¶ E1 - Introduction to electricity and electronics
¶ What this course aims
Right now you might know what has been taught in school. If you're among the lucky few, electricity was taught well to you. If you're not, you probably can't even name what a resistor is...in fact, let's try this as a first experiment: Try to articulate a good definition of a resistor, make it as complete as possible and try to really get at the core of how one of those components might work.
Exercise: Explain in detail what a resistor is and how it works; try to really get deep into the details and see if you get stuck.
This course aims to give you the insight needed to answer the aforementioned question and many more like it regarding electricity and electronics.
Call it "giving you a strong foundation" if you will.
¶ Where the study of electricity started
The story starts with a puzzle the ancient Greeks noticed around 600 BC: rub a piece of amber with fur, and it attracts small, light objects, say, a hair, a feather. The Greek word for amber was ēlektron and that single word is where "electricity" comes from. For over two thousand years, though, this stayed a curiosity with no explanation.
Real study began around 1600, with William Gilbert. He became the first to investigate the effect systematically (and to separate it from magnetism). Over the next century the pieces filled in: that electricity could flow through some materials but not others and that there seemed to be two opposite kinds of charge, with Charles-Augustin de Coulomb later measuring the force between them. In the 1750s, Benjamin Franklin famously flew a kite in a thunderstorm to show that lightning itself is electrical, and gave us the labels we still use: positive and negative. (That's why we draw current flowing from + to −; this convention will haunt us until the end of our days and beyond =)) )
All of this was static electricity. The turning point came in 1800, when Alessandro Volta invented the battery (the "voltaic pile"). For the first time, scientists had a steady, controllable current to work with. In the decades that followed, a handful of researchers worked out the precise rules of how electricity behaves:
- André-Marie Ampère studied how currents create magnetism;
- Georg Ohm found the relationship between voltage, current, and resistance (you'll meet his law in a few minutes);
- Michael Faraday discovered how to generate electricity from magnetism; the basic principle behind every power station and electric motor;
- James Clerk Maxwell (the absolute OG of electricity) tied electricity and magnetism together into a single theory. The laws that tied those things together were discussed in physics, they're called Maxwell's laws.
Fun fact: Maxwell's laws were first expressed in quaternions. But they were pretty complicated to understand at a first glance (Maxwell found 20 equations with 20 variables). Lord Kelvin (the biggest hater in physics history and the biggest loser in physics history at the same time) called quaternions: "an unmixed evil to those who have touched them in any way, including Clerk Maxwell" (in a letter in 1892). Oliver Heaviside (an absolute beast that taught himself calculus and electricity from scratch) was, thus, determined to simplify them; and simplify he did: all the way down to 4 equations with 4 variables expressed in vector notation. This helped vector notation gain traction too.
Please do read this article about Heaviside; I promise it's inspirational, informative and well-written.
Fun fact: Maxwell has his own demon! We wrote about it on our blog in this article.
¶ What even is "an electricity"? Can I have a few?
¶ What is charge?
For the comprehensive answer, you can re-visit the lecture notes on the Physics class on this Wiki.
If you didn't really get along with those explanations, here is a basic and friendly explanation and here is the more in-depth one. I like the second one better but feel free to use either.
For our purposes, three ideas matter most:
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Charge comes in two flavors: negative ("-"; the one that electrons have) and positive ("+"; the one that protons have). Like charges repel and opposite ones attract.
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Charge is a FUNDAMENTAL property of particles. Fundamental means that it sits at the basis of how the Universe works and nobody was asked (not even me) when they were "made" =). They just work that way (that doesn't mean we can't challenge them, their existence or the way they are formulated, this is what science is all about; but for the purposes of electronics...well for any purposes whatsoever they work as explained).
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Charge is measured in coulombs (C). One coulomb is a fixed amount of charge, specifically, the combined charge of about 6.24 × 1018 electrons (over six billion billion of them). Put the other way round, a single electron carries an almost unimaginably tiny charge of about 1.602 × 10-19 coulombs. So a coulomb is a lot of electrons' worth of charge. (The quantity of charge is usually written with the symbol Q.)
¶ What is current?
Start with the picture, not the formula. Inside a metal wire there's a vast crowd of free electrons; charge that's able to move. Left alone, they just jiggle about randomly, going nowhere in particular. But give them a push (that's what a battery does, more on that when we reach voltage), and the whole crowd starts drifting in the same direction. That organised flow of charge is what we call electric current.
Therefore, when we talk about CURRENT, we are refering to that flow. Without electrons moving from one place to another, there is NO FLOW.
So how do you measure a flow? The same way you'd measure any flow: pick a spot and count how much goes past it each second. Stand beside a river and measure how many litres of water flow past per second; stand by a motorway and count the cars passing a point each minute. Current is exactly that idea, for charge: how much charge flows past a point in the wire each second. More charge per second means a bigger current.
(now first excuse my artistic skill - or lack thereof - and second don't imagine electrons are that big compared to a wire or that you will see so little of them if you were to shrink to that level; you would see billions upon billions of electrons at each point)
This way of expressing electricity by referencing water is called the water analogy, and it's worth holding onto: if charge is the water, current is the flow rate, not how much water exists in total, but how fast it's streaming past you.
Although the water (or hydraulic) analogy is useful for demonstrating basic phenomena, it really does come short when explaining more complicated stuff.
Read here about how the hydraulic analogy is lacking in some areas.
One neat consequence: in a simple circuit it doesn't matter where along the wire you stand, the same amount of charge flows past every point, because charge isn't created or piling up anywhere inside the wire. Whatever flows in, flows out, just like water in a pipe. (Hold that thought because it comes straight back when we meet Kirchhoff's laws.)
Now the formula. If an amount of charge ΔQ flows past a point in a time Δt, the (average) current I is just that charge divided by the time it took:
I = ΔQ / Δt
Current is measured in amperes (A), and the definition is exactly what the picture suggests: one ampere is one coulomb of charge passing a point every second (1 A = 1 C/s). So if 2 coulombs drift past a point in one second, that's a current of 2 amps. (The symbol for current is I, from the French intensité.)
If you meet the textbook version: some books define current as "the charge passing through a cross-sectional area per unit time." Don't let that throw you off guard, a "cross-sectional area" is just the formal name for a slice straight across the wire. Pick a slice, count the charge flowing through it each second, and you're measuring the exact same thing.
The big idea: electric current is simply charge on the move. A conductor (copper, iron, any metal) is full of free electrons that are able to flow. Give them a push (a voltage), and they all start drifting through the material together. That flowing charge is the current. So the whole thing boils down to one sentence: get the electrons moving, and you've got a current. Simple as that.
Wait...I just used the magic word: VOLTAGE. That without which no electron moves through a conductor; that which harnesses the current =)).
But what is it?
¶ What is voltage?
This one is funny because this is perhaps the least well explained concept in electricity.
The term "difference of potential" gets thrown around to explain voltage as if they're not synonyms. Explaining "voltage" as "difference of electric potential" is like explaining "to start" as "to begin".
They, indeed, mean the same thing. But one does very little to explain the other and vice-versa.
To understand what voltage is, let's look at the thing that makes the electrons move. A voltage source. A battery!
¶ The battery as a "pump".
The battery DOES NOT supply electrons. It's not an electron factory. It's an "electron pump".
A pump does not supply water, it just pushes water that already exists down the pipes.
So if a battery doesn't supply the electrons, what is it doing?
You probably either know this from Physics or will learn about it in Chemistry soon but the wire is already packed with free electrons, sitting there ready to move. The battery never had to hand any over; they were there all along (this is the main property of a material that conducts electricity, it has FREE ELECTRONS). What it does is more interesting: it builds and maintains a gradient of electrons.
Using the chemical energy stored inside it (those reactions are the battery's entire job and the reason it eventually goes flat), a battery drags electrons onto one terminal and strips them off the other. The result is:
- the negative (−) terminal ends up crowded with electrons —> electron-rich;
- the positive (+) terminal ends up starved of them —> electron-poor.
We call each "access point" into ANY electronic component a terminal.
The + and - "sides" of a battery are the positive and negative terminal respectively.
