Foundations — FEEP, MEMS thrusters — micro-propulsion

This page assumes you know nothing. We build every letter the parent topic uses, one at a time, each earning its place before the next arrives.

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0. The picture everything hangs on

Before any symbol, look at the machine we are describing: a container of "stuff" with a hole, and the stuff shooting out.

Notice three things in that picture, because they become our first three symbols:

• a little blob of mass leaves per second,
• it leaves at some speed,
• and the ship feels a push the opposite way.

That is the whole subject. Now we name each piece.

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1. Mass — the symbol $m$

Picture: a lump on a scale. Bigger lump, bigger number.

Why the topic needs it: an ion (a single charged atom) has a mass — for indium it is a mind-bendingly small $m=1.9\times 10^{-25}$ kg. When we ask "how fast does one ion fly?", the answer depends on how heavy that ion is: a heavy ion is harder to fling fast, just like it is harder to throw a rock than a pebble.

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2. Speed and velocity — the symbol $v$ (and $v_e$)

Picture: an arrow whose length is the speed and whose point shows the direction of travel.

The special one — exhaust velocity $v_e$: the little $e$ is just a label meaning "exhaust", i.e. the speed of the stuff as it leaves the nozzle. This is the single most important number in the whole topic. Big $v_e$ = the flag "we barely use any fuel".

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3. Rate of change — the dot, $\dot{m}$

Here a notation enters, not just a quantity, so we slow down.

Picture: a tap dripping into a bucket. $m$ is how much water is in the bucket; $\dot{m}$ is how fast the drips add up — the flow.

Why this tool and not just $m$? Thrust is a continuous push. It does not depend on the total fuel; it depends on how fast fuel streams out right now. To talk about "per second" we need a rate, and the dot is physics' shorthand for a rate. We could write "kg per second" in words every time, but the dot keeps formulas short.

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4. Momentum and force — the symbol $F$

The link the topic lives on: force is momentum handed over per second. Fling mass $\dot{m}$ each second, each kg carrying speed $v_e$, and the push you feel is

$F=\dot{m}{v}_e.$

Picture: stand on a skateboard and throw baseballs forward — each throw shoves you backward. Throw them faster ($v_e$ up) or throw more per second ($\dot{m}$ up), and you roll away harder ($F$ up).

Why the topic needs it: this is literally what a thruster does. $F=\dot{m}{v}_e$ is the heart of every rocket, big or micro.

Recall Why can't a rocket just "push against" empty space?

It doesn't push against anything — it throws mass one way and the recoil (Newton's third law: every push has an equal opposite push) moves it the other way. ::: The exhaust is the thing it pushes against.

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5. Micro-prefixes — $\mu$, m, and scientific notation

The parent page keeps saying "µN" and "$10^{-25}$". These are just size labels.

Picture: a ruler zooming in. A newton is an apple; a milli-newton is a mosquito landing; a micro-newton is a single grain of pollen settling. That "grain of pollen" push is exactly the gentleness CubeSat attitude control needs.

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6. Charge and electric field — $q$, $V$, $E$, ${\epsilon }_0$

FEEP moves charged atoms, so we need the language of electricity.

Picture: voltage is a hill; charge is a ball on it. Steeper/taller hill ($V$ big) → faster ball at the bottom.

Why the topic needs all this: FEEP's trick is to make $E$ so enormous at a sharp tip that it literally rips charged atoms off the liquid metal and then uses voltage $V$ to accelerate them. No charge, no grip; no voltage, no speed.

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7. Energy conservation — the equation $qV=\frac{1}{2}m{v}_e^2$

This is the single most-used derivation on the parent page, so let us earn every symbol in it.

The idea: energy is never lost, only converted. A charge $q$ sliding down a voltage hill $V$ gives up electrical energy $qV$. Where does it go? Into motion. So

$\underset{\text{electrical energy spent}}{\underbrace{qV}}=\underset{\text{motion energy gained}}{\underbrace{\frac{1}{2}m{v}_e^2}}.$

WHAT we do next: solve for the speed. WHY: we want $v_e$; the equation currently hides it inside a square. Undo the square with a square root:

$v_e=\sqrt{\frac{2qV}{m}}.$

WHAT it looks like: a heavier ion (big $m$ on the bottom) comes out slower; a bigger voltage (big $V$ on top) comes out faster — exactly matching the "steeper hill = faster ball" picture. This same reasoning powers Ion Thrusters and Electrospray & Colloid Thrusters.

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8. The exponential $e^x$ — for the rocket equation

The parent's fuel formula $m_p/m_0=1-e^{-\Delta v/v_e}$ uses a strange letter $e$.

Picture: a hot cup of coffee cooling. It drops fast at first, then slower and slower, easing toward room temperature — that curve is $e^{-x}$.

The magic shortcut we actually use: when $x$ is tiny, $e^{-x}\approx 1-x$. So for a CubeSat where the needed speed change $\Delta v$ is tiny compared to the huge $v_e$,

$1-e^{-\Delta v/v_e}\approx \frac{\Delta v}{v_e}.$

That is why "$m_p\approx m_0\Delta v/v_e$" appears — it is the small-$x$ face of the exponential. See Rocket Equation and Specific Impulse for the full story.

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9. Gas symbols — $p$, $T$, $R$, $M$, $\gamma$, $A$, $\rho$, $\mu$

The MEMS section swaps ions for hot gas. New cast, same job.

Picture: a shaken fizzy bottle. High pressure and temperature inside; open the cap and the gas rushes out, converting its stored "push" into a fast jet — that is the nozzle turning enthalpy into $v_e$.

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10. Reynolds number — $Re=\frac{\rho vL}{\mu }$

The parent says micro-nozzles suffer because "$Re$ is small". Here is what that means.

Why this tool and not another? Engineers need one number to predict "will this flow behave nicely or get bogged down by wall friction?" $Re$ is exactly that number. In a micro-nozzle $L$ is tiny, so $Re$ is tiny — friction dominates and the ideal-gas formula over-promises. Full story: Reynolds Number.

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The prerequisite map

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Equipment checklist

Test yourself — cover the right side and answer before revealing.