein ghoomata hai bajaaye bhagne ke. Trapped bees hawa banate hain aur gas atoms se takraa kar aur marbles banana mein madad karte hain. Bhaare marbles ke liye fence bahut bada hai, isliye ye seedha bahar nikal jaate hain — aur vo push hi rocket thrust hai.

Flashcards

Hall thruster channel ki kya geometry hoti hai aur kyun?

Annular (ring) taaki

E \times B

electron drift ek loop mein close ho sake (Hall current).

Hall thruster mein E aur B ki orientation kya hai?

E axial hai (thrust axis ke along), B radial hai annular gap ke across; ye crossed hain (

E ⊥ B

Larmor radius ka formula aur ions magnetized kyun nahi hote?

r_{L} = m v_{⊥} / (q B)

; kyunki

r_{L} \propto m

hai, bhaare ions ka

r_{L}

channel se bada hota hai aur ye almost seedha udte hain.

E \times B

drift velocity derive aur state karo.

Force balance

q (E + v_{d} \times B) = 0 \Rightarrow v_{d} = (E \times B) / B^{2}

, magnitude

E / B

, azimuthally directed.

Hall thruster mein ions ko kya accelerate karta hai?

Axial electric field (potential drop

Δ V

); B unpar koi kaam nahi karta.

Hall thruster gridded ion engines ki space-charge limit ko kyun beat karta hai?

Acceleration region quasineutral plasma hai (electrons present hain), isliye koi net space charge ion current density ko cap nahi karta.

Voltage

Δ V

se fall karne wale ions ki exhaust speed?

v_{e} = 2 q Δ V / m_{i}

q Δ V = \frac{1}{2} m_{i} v_{i}^{2}

se.

B

ke across axial electron current ko kya sustain karta hai?

Collisions aur anomalous (turbulent) cross-field transport, guiding centres ko har collision mein ~ek gyroradius hop karte hain.

Cyclotron frequency formula aur magnetization criterion?

ω_{c} = q B / m

; magnetized agar

ω_{c} τ ≫ 1

(har collision mein bahut saare orbits).

Thrust aur specific impulse ke formulas?

F = \overset{m}{˙} v_{e}

I_{s p} = v_{e} / g_{0}

Connections

Ion Thruster (Gridded) — space-charge / Child–Langmuir limit jo ye design bypass karta hai
Lorentz Force — gyromotion aur $E \times B$ drift ka origin
Larmor Radius and Cyclotron Motion
Plasma Quasineutrality
Tsiolkovsky Rocket Equation — kaise $v_{e}$ ( $I_{s p}$ ke zariye) $Δ v$ budget set karta hai
Magnetic Mirror & Guiding-Centre Drifts
Child–Langmuir Law