The little "legs" of an LED are its terminals.
Terminal in the context of electronics means just a way through which the component is accessible for the current to flow. It's the way a component interacts with the outside world.
That imbalance (crammed at one end, empty at the other) is the gradient. And a gradient is a slope: the electrons piled up at the terminal sit at a higher energy "level" than the empty + terminal, and like anything at the top of a slope, they're driven to move toward the bottom.
Connect a wire between the terminals and you hand them a path. Now the gradient goes to work: it pushes the free electrons already in the wire, nudging the whole crowd to drift from the electron-rich − side, through the circuit, toward the electron-starved + side. That drifting charge is your current.
Here's the clever bit — why a battery keeps working instead of evening out in an instant. As electrons drain off the − terminal and pile onto the +, the battery's chemistry immediately pumps them back the other way, internally, from + to −. It re-steepens the slope exactly as fast as the circuit tries to flatten it. So picture a battery as a pump that maintains an electron gradient: it never runs out of electrons to push (the wire provides those) —> it runs out of the chemical energy needed to keep pumping, and that is the moment the battery dies.
So, to finally answer the question: a battery doesn't supply electrons. It supplies the push. It spends chemical energy to hold a steep electron gradient between its terminals and that gradient is precisely what we've been calling voltage.
Electrons move from a "crammed place" (high electron density) to a more "free space" (low electron density) because they want to spread out nicely. This is fundamental to physics (particles, matter, energy, all wanting to be spread out). You can read more about why electrons don't really want to "meet" here (to call this article by the Perimeter Institute "excellent" would be an understatement).
You can read more about why matter wants to spread out here.
¶ Finally: the difference in potential.
Perhaps by this point you might've caught on to what people mean when they say "difference in potential". If not, let's break it down thoroughly.
As we mentioned above, the battery effectively pushes electrons from its + terminal to its "-" terminal, making the "-" side saturated. All that density of electrons creates a push amongst them, electrons don't like when they're crammed, they want to be evenly distributed and so they get pushed both by the battery's pumping and by one another.
Voltage IS THE FORCE OF THAT PUSH.
Right?
Wrong!
VOLTAGE is not a FORCE.
You might've seen it named as "electrical tension" and you might've (understandably) thought that it's a force, just like mechanical tension.
Here's the part that finally makes "difference of potential" mean something, instead of being a fancy synonym for "voltage."
When the battery crams an electron onto that crowded "−" terminal, it isn't just moving the electron, it is effectively pouring/stamping energy onto it. Real joules. Shoving one more electron in among a crowd that's all repelling it takes a lot of work (and work in the literal physics sense), and that work gets stored in the electron as potential energy, much like winding up a spring or drawing back a slingshot. The electron comes out of the battery loaded.
How much energy gets stamped onto each unit of charge? That number is the voltage:
V = U / q (energy / charge; call it "how many units of energy I can carry per unit of charge)
Thus 1V = 1J/1C. To have a voltage of 1V is literally to imprint one whole joule of energy upon one whole Coulomb worth of electrons.
A 1.5V, AAA-battery imprints 1.5 joules onto every coulomb of charge it pushes out. That's the entire meaning of "1.5 volts" -> 1.5 joules handed to each coulomb.
¶ Okay, let's prove it. Let's count the joules in one electron.
Talk is cheap, send calculations. Let's actually compute the energy a 1.5 V battery pours into a single electron as it travels around a simple LED circuit, from the − terminal, through the LED, to the +.
The tool is the formula from a moment ago, V = U / q, just rearranged to solve for the energy:
U = q × V -> energy poured into a charge = that charge × the voltage it travels through
Now we plug in two numbers. The charge is one electron; the voltage is 1.5 V (and remember, a volt is a joule per coulomb):
- charge of one electron: q = 1.602 × 10-19 C
- battery voltage: V = 1.5 V = 1.5 J/C
So:
U = (1.602 × 10-19 C) × (1.5 J/C)
Watch the units do the work , the coulombs cancel, leaving pure joules:
U = (1.602 × 1.5) × 10-19 (C × J/C) = 2.4 × 10-19 J
There it is. One electron gains about 2.4 × 10-19 joules crossing a 1.5 V battery. Not a force. Not a push. A literal, countable scrap of energy, handed to one electron.
It's like the battery is giving the electrons each a small sip of White Monster and they just want to run to burn it.
"But that's basically nothing!" ... correct. It's an absurdly tiny number. The whole trick is in how many electrons do it. Say our LED is drawing a typical 20 mA (0.02 amps). How many electrons is that per second?
electrons per second = I ÷ q = 0.02 A ÷ (1.602 × 10-19 C) ≈ 1.25 × 1017 electrons every second
So the energy delivered each second is:
(2.4 × 10-19 J per electron) × (1.25 × 1017 electrons/s) ≈ 0.03 J/s = 0.03 W
And here's the satisfying part, we can check it against power = voltage × current:
P = V × I = 1.5 V × 0.02 A = 0.03 W
Exactly the same answer. Counting joules one electron at a time gives the identical power to the textbook formula P = V × I, because that's all that formula really is: the energy per charge, times the charges per second.
A real-world honesty note: a typical red LED actually needs about 1.8–2 V to light, so a single 1.5 V cell on its own usually won't quite manage it, in a real build you'd use something like a 3 V supply with a current-limiting resistor (exactly what we'll wire up in the homework). None of the math above changes; you just plug in your real voltage. At 3 V, each electron would gain 1.602 × 10-19 × 3 ≈ 4.8 × 10-19 J instead.
One very important consequence of what we discussed above is this: there is NO voltage at a SINGLE point. Voltage must always be relative between two points.** So there is no voltage at point A** because there is no difference in the energy of electrons from point A relative to point A; energy can't be created out of thin air, electrons don't accelerate for no reason. You can say there is a voltage between point A and B. Or, and that's the standard when you actually start working in electronics and is what is assumed you should know when you read a datasheet: voltage at point A means the difference of potential between point A and another point on the circuit we agree to consider as electrically neutral, called the GND! More on that later
Wait...power? We haven't discussed it yet. So there ya' go:
¶ What is power?
Power is the rate at which energy gets used or delivered: how many joules per second. Its unit is the watt (W), and a watt is exactly that, one joule of energy per second (1 W = 1 J/s). A 60 W bulb burns 60 joules every second; a 5 W phone charger delivers 5. (The symbol for power is P.)
Notice that's a rate, not a total. A 60 W bulb left on for an hour uses far more total energy than one left on for a minute, but its power is 60 W the entire time. Power is the speed of the spending, not the size of the final bill (just like me drinking 3 Monsters worth of energy per day when I work; if I'm working for 3 days, it'll be 9 Monsters; if I'm working for 5, I'll need 15; if it's the retake exam session that'll be like 1000).
In a circuit, power comes with one formula you'll use constantly:
P = V × I (power = voltage × current)
And here's the lovely part, you already have everything you need to see why it's true. Just look at the units:
- voltage is energy per charge —> joules per coulomb (J/C)
- current is charge per second —> coulombs per second (C/s)
Multiply them, and the coulombs cancel:
(J/C) × (C/s) = J/s = watts
That's the whole thing. Energy-per-charge times charges-per-second is energy-per-second, which is power. So P = V × I isn't a formula to memorize; it's just energy per charge × charges per second, which can't be anything other than the rate of energy use.
Our LED from before, one last time: P = 1.5 V × 0.02 A = 0.03 W. The battery is pouring energy into that circuit at three-hundredths of a joule every second and nearly all of it leaves the LED as light and heat.
¶ Clarification: how fast does it all happen?
We've been saying current is the whole crowd of free electrons drifting through the wire together. Fair thing to pin down before we move on: how fast are they actually going?
Your gut probably says fast. You flip a switch on the wall and the light across the room turns on instantly (which is almost as fast as Muhammad Ali at his prime; you can skip to 0:56 if you want to get the joke =) ), no waiting, no delay, no matter how far away the bulb is. So the electrons must be screaming through the wire at something close to the speed of light, right?
