SME
Propulsion · Problem type
Ion thruster exhaust and beam thrust
Use this for “Xenon ions (131.29 u) leave through a 1100 V grid in a 1.5 A beam. Find vₑ, ṁ, F and Iₛₚ.”
Diagram
K = qΔV = 1 × 1,100 = 1,100 eV
K
=
1.76 ×
10
-16
J
v
=
√
2K/m
v
=
√
2 × 1.76 ×
10
-16
/2.18 ×
10
-25
v
=
4.02 ×
10
4
m/s
The + charge is pushed from the + plate toward the −, speeding up all the way.
This is an example. Type your own values to start a new problem.
Vb
Beam voltage
example
m
Ion mass
example
K
Ion energy
calculated · eV
vₑ
Exhaust speed
calculated
Ib
Beam current
example
ṁ
Propellant flow
calculated
F
Thrust
calculated
Iₛₚ
Specific impulse
calculated · s
Formulas
K
=
1
×
Vb
1,100 = 1 × 1,100
vₑ
=
√(2
×
K
×
1.602 ×
10
-19
m
)
40,207.0512
=
√(2
×
1,100
×
1.602 ×
10
-19
2.1801 ×
10
-25
)
ṁ
=
Ib
×
m
1.602 ×
10
-19
2.0413 ×
10
-6
=
1.5
×
2.1801 ×
10
-25
1.602 ×
10
-19
F
=
ṁ
×
vₑ
0.08208
=
2.0413 ×
10
-6
×
40,207.0512
Iₛₚ
=
vₑ
9.81
4,098.5781
=
40,207.0512
9.81
Formulas are worked in kg, N. The step-by-step shows the conversions.
Assumptions
•
Singly charged ions (q = 1 e), all accelerated from rest through the full beam voltage Vb.
•
Every ion in the beam is propellant: no neutral atoms leak out unused.
•
1 eV = 1.602 × 10⁻¹⁹ J; xenon, the usual propellant, is 131.29 u.
•
g = 9.81 m/s² defines specific impulse, so Iₛₚ = vₑ ÷ g.
Step-by-step
WE KNOW
Vb = 1,100 V, m = 131.29 u, Ib = 1.5 A
FIND
ion energy (K), exhaust speed (vₑ), propellant flow (ṁ), thrust (F), specific impulse (Iₛₚ)
Step 1 · Convert to the formula’s units
The formulas don’t work directly in the units you chose, so convert first (to kg, N).
m = 131.29 u = 2.1801 × 10⁻²⁵ kg (1 kg = 6.02214 × 10²⁶ u)
Step 2 · Find ion energy (K)
Use K = 1 × Vb
An ion of charge 1 e gains 1 eV for each volt it falls through.
K
=
1
×
1,100
K
=
1,100
eV
Step 3 · Find exhaust speed (vₑ)
Use vₑ = √(2 × K × 1.602 × 10⁻¹⁹ ÷ m)
Change
eV
to
joules,
then
solve
K
=
1
2
mvₑ²
for
the
speed.
vₑ
=
√(2
×
1,100
×
1.602 ×
10
-19
2.1801 ×
10
-25
)
vₑ
=
√(2,200
×
1.602 ×
10
-19
2.1801 ×
10
-25
)
vₑ
=
√(
3.5244 ×
10
-16
2.1801 ×
10
-25
)
vₑ
=
√
1.617 ×
10
9
vₑ = 40,207.051 m/s
Step 4 · Find propellant flow (ṁ)
Use ṁ = Ib × m ÷ (1.602 × 10⁻¹⁹)
Each
ion
carries
charge
e,
so
Ib
e
ions
leave
each
second,
each
of
mass
m
.
ṁ
=
1.5
×
2.1801 ×
10
-25
1.602 ×
10
-19
ṁ
=
3.2701 ×
10
-25
1.602 ×
10
-19
ṁ
=
2.04131 ×
10
-6
kg/s
Step 5 · Find thrust (F)
Use F = ṁ × vₑ
Thrust is the momentum carried off each second.
F
=
2.04131 ×
10
-6
×
40,207.051
F
=
0.08208
N
Step 6 · Find specific impulse (Iₛₚ)
Use Iₛₚ = vₑ ÷ 9.81
Divide
the
exhaust
speed
by
g
=
9.81
m/
s
2
.
Iₛₚ
=
40,207.051
9.81
Iₛₚ = 4,098.578 s
Step 7 · Convert the answers
Change each answer back to the unit you chose.
F = 0.08208 N = 82.0752 mN (1 N = 1000 mN)
Check
Put the numbers back in. Both sides should match.
1,100 = 1 × 1,100 ✓
40,207.051
=
√(2
×
1,100
×
1.602 ×
10
-19
2.1801 ×
10
-25
)
✓
2.04131 ×
10
-6
=
1.5
×
2.1801 ×
10
-25
1.602 ×
10
-19
✓
0.08208
=
2.04131 ×
10
-6
×
40,207.051
✓
4,098.578
=
40,207.051
9.81
✓
Topic check
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