The electrons are NOT racing along anywhere near the speed of light. Not even in the same ballpark.
I'm not kidding, the difference at which electron cruise through the conductor VS the speed of light is like comparing a turtle to a bullet.
They're snails. Absolute snails. The thing that crosses the room at near light-speed is something else entirely, and it's something you've already met in Physics. Hang on, we'll get there.
First, how slow is "snail," exactly? Two completely different motions are going on for any one electron in a wire:
The following are AI-made animations. They have their defects, don't take them ad-literam. They illustrate nicely the point I'm trying to make but they're not perfect. Treat them as a small help towards your mental model, NOT as scientific reference.
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Random thermal jiggling. Even with no battery connected at all, the free electrons are tearing around in random directions ridiculously fast (think 106 m/s, over a million meters every second). But it's random: for every electron darting left there's one darting right, so it all cancels and the crowd as a whole goes nowhere.
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Drift. Switch the battery on and you tilt the whole game, very slightly, toward the + terminal. On top of all that frantic random motion, the crowd now creeps, as a whole, in one direction. That slow net creep is the drift velocity, and it's the only part that actually counts as current.
And the drift velocity is tiny. For an ordinary copper wire we can just compute it:
The drift-speed formula. Picture first: push more current through and the electrons hustle faster; pack more free electrons into the metal (or use a fatter wire) and they can dawdle along more slowly for the same current. As a formula:
-v = I / (n × q × A)
-> I: the current (amps
-> n: how many free electrons are packed into a cubic meter of the metal (for copper, a measured ~8.5 × 1028 per m3; eighty-five thousand billion billion billion of them, per cubic meter)
-> q: the charge on one electron, our old friend 1.602 × 10-19 C
-> A: the wire's cross-sectional area (that "slice straight across the wire" from the current section)
Plug in a perfectly ordinary case. Say 1 amp flowing through a copper wire of cross-section 1 mm2 (that's 1 × 10-6 m2):
v = 1 ÷ (8.5 × 1028 × 1.602 × 10-19 × 1 × 10-6) ≈ 7 × 10-5 m/s
That's about seven hundredths of a millimeter every second. Read that again: not meters, not even whole millimeters. At that crawl, a single electron would take nearly 4 hours to travel one meter down the wire. (Drop the current to our LED's 20 mA in the same wire and it's another ~50× slower still. The poor things are basically parked.)
So here's the paradox, stated bluntly: the electrons crawl, but the light comes on instantly. How?!
So what actually moves at light-speed? The light comes on right away not because an electron sprinted from the switch to the bulb, but because the electron that was already sitting at the bulb got its push almost immediately.
Remember the battery-as-a-pump section: the wire is already packed full of free electrons, everywhere, all along its length, before you connect anything. When you close the circuit, the battery sets up its electric field (the "push", the gradient), and that field does NOT crawl along at electron-speed. It sweeps down the entire wire at a huge fraction of the speed of light (order 108 m/s: nearly all of c, depending on the wire and what's around it).
So the push reaches every electron in the circuit at very nearly the same instant, and the whole crowd (the ones sitting right at the bulb included) lurches into its slow drift together. The bulb lights immediately; each individual electron has barely twitched.
And that fast-traveling push, the electric field, is the very thing you met in Physics, with Maxwell's equations (the same Maxwell from our history section up top). The headline result of Maxwell's whole theory is that electric and magnetic fields can ripple outward as waves that travel at the speed of light, and that light itself is one of those waves. The "signal" that flies down your wire the moment you flip the switch is one of these electromagnetic disturbances, with the conductor mostly just guiding it along. So whenever something in a circuit genuinely moves at light-speed, it's the field, the Maxwell thing, doing it. Never the electrons.
The big idea: the electrons themselves crawl, fractions of a millimeter per second. What travels at nearly the speed of light is the electric field, the push, which reaches every electron in the circuit almost at once and sets the whole crowd drifting together. That's why the bulb lights the instant you flip the switch, even though no single electron has actually gotten anywhere. The charges are slow; the signal is fast.
For the brave, the rabbit hole goes deeper. Strictly speaking, the energy doesn't even ride inside the wire alongside the electrons at all. It travels in the electromagnetic field in the space around the wire, and the wire mostly just guides it. This is genuinely mind-bending, and exactly how to teach it honestly is a little contested. Derek at Veritasium kicked off a famous argument with this video; then Brian at AlphaPhoenix went and built the experiment with 1,000 meters of wire and an oscilloscope and actually measured what happens: a tiny current shows up almost immediately (the field jumping straight across the gap), but the full current only builds up once the signal has run the entire length of the wire. Totally optional, but if this section scratched an itch, that's where the itch leads.
¶ What is resistance?
Cast your mind back to those jiggling electrons from a moment ago, ricocheting off all the fixed metal atoms. Hold onto that picture, because it is the whole secret of resistance.
When a voltage finally gets the crowd drifting, the electrons do not get a clean, free run down the wire. They are constantly crashing into the vibrating atoms of the metal lattice (lattice being the very cutely-aranged-looking atomic structure of metals, as was stated in Physics), they're knocked sideways, slowed, deflected, then shoved onward again by the field. Every one of those collisions steals a little of their energy and dumps it into the lattice as heat (every impact releases heat; you can take a look here to see how a piece of iron glows red-hot from repeatedly striking it with a hammer). That constant, built-in opposition to the flow of charge is exactly what we call resistance.
So resistance is not some exotic thing. It is just how hard a material fights the current trying to pass through it. Some materials fight a lot (poor conductors, or outright insulators); some barely fight at all (good conductors, like the copper in our wire).
Resistance (R) is measured in ohms, symbol Ω (the Greek capital omega). One ohm is defined so that 1 Ω = 1 volt per amp (1 V/A): it tells you how much voltage you have to push with to drive one amp through the thing. More ohms means more push needed for the same current, which means a stingier flow.
And that little tug-of-war between volts, amps, and ohms is not a coincidence. It has a name, and it is the very next thing we will nail down: Ohm's law.
The resistance of a material depend on two things: the type of material itself AND how good of a "highway" for current a specific piece of material is. In a very short and thick piece of copper wire, electrons can travel really easily, making the resistance very low. If you take a longer and narrower piece of copper wire (same material), that will have a HIGHER resistance because the electrons have to make their way across a longer highway to get to the other end of the circuit.
Thus the formula for resistance is: R = ρ × L/A, where
- ρ is the resistivity of the material (a literal characteristic that the material is -and excuse my poor choice of words here- "born" with; just like your eye color or your skin color)
- L is the length of the conductor
- A is the cross sectional area of the conductor (if we cut a slice of the wire, how many square meters does it have?).
As you can see from the diagram:
¶ Ohm's "law"
Now that you know what resistance is, here is the rule that lets you actually do something with it.
Back in the 1820s, a poor son of a locksmith, Georg Simon Ohm, (yes, the ohm is named after him) ran a pile of careful experiments and noticed something beautifully simple about a lot of everyday materials: the current flowing through them is directly proportional to the voltage across them. Push twice as hard and you get twice the flow. Triple the voltage and you triple the current. The thing setting the exchange rate between "how hard you push" and "how much flows" is, you guessed it, the resistance.
The story of how Ohm's father taught him and his brother mathematics is beautiful and remarkable as is the story of how Ohm was actually ostracized for his new discovery. Read more about him here.
Write that down and you get the single most useful equation in all of basic electronics:
I = V / R (current = voltage / resistance).
And because it is just a product of three quantities, you can shuffle it into whichever form the problem hands you:
- V = I × R -> the voltage needed to push a given current through a given resistance.
- I = V / R -> the current you actually get: more push (V) means more current; more resistance (R) means less. A bigger fight, a stingier flow. (although I much prefer this form, for reasons that were discussed for reasons discussed by former NASA Postdoctoral Fellow Ali Alqaraghuli here and here; please watch them both, he has some really interesting perspectives on how physics and engineering should be taught ).
- R = V / I -> and this is literally how you measure a resistance: apply a known voltage, measure the current that flows, divide one by the other.
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The main idea: Ohm's law is just current = push divided by stinginess. Crank the voltage and the current climbs; crank the resistance and the current gets choked. That is the whole relationship, and you will reach for it more than any other formula in this course.
Careful: Ohm's law is not a law of the Universe. Unlike charge or energy, which genuinely always behave themselves, Ohm's law is an empirical rule, something many materials happen to obey, not something everything must. Materials that follow it (metals, and ordinary resistors at a steady temperature) are called ohmic. Plenty of components are non-ohmic and flatly ignore it. The big example is sitting right in our circuit: an LED's current does not climb in neat proportion to voltage at all, which is exactly why you can never tame an LED with a single resistance value, and exactly why it needs a current-limiting resistor babysitting it. Even a plain wire drifts off perfect ohmic behavior once it heats up, because R itself creeps up with temperature.
Moreover, even for ohmic materials this is more something "right on average" so to speak. It's a macro effect. Current still behaves according to Maxwell's laws. Ohm's "law" is less of a law and more of a phenomenological "rule" that works in the case of some materials and with a lot of restrictions (for example, Ohm's law completely falls apart in cases in which the electric field is too strong; in power electronics .
Which raises the obvious question: if a resistor is the component we keep reaching for to set a current, what actually is one, and how do we pick the right value? That is next.
¶ The resistor (the component)
Remember that very first exercise, the one where I asked you to define a resistor and then watch yourself get stuck? Time to pay it off properly.
A resistor is, at heart, the most honest component in electronics. Its entire job is to have a specific, deliberate, stable resistance, and nothing else. Where a wire tries its best to have no resistance (i.e. leave electricity to its own devices and not interfere), a resistor is the dependable fellow: hand it a voltage and it passes a current in the exact, predictable proportion Ohm's law promises. A controlled lump of stinginess. That is the whole idea. Controlled current.
What it's actually made of. You already know, from the resistance section, where resistance comes from: electrons fighting their way past the atoms of a material and losing energy to collisions (remember the hammer hitting the iron rod). A resistor is just a material that makes electrons bump into its atoms a certain, controlled, amount. The more the electrons bump and lose energy (we call this a VOLTAGE DROP across the resistor; remember voltage is just the difference in energy from one point to another; a voltage DROP across a resistor just means the electrons are losing some of those Joules when passing through the resistor) . The most common modern type is a film resistor: a thin film (literally a very thing pelicule) of carbon (a carbon-film resistor) or a metal alloy or metal oxide (a metal-film resistor) deposited onto a tiny ceramic rod. To dial in the exact value, a fine helical groove is cut into that film, forcing the current to spiral the long way around the rod (a longer, narrower path means more collisions, which means more resistance, exactly the lever you'd expect; you can recall R = ρ × L / A). The whole thing is sealed in an insulating coating with a metal lead poking out of each end. Metal-film parts are prized for being more precise and quieter; carbon-film is the cheap-and-cheerful default. For jobs that have to shrug off serious heat you will also meet wirewound resistors: literally a length of resistance wire (an alloy like nichrome) coiled around a ceramic core. Burlier, handles real power, used where the little film types would simply cook.
Here's a look inside a simple carbon-film resistor:
Its main characteristics. When you reach into a parts drawer, these are the numbers that actually matter:
- Resistance value, in ohms (Ω): the headline figure, the "how stingy is it" number. Resistors don't come in every conceivable value; they're sold in standardized steps (the E-series), so you pick the nearest standard value to whatever your math asked for.
- Tolerance: how close the real part is to its printed value. A "±5%" 100 Ω resistor is only guaranteed to land somewhere between 95 and 105 Ω. Precision metal-film parts hit ±1% or tighter.
- Power rating, in watts: how much heat it can safely get rid of before it damages itself. Important enough that it gets its own paragraph in a second.
- Temperature behavior: just like that overheating wire from the Ohm's law warning, a resistor's value drifts a little as it heats up. Ignorable for most beginner work; not ignorable for precision work.
- Non-polar: it has no orientation. Unlike the LED with its fussy long-leg, short-leg business, a resistor drops in either way round. It does not care.
Those little colored stripes. The bands painted around a through-hole resistor aren't decoration, they're the value and tolerance written in a color code (each color stands for a digit, with one band reserved for the tolerance). You end up reading them so often that people just memorize the code, but for now all you need to know is: the stripes tell you the resistance and how trustworthy it is. A quick lookup chart (or any phone app) decodes them in seconds.
Here is the example of a 220 Ohm resistor:
Stringing them together: series and parallel. You will almost never use a single resistor in isolation, so you need the two ways they combine.
In series means end to end, in a single-file line (like putting pearls on a necklace), so the same current has to fight its way through every one of them in turn. The obstacles simply stack up, so series resistances just add:
total R = R1 + R2 + R3 + ...
It's a longer obstacle course: the current meets each resistor's stinginess one after another, so the total is always bigger than any single one. (And recall from the current section that the current is identical at every point of a single path, it has nowhere else to go.) The supply voltage gets shared out among them.
In parallel means side by side (like the steps on a ladder), both terminals of each resistor tied to the same two points, so the current arrives and splits between several paths at once. Here is the bit that trips everyone up:
Myth: "more resistors means more resistance." WRONG, at least in parallel. Adding a parallel resistor hands the current an extra lane to flow through, so the crowd gets through more easily, not less. Parallel resistance always comes out lower than the smallest single resistor in the bunch.
The rule for parallel is a reciprocal sum (you add them "upside down"):
1 / (total R) = 1/R1 + 1/R2 + 1/R3 + ...
and for the everyday case of just two resistors, that tidies up to:
total R = (R1 × R2) / (R1 + R2)
The intuition: think extra checkout lanes at a supermarket, or more exit doors on a stadium. Every path you add lets more charge through for the same push, so the combined resistance drops. This time the voltage across each branch is the same, and it's the current that divides.
Rule of thumb: series adds, parallel reduces. Line resistors up one after the other (like pearls on a necklace) and their resistances sum (more total opposition); wire them side by side (like the wooden planks in a rail track) and the total falls below the smallest one (more total flow). Almost every resistor network you will ever meet is just these two moves, used over and over.
Power rating: why a resistor has a wattage. Now the characteristic worth slowing down for, and the one that sends you straight back to the power section.
Back there we said power is energy per second, P = V × I, measured in watts. And from the resistance section we know exactly what a resistor does with that power: every one of those electron-versus-atom collisions dumps energy into the material as heat. So a working resistor is, quite literally, a tiny heater, constantly turning electrical energy into heat at a rate of P watts.
Using Ohm's law, that same P = V × I can be rewritten into two forms that are far handier for a resistor, because you usually know its resistance:
- start from P = V × I, substitute V = I × R, and you get P = I2 × R
- or substitute I = V / R, and you get P = V2 / R
All three give the identical answer; you just use whichever quantities you happen to have in front of you.
So what is the power rating? Simply the maximum power (in watts) the resistor can turn into heat without damaging itself. That heat has to escape into the surrounding air, and a small component can only shed so much before it overheats, drifts way off value, scorches, or fails outright. Common through-hole ratings are 1/8 W, 1/4 W, 1/2 W, 1 W and 2 W, with chunky wirewound parts climbing far higher. And here's a useful tell: physically bigger resistor, higher power rating, because more surface area means more heat shed per second.
Careful: always check the power rating. Choose a resistor for its resistance and confirm that the power it will actually dissipate sits comfortably under its rating. A sensible habit is to run a resistor at no more than about half its rating, so a part expecting to burn 0.1 W wants at least a 1/4 W resistor, not a 1/8 W one cut right to the wire. Ignore this and the cheapest component in your circuit becomes the one that goes up in smoke.
Let's make it concrete with the LED circuit we keep promising to build. Say a 5 V supply, a red LED dropping about 2 V, and a target current of 20 mA. The resistor has to soak up the leftover voltage, 5 V - 2 V = 3 V, at 20 mA, so by Ohm's law:
R = V / I = 3 V / 0.02 A = 150 Ω
And the power it has to get rid of as heat:
P = I2 × R = (0.02)2 × 150 = 0.06 W (cross-check the other way: P = V2 / R = 32 / 150 = 0.06 W, the same 60 mW)
So our resistor turns 60 mW into heat. Drop in a bog-standard 1/4 W (250 mW) resistor and it's barely breaking a sweat, running at under a quarter of its limit. Notice too what this quietly says about energy: the 3 V across the resistor becomes heat, while the 2 V across the LED becomes light (and a little heat). The energy from the battery splits between the two; the resistor's share is just the price of keeping the LED's current gentle.
What you want to keep in mind: a resistor is a deliberate, stable, non-polar lump of resistance that obeys Ohm's law, and it pays for the privilege by turning P = I2 × R = V2 / R watts into heat. Its power rating is the most heat it can shed before it cooks itself, so you size a resistor for both the resistance you need and the wattage it will burn. Respect that, and the resistor is the most reliable friend in your circuit.
With single resistors, series, parallel, and power all in hand, the next move is to handle the currents and voltages across a whole tangled network at once, not just one resistor at a time. That is exactly the job of Kirchhoff's laws, coming up next (after I give you a nice little example tho)
¶ Resistors: an interesting case
Crack open a budget multimeter sometime and you may spot something that looks like a manufacturing mistake: a bare little copper wire (or a copper track on the board) with a few small cuts or notches snipped into it. It is not a mistake. It is the meter's current shunt, and those cuts are how it was calibrated.
Here's the thing a meter never does: measure charge-per-second directly. If you expected some sort of apparatus that counts literal electrons going through an infinitesimal point in the wire, I just popped your bubble. To read a current it instead sends that current through a tiny, known resistance and measures the voltage across it, then runs Ohm's law backward: I = V / R. So the entire accuracy of the current range rests on that one resistance being exactly right. Pricey meters use precision alloy shunt resistors. Cheap ones just grab a length of ordinary copper wire and trim it by hand: a notch thins the wire (less cross-section -> more resistance, remember the formula), or snipping one of several parallel strands kills a path (and you already know parallel paths lower resistance, so removing one raises it). Someone literally sat there cutting until the display matched a reference meter. That is R = V / I being dialed in with a pair of snips.
The catch: copper's resistance drifts noticeably with temperature (remember the Ohm's law warning). A hand-trimmed copper shunt is cheap, but it wanders as things warm up, which is a big part of why the current ranges on a budget meter are the least trustworthy numbers it gives you.
Here's a photo I took with one of the microscopes in the Electronics Lab at our Education Center of a 17 lei Multimeter.
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You can see the cuts quite clearly (I had to flex that 4k resolution of the microscopes we use every day)
If you apply the formula for resistance and assume the wire is 10cm long and that the cuts get rid of roughly half of the cross sectional area, while being 3mm long (not quite true but we'll go with it), you can try and calculate the difference in resistance between a wire that has those notches and one that doesn't.
Here is a table of resistivities to help you with your calculations.
¶ Choosing a resistor.
Already we said that an LED is a fragile thing that can't survive on its own (much like humans in a sense). The reason why this happens is that the dumb LED is too greedy for its own good and has a tendency to want to suck in more corrent than it can handle.
This, in the best case, results in LEDs changing colors (Green LEDs can become Red from too much current).
Now notice my wording: an LED WILL TRY and draw more current than neccesary. That means that even if you put the LED at its required voltage, it still has a chance to burn without a resistor.
We need a way to limit the current going through to the maximum allowed value the LED can handle (something we will see in its datasheet, more on datasheets later).
If only the component we just spent 20 minutes talking about was made SPECIFICALLY to limit current...
You guessed it, we need a resistor.
Let's take the following circuit as an example:
This is just a hand-drawn schematic. In reality, we use schematic editors like KiCAD; KiCAD can also be used for making PCB designs and is one of the best tools for doing so; it's free and open source.
Bellow is the same schematic drawn in KiCAD. Looks more like what you would see in the real world.
Notice how: 1. According to the convention, current goes from "+" to "-", although in reality electrons go from the negative terminal to the positive terminal; 2. Each component has a specific symbol: the LED is that triangle thingy; 3. Some components (like the resistor) are unpolarized (it's just a piece of metal with an intentionally lower ability to conduct electricity, of course current can go both ways through it), while some components (like the LED) are polarised (for reasons far beyond the scope of this course; a bit of history and the exact way it works here.
Let's say that by reading the LEDs datasheet (again, an entire section dedicated to datasheets later) we have determined it needs around 20mA of current for proper function.
We also know that our USB provides 5V for us. We have V and I, finding R is trivial:
This DOES NOT take into account the voltage drop on the LED. For this purpose, we'll assume the voltage drop to be the usual 2V. This gives us: R = (5-2)/0.02 = 150Ω (also see the exact same example in the section above)
¶ Kirchhoff's two laws
Up to now we've worked one piece at a time: Ohm's law on a single resistor, plus the series and parallel shortcuts for stringing a few together. But a real circuit is a tangle of junctions and loops, and you need a way to pin down every current and voltage in it at once. That is exactly what a German physicist named Gustav Kirchhoff handed us back in 1845, in the form of two rules so simple they almost feel like cheating.
And here's the lovely part: they are not new physics. They are just conservation of charge and conservation of energy, the two ideas you already trust, wearing circuit clothes. One rule for currents at junctions (KCL), one for voltages around loops (KVL).
Two words you'll need. A node (or junction) is any point where two or more wires meet, a place where current can split or merge. A loop is any closed path you can trace around the circuit that brings you back to where you started. That's it. Kirchhoff's laws are just one statement about nodes and one about loops.
¶ Kirchhoff's Current Law (KCL)
Whatever flows in must flow out. Picture a junction where three wires meet, like a fork in a river or a road intersection. Charge cannot pile up at a single point, and it certainly can't vanish (remember, charge is conserved, it's one of those fundamental things nobody gets to break). So every last electron that flows INTO that junction has to flow back OUT of it through some other wire.
You've actually met this already. Back in the current section we said that in a single unbranched wire, the same amount of charge flows past every point, whatever flows in flows out. That was KCL for the boring case with no forks. Add some forks and it generalizes to the full rule:
The sum of the currents flowing into any node equals the sum flowing out. Write it as ΣI(in) = ΣI(out) (or, if you like signs, the currents at a node add up to zero). So if 100 mA flows into a junction and one branch carries 60 mA away, the other branch must carry exactly 40 mA. No bookkeeping tricks, no exceptions.
One word on signs (so a minus sign never spooks you). When you don't already know which way a current flows, just pick a direction for it and go (remember what we said in the introduction on the first day of the bootcamp when some of you said you had problems starting? The solution is to just pick something and start =)) ). Guess right and the number comes out positive; guess wrong and it comes out negative, which is not a mistake at all: the minus sign is simply the math telling you the current really flows the other way. So you genuinely cannot pick wrong. Assign a direction to every unknown current, do the algebra, and let the signs sort out reality for you.
Here is a resource to help you on that. PLEASE work at least two examples (after reading this whole section) before attempting the homework.
Myth: "the current gets used up as it goes around the circuit, so less comes back to the battery." WRONG. By KCL, exactly as much current returns to the battery as left it. What the circuit drains is energy, not current (the battery's chemical energy is what runs out, remember the pump). Current is not a fuel that gets consumed on the lap; it's a flow that goes all the way round.
And this is precisely the rule hiding behind parallel resistors: at the node where the current splits between the branches, KCL says those branch currents must add up to the total going in. That is why the current divides the way it does.
¶ Kirchhoff's Voltage Law (KVL)
Around any loop, it all sums to zero. For this one, lean on what voltage actually is: energy per charge (V = U/q, from way back). It helps to picture voltage as altitude in an energy landscape. Now take a single charge and walk it all the way around a closed loop, back to the exact point it started from. Since it has returned to where it began, its net change in energy must be zero, you cannot end up higher or lower than your own starting point. Conservation of energy, plain and simple.
So what happens on that lap? The battery is the climb: it spends chemical energy to lift the charge up, handing it energy (a voltage rise). Every component the charge then passes through is a descent: the resistor spends that energy as heat, dropping the charge back down (a voltage drop). Go all the way around and the one big rise has to be exactly canceled by all the drops:
Here is another diagram with a nice step scale kindly generated by Gemini (because I'm sick of writing AND drawing all this by hand)
As you can see, we start from low potential, the 9V battery give us a boost (a voltage rise) , then each resistor is just a step down until we eventually reach the other end.
The sum of all the voltages around any closed loop is zero, which is the same as saying the source voltage equals the sum of the drops across everything else in the loop. Σ(rises) = Σ(drops). (And exactly like KCL, the signs keep you honest: pick a direction to walk the loop, count every rise as positive and every drop as negative (or the reverse, as long as you stay consistent), and if some voltage lands negative it simply means you crossed that component against the polarity you'd assumed.)
This is the rule behind series resistors and the famous voltage divider: in a series loop the supply voltage gets shared out among the components, and those shares always add back up to the source. That is why series voltages add.
For extra points: prove the formulas for series and parallel resistors using Kirchoff's laws.
Careful, the simple form has fine print. "Voltages around a loop sum to zero" is rock-solid for the lumped, low-frequency circuits in this course. The catch only shows up at high frequencies, or when a changing magnetic field threads through the loop: by Faraday's law that changing field induces its own voltage, and the naive sum stops being zero. Same Maxwell-and-fields story from earlier. You can safely ignore it here, but file it away so it doesn't ambush you later.
Let's put both to work. Take a 12 V battery driving two resistors in series, R1 = 1 kΩ and R2 = 2 kΩ:
(I swear to God, it's the last AI diagram =)))); pinky promise)
KCL is almost trivial here: it's a single loop with no forks, so the same current I flows through every part of it. KVL then says the two drops must add up to the 12 V the battery supplies:
12 V = V1 + V2 = I × R1 + I × R2 = I × (R1 + R2)
The resistors are in series, so they add to 3 kΩ, and we can solve for the current:
I = 12 V / 3 kΩ = 0.004 A = 4 mA
Now back-substitute with Ohm's law to get each drop:
- V1 = I × R1 = 4 mA × 1 kΩ = 4 V
- V2 = I × R2 = 4 mA × 2 kΩ = 8 V
And the check that should make you smile: 4 V + 8 V = 12 V, exactly the battery voltage, just as KVL promised. That little calculation, the source voltage splitting across series resistors in proportion to their size, is the voltage divider, and you will use it constantly. It is nothing more than KVL and Ohm's law shaking hands.
The big idea: Kirchhoff gives you two bookkeeping rules (think of it like accounting), and they are just conservation laws in disguise. KCL (charge is conserved): at every node, current in = current out. KVL (energy is conserved): around every loop, the voltages sum to zero. Arm yourself with these two plus Ohm's law and you can untangle any DC resistor network, no matter how many junctions and loops it has. Bigger circuits just mean more equations, never new ideas.
So far every component has been happy to respond instantly, you change the voltage and the current obeys at once. Next we meet a component with a sense of timing, one that stores energy and takes a moment to react: the capacitor, and the RC circuits that make it tick. We're finally stepping into territory where time and timing are important and affect our circuits.
¶ The capacitor (the component)
That component that brings more timing into our world is the capacitor. Where a resistor reacts the moment you change the voltage, and where a wire just passes everything through, a capacitor does something stranger: it stores charge, takes time to fill and empty, and pushes back harder the more you stuff into it.
What it physically is. A capacitor is almost insultingly simple: two conductive plates with an insulating gap between them. That insulating layer in the middle is called the dielectric, and its whole job is to stop charge from crossing. That's it. Two plates, a gap, no path across.
This video does an DELIGHTFUL job at describing capacitors. But you might wonder: "the hell? that thing with two huge plates doesn't look like any capacitor I've ever seen in my netire f-in life". First, if you really asked the question like that, take some anger management classes; second...yeah, true, capacitors in practice don't look like that.
This is a plate capacitor...too big for any practical electronics:
In practice we need things to be small; we don't need that much capacity so we use less metal. But at that scale, air is not a good insulating layer anymore. We can't use it as a dielectric. So we just use something akin to paper soaked in a magic liquid in the case of electrolytic capacitors (so magic, companies that make capacitors guard the exact recipe with their life; that liquid is called the electrolyte, hence the name) OR some badass material like ceramic or tantalum to separate the pieces of metal.
Here is the inside of an electrolytic capacitor:
Actually cutting into one:
And here is inside a ceramic capacitor:
So what good is a component you can't push current through? Cast your mind back to the voltage section, to the battery cramming electrons onto its crowded − terminal. A capacitor does that on purpose, and then keeps it. Wire one up to a battery and the battery goes to work: it drags electrons onto one plate (electron-rich, just like the "−" terminal) and strips them off the other (electron-poor, like the +). The electrons would love to rush across to the empty plate and even things out, but they can't: the dielectric blocks them. So they just sit there, crammed, stored.
Nothing crosses the gap, yet the wire still sees a current. This trips people up, so let's nail it now. While a capacitor charges, no electron ever jumps the dielectric (that's the insulator's entire purpose). The "charging current" is electrons crowding onto one plate while an equal number are pulled off the other plate. The external wire carries a real current, even though nothing actually crosses the middle. It looks like current flows "through" the capacitor; really it's charge piling up on one side and draining off the other.
And here is the behavior that makes a capacitor a capacitor. Remember from the voltage section: electrons hate being crowded, they repel each other. So the first few electrons you push onto an empty plate go on easily. But as the plate fills, the crowd already sitting there shoves back against every new arrival, harder and harder. That growing push-back is a voltage building up across the capacitor, and it climbs as the charge climbs. The capacitor fights you more the fuller it gets. (It's the spring from the voltage section all over again: the more you compress it, the harder it pushes back.)
¶ Capacitance: how much charge per volt (Q = CV)
If a capacitor's voltage rises as you add charge, the obvious question is: how much charge does it take to raise the voltage by one volt? That number is the capacitor's capacitance, and it's the headline spec of the part.
Capacitance (C) measures how much charge a capacitor stores per volt across it. Big capacitance means it swallows a lot of charge for only a small rise in voltage (a roomy bucket); small capacitance means even a little charge shoots the voltage right up. The defining relationship is dead simple:
Q = C × V (charge stored = capacitance × voltage across it)
Capacitance is measured in farads (F), and a farad is exactly one coulomb per volt (1 F = 1 C/V).
Heads-up on a clashing letter (sorry, not my fault). We now have C meaning two different things: italic C the quantity (capacitance, in farads) and plain C the unit (coulombs, the charge). Physics has lived with this collision for a century and so will we. Context tells you which: "a 10 µF capacitor" is capacitance; "a charge of 10 C" is coulombs.
One farad turns out to be an enormous amount of capacitance, so in real life you'll almost always see fractions of it: microfarads (µF, 10-6 F), nanofarads (nF, 10-9 F), and picofarads (pF, 10-12 F). A typical capacitor on a circuit board is a few µF or a few nF; a chunky power-supply capacitor might be hundreds or thousands of µF.
What sets the capacitance (and notice the lovely echo of resistance). Just like resistance, capacitance comes down to geometry plus the material in the middle:
C = ε × A / d
- A: the plate area. Bigger plates mean more room for charge to spread out, so more capacitance.
- d: the gap between the plates. A smaller gap means the opposite charges on the two plates pull on each other more strongly across the gap, which lets more charge pack on for the same voltage, so more capacitance.
- ε (epsilon): the permittivity of the dielectric, a property of the insulating material itself (exactly like how resistivity ρ was a property of the conductor). Some dielectrics let you store far more charge than others.
Compare this to R = ρ × L / A from the resistor section and enjoy the symmetry, with one satisfying twist: for a resistor, more cross-sectional area meant less resistance, but for a capacitor, more plate area means more capacitance. Same geometric ingredients, opposite roles. Both come down to "what's the stuff, and what's its shape."
The energy a capacitor stores. A charged capacitor is a little tank of energy (that's why it stores charge in the first place). But there's a subtlety worth savoring. Back when one electron crossed a battery, every coulomb met the same fixed voltage, so the energy was simply U = q × V. A capacitor is different: its voltage grows from zero up to V as you fill it. The first charge goes on while the voltage is near zero (cheap), the last charge goes on at the full voltage (expensive). So the energy stored is the average voltage (half of the final value) times the total charge:
U = ½ × Q × V, and substituting Q = C × V gives the form you'll see most: U = ½ × C × V2
That ½ is not a fudge factor, it's the whole "voltage climbs as you fill it" story written as a number. (Units check, as always: farads × volts2 = (C/V) × V2 = C × V = coulombs × joules/coulomb = joules. Energy, as promised.)
A capacitor is the sprinter; a battery is the marathon runner. A battery holds a lot of energy but hands it over slowly. A capacitor holds comparatively little energy but can dump it (or soak it up) almost instantly. That "fast in, fast out" is exactly the superpower we're about to use. Don't expect a capacitor to run your phone for hours; expect it to react in a flash.
¶ Charging, discharging, and the time constant (RC)
Now the main event, and the reason capacitors are interesting: what happens over time when you charge and discharge one through a resistor. A resistor plus a capacitor is called an RC circuit, and its behavior is the single most important pattern in this whole section.
Here is the schematic of the RC circuit:
Charging starts fast and finishes slow. Let us close the switch on an empty capacitor and follow the logic, because every bit of it is something you already know:
- At the very first instant, the capacitor is empty, so it has 0 V across it. By KVL, that means the entire battery voltage Vs sits across the resistor. So Ohm's law gives the biggest current the circuit will ever see: I = Vs / R. Right at switch-on, an empty capacitor offers no opposition at all, it behaves exactly like a plain wire (a short circuit). Current floods in.
- As charge piles onto the plates, the capacitor's voltage climbs (that's Q = CV at work). And by KVL, every volt that appears across the capacitor is a volt removed from the resistor, so the current drops. Less current means charge arrives more slowly, which means the voltage rises more slowly, which drops the current further. The thing throttles itself.
- After a long time, the capacitor voltage has climbed almost all the way up to Vs, leaving almost nothing across the resistor, so the current fades to essentially zero. The capacitor is "full," and now it behaves like a break in the wire (an open circuit). This is the headline fact you'll use forever: a fully charged capacitor blocks steady DC.
So the current doesn't switch off, it just eases off, and the voltage doesn't jump up, it glides up. The shape of that glide is a curve called an exponential, and the equations are:
- capacitor voltage climbing: V(t) = Vs × (1 - e(-t/RC))
- current fading: I(t) = (Vs / R) × e(-t/RC)
Let's read that equation out loud, piece by piece, because it looks scarier than it is:
- V(t) : the capacitor's voltage at a particular moment t. The little "(t)" just means "this value depends on time", feed in a time, get back the voltage right then. It's not V multiplied by t.
- t : time, in seconds, counted from the instant you flipped the switch (t = 0 is switch-on).
- Vs : the supply voltage, the battery's value. This is the target, the ceiling the capacitor is climbing toward but never quite passing. Everything in the formula is really just answering "what fraction of Vs have we reached so far?"
- RC : the time constant τ. It sets how fast all of this happens. (Same RC, nothing new.)
- t/RC : time measured in units of time constants. This is the real engine of the formula. Don't read t as "seconds", read t/RC as "how many τ's have gone by." At t = one τ, t/RC = 1; at t = five τ's, t/RC = 5. The circuit doesn't care about raw seconds, it cares how many of its own ticks have elapsed.
- e(-t/RC) : Euler's number e (≈ 2.718, the smooth-curve number, NOT the electron charge) raised to the negative of that. This is the "how much is still LEFT to do" term. At the very start (t = 0) it equals 1, meaning 100% of the climb still remains. As time passes it shrinks toward 0, meaning less and less left to go. After 1τ it's down to ~0.37 (37% left); after 5τ it's ~0.01 (1% left). It's the shrinking-sliver from intuition #3, written as a number.
- (1 − e(-t/RC)) : flip "what's left" into "what's DONE." If e(-t/RC) is the fraction still remaining, then 1 minus it is the fraction already achieved. At t = 0 that's 1 − 1 = 0 (empty, just starting). After 1τ it's 1 − 0.37 = 0.63 (the famous 63%). After 5τ it's 1 − 0.01 = 0.99 (basically full). This bracket is the whole curve, sweeping from 0 up to 1.
So the sentence the whole formula is saying: "take the target Vs, and multiply it by the fraction of the journey completed so far." Voltage now = ceiling × (how-far-along you are). The (1 − e…) part rises from 0 to 1, dragging V(t) from 0 up to Vs along exactly the easing-off curve you'd expect.
The other clashing letter (last one, I promise). That e in the formula is not the electron charge from earlier (1.602 × 10-19 C). It's Euler's number, e ≈ 2.718, the natural base for anything that grows or decays smoothly (populations, radioactive decay, cooling coffee, and yes, charging capacitors). Same symbol, totally different beast. If you have not done calculus yet,as many of you have not up until now, you can take it as it is. Here is a nice read about Euler's number. In this context you just read "e(something)" as "the smooth-curve function."
The time constant: τ = R × C. Look at those formulas and notice the combination RC sitting in the exponent. That product has a name, the time constant, written τ (the Greek letter tau), and it is the single number that tells you how fast the whole thing happens.
First, the part that should make you smile. R × C has units of seconds. Watch them cancel, the same way we did for power:
ohms × farads = (volts / amp) × (coulombs / volt)
The volts cancel, leaving coulombs / amp. And an amp is a coulomb per second (1 A = 1 C/s), so coulombs / amp = coulombs / (coulombs/second) = seconds. So τ = RC is genuinely a time, falling straight out of the units. Beautiful.
What does that time mean? It's the natural "tick" of the circuit:
- After one time constant (t = τ = RC), a charging capacitor has reached about 63% of the way to full. A discharging one has fallen to about 37% of where it started.
- After about five time constants (5τ), it's over 99% of the way there, which everyone agrees to call "done."
The full picture, worth memorizing the shape of:
- after 1τ: ~63% charged (~37% left to go)
- after 2τ: ~86%
- after 3τ: ~95%
- after 4τ: ~98%
- after 5τ: ~99% (treat as fully charged / fully discharged in practice)
Here is a desmos graph with all the values for charge Voltage. The orange dashes are the "tau" constant multiples. You can play along with the values and see where they lead you.
Direct link in case the embedded version fails.
And why does τ depend on both R and C? Pure intuition:
- Bigger C means more charge has to be shuffled to change the voltage, so everything takes longer. (Bigger bucket, longer to fill.)
- Bigger R means the resistor chokes the current down, so charge trickles in more slowly, so everything takes longer. (Narrower pipe, longer to fill.)
τ = RC rolls both effects into one number. Want it faster? Shrink R or C. Want a slow, lazy charge? Make them big. This one knob, RC, is how you design timing into a circuit.
Discharging: the mirror image. Now take that fully charged capacitor, disconnect the battery, and connect the charged capacitor straight across a resistor. The capacitor is now the source. The exact same logic runs in reverse:
- At the first instant it's full, sitting at its starting voltage V0, so it drives the biggest current: I = V0 / R (flowing the opposite way to before, since the capacitor is now pushing out).
- As it empties, its voltage falls, so the current falls, so it empties more slowly. Self-throttling again.
- After about 5τ it's essentially flat at 0 V, fully drained.
Same curve shape, same time constant:
- capacitor voltage decaying: V(t) = V0 × e(-t/RC)
- current decaying: I(t) = (V0 / R) × e(-t/RC)
And where does the stored energy go when it discharges? Straight back to the resistor section: the resistor turns it into heat. Every joule the capacitor banked (that ½CV2) gets dumped into R as warmth as the charge drains through it. Nothing is lost, it's just converted, exactly as a resistor always does.
The main idea (charge/discharge in one breath): an RC circuit charges and discharges along a smooth exponential curve, fast at first and slowing as it approaches its target, and the speed of that curve is set entirely by τ = R × C. One time constant gets you ~63% of the way (or down to ~37%); five time constants is "done." Bigger R or C means a slower, lazier curve. That single number, RC, is the heartbeat of the circuit.
What a capacitor really does, in one sentence. Step back from the curves and here's the deep truth they're all pointing at: a capacitor resists changes in voltage. Its voltage can't jump instantly, because moving charge takes time (and time is set by τ). Push a steady, unchanging DC voltage at it and, once charged, it just blocks (no current, an open circuit). But hit it with a fast-changing voltage and it passes current readily, gulping charge in and out to fight the change. We can even write that as a cousin of the current formula you already know:
I = C × (ΔV / Δt) (the current into a capacitor depends on how fast its voltage is changing)
Read it off: if the voltage isn't changing (ΔV/Δt = 0, steady DC), the current is zero -> it blocks DC. If the voltage is changing fast, the current is large -> it passes the wiggles. A capacitor blocks the steady and passes the changing. Hold that sentence, because it's the entire reason capacitors are about to save your power supply.
¶ Why capacitors are everywhere: ripple, smoothing, and power supplies
Time to cash all of this in on the single most common job a capacitor does in the real world: smoothing out a bumpy power supply. This is where charge, discharge, and τ stop being theory and start being the reason your phone charger doesn't hum and flicker.
First, the problem: the wall gives you the wrong shape of electricity. The mains socket in your wall (here in Romania, 230 V at 50 Hz) supplies AC, alternating current: a voltage that smoothly swings positive, back through zero, negative, and back again, fifty times every second. But the chips inside your electronics want DC: a steady, unchanging voltage to sit on. So a power supply's job is to turn that oscillating AC into flat DC.
The first step (using components called diodes, arranged as a rectifier, which you'll meet properly in the digital lesson) is to flip all the negative swings up so everything points the same way. But that doesn't give you flat DC. It gives you a row of humps: the voltage still rises to a peak and falls back toward zero between each hump. It's all one polarity now, but it's pulsating, not steady. This leftover bumpiness is called ripple, and a chip running off it would behave erratically.
[NOTE: add diagram here] Description: a three-panel vertical stack sharing a time axis. Panel 1, "from the wall (AC)": a smooth sine wave swinging above and below zero. Panel 2, "after the rectifier": the same wave with all negative humps flipped up to positive, giving a row of bumps that dip back down to (near) zero between each hump, labeled "pulsating DC / ripple". Panel 3, "after the smoothing capacitor": a nearly flat line near the peak voltage, with only a small sawtooth wiggle on top, labeled "smoothed DC, small ripple".
Now the fix, and it's pure charge/discharge. Connect a big capacitor across that bumpy output (this is called a reservoir or smoothing capacitor), in parallel with whatever you're powering (the load). Watch what it does:
- On the way up to each hump's peak, the rising voltage charges the capacitor (charge flows in, the cap stores energy). The source has low resistance here, so this charging is fast.
- As the hump falls away, the rectifier's diodes block the capacitor from draining backward into them. So instead, the capacitor discharges into the load, holding the voltage up while the source momentarily sags. The cap acts as a backup tank, dumping its stored charge to keep the load fed until the next hump arrives and tops it back up.
So the output no longer crashes to zero between humps. It droops only slightly before the next peak refills the capacitor. That small remaining droop is the ripple voltage, and it's vastly smaller than the raw bumps were. The capacitor has filled in the valleys using charge it banked at the peaks. Charge on the peaks, discharge in the valleys, over and over, fifty or a hundred times a second.
[NOTE: add diagram here] Description: a zoomed-in view of the smoothed output over two humps. Show the faint dashed outline of the original rectifier humps underneath, and over it the actual capacitor voltage: rising sharply to follow each peak (labeled "cap charging"), then sloping gently downward in the gap (labeled "cap discharging into load, holding the rail up"), then snapping back up at the next peak. Mark the small vertical height of the droop as "ΔV = ripple voltage".
Here's the payoff that ties the entire section together. How big is that ripple droop? Use exactly what you already know. Between refills, the load draws a current I for the gap-time Δt, so the charge the capacitor loses is ΔQ = I × Δt. And from Q = C × V, losing a charge ΔQ drops the voltage by ΔV = ΔQ / C. Put them together:
ripple voltage: ΔV ≈ (I × Δt) / C
Stare at that and read off everything a power-supply designer cares about:
- Bigger capacitor C -> smaller ripple. This is the reason reservoir capacitors are physically fat (hundreds or thousands of µF). More farads, flatter rail.
- More load current I -> more ripple. A hungrier load drains the tank faster between refills.
- Longer gaps between humps -> more ripple, which is part of why supplies rectify cleverly to refill more often.
That single formula is nothing but Q = CV and charge/discharge wearing a power-supply hat. And you can say the same thing in time-constant language: you want the discharge time constant (the load's resistance × C) to be long compared to the gap between humps, so the capacitor barely empties before it's topped up again. Long τ -> flat rail. The very same τ from the charging section, now doing a real job.
Honest real-world note. A reservoir capacitor makes the rail much flatter, but never perfectly flat, there's always some ripple. For electronics that need a rock-steady rail, the smoothed-but-still-ripply voltage is fed into a voltage regulator (a later-lesson component) that irons out the rest. The capacitor does the heavy lifting; the regulator does the finishing. Pretty much every wall charger and power supply you own is some version of: transform down -> rectify -> smooth with a big capacitor -> regulate.
The same trick, shrunk down: decoupling capacitors. Look at any circuit board, anywhere, and you'll see tiny capacitors sprinkled right next to every chip. Same idea as the reservoir cap, just local and fast. When a chip suddenly gulps a burst of current (digital chips do this constantly as they switch), the long wires back to the power supply can't deliver that spike instantly, so the chip's local voltage would dip. A small decoupling (or bypass) capacitor sitting right beside the chip acts as a tiny on-the-spot reservoir, supplying that sudden gulp from its own stored charge and keeping the chip's supply rock-steady. It's ripple-smoothing in miniature, and it's one of the most common reasons capacitors exist on a board at all.
A few real-world cautions (because capacitors have teeth, and the resistor section taught you to respect ratings):
Some capacitors have a + and a − leg, and they care a LOT. The big high-capacitance types, electrolytic capacitors (the little cans you'll see in power supplies), are polarized, exactly like the LED with its long-leg/short-leg fussiness. Wire one in backwards, or push it past its rated voltage, and it can heat up, vent, or even burst. Smaller ceramic capacitors are non-polarized and don't care which way round they go. When in doubt, check the markings.
Every capacitor has a voltage rating, and it is not a suggestion. Just like a resistor has a maximum wattage, a capacitor has a maximum voltage its dielectric can withstand. Exceed it and the insulating gap breaks down, killing the part (sometimes loudly). Pick a capacitor rated comfortably above the voltage it'll actually see.
A charged capacitor stays charged, and a big one can bite. Because it stores energy and only discharges as fast as τ allows, a large capacitor can hold a hefty charge for a long time after the power is switched off. Small ones are harmless, but the big reservoir capacitors in mains power supplies (and the truly dangerous ones in things like microwave ovens and camera flashes) can deliver a serious, even lethal, jolt long after unplugging. Rule for now: do not go poking around inside mains-powered equipment, and know that "unplugged" does not always mean "safe to touch."
The big idea: a capacitor stores charge (Q = CV) and energy (½CV2), it resists changes in voltage, and through an RC circuit it charges and discharges on a smooth exponential curve whose speed is the time constant τ = RC. That single behavior, banking charge on the peaks and releasing it in the valleys, is what smooths a bumpy supply into clean DC, with the ripple set by ΔV ≈ I × Δt / C (bigger C, smaller ripple). From giant reservoir cans to the tiny decoupling caps beside every chip, it's all the same charge-and-discharge story you just learned.
With charge, current, voltage, power, resistance, Ohm's law, resistors, Kirchhoff, and now capacitors and RC all in your toolkit, you finally have everything you need to stop drawing circuits and start building one. Next we get our hands dirty: the breadboard, the multimeter, and the lab you've been promised since the very first section, lighting an LED with a current-limiting resistor you size yourself.