Resistors and Ohm's Law

A resistor is a device built to do exactly one thing: restrict the flow of charge through it. Physically, it's a slug of slightly conductive material — usually a thin film of carbon or metal oxide deposited on a ceramic core, with wire leads — engineered so that pushing current through it requires a definite, predictable amount of work.

You'll see resistors everywhere on a circuit board: small cylindrical things with colored bands (the bands encode the resistance value), or tiny black rectangles in surface-mount form. They're cheap, passive (no power needed to make them work), and one of the three foundational building blocks of every analog circuit.

Why does current have a hard time getting through?

The same reason a marble has a hard time crossing a patch of carpet: friction.

Inside the resistive material, the moving charge carriers (electrons, in the simplified picture) keep bumping into atoms in the lattice. Each collision converts a little bit of the charge's directed motion into random vibration of the atoms — which is what we measure as heat. The "harder" the material is to push through, the more collisions per unit length, and the higher the resistance.

Picture this. Imagine a marble starting at rest at the top of a smooth ramp 12 cm tall. Released, it accelerates down the ramp and zooms off the bottom at full speed.

Now glue a strip of carpet across the middle of the ramp.

Two things change at once:

1. The marble moves slower — friction with the carpet drags it down. (In the circuit: less current.)
2. The marble has less energy when it leaves the carpet patch — some of its potential energy got absorbed by the carpet as heat. (In the circuit: the charge has less potential left, so the voltage across the carpet patch is lower than the voltage above it.)

These aren't two separate effects. They're two consequences of the same physical thing: the carpet absorbs energy from the marble's motion. That same energy absorption simultaneously slows the marble down (less current) and robs it of height (voltage drop). Friction does both jobs at once because friction is the conversion of orderly motion into disorderly heat.

This is the single most important intuition in the rest of this course. Whenever you see a resistor in a circuit:

• The current through it is throttled compared to a plain wire.
• The voltage across it equals how much energy each unit of charge gave up to the carpet (heat).

Both effects, one mechanism. The next section gives the formula that ties them together quantitatively — but the formula is just bookkeeping for what the marble already shows you.

(Aside: not every voltage drop is a marble-loses-energy-to-carpet drop. Transformers and switching regulators can lower voltage by trading it for current — energy is stored and re-released, not dissipated as heat. We'll meet those in the AC fundamentals lesson. For now, every drop you see in a DC circuit with just resistors is a friction drop.)

Now we put numbers on the marble-and-carpet picture from the previous section. The relationship is straightforward: voltage across a resistor equals current through it times its resistance. This is Ohm's Law: $V = IR$^[1].

Why "I" for current? It stands for intensity — short for the French intensité du courant ("intensity of current"), the term André-Marie Ampère used in the 1820s when he was inventing the field of electrodynamics. His name lives on in the unit "ampere" (the unit of current) — the same person gave us both the symbol and the unit. So the trio is: V = Voltage, I = Intensity (of current), R = Resistance. Mnemonic if you need one: current is intense flow.

Principle — V = IR is a relationship, not a recipe. It's tempting (especially if you come from programming) to read V = I * R like an assignment: "set I and R, get V." Physics doesn't work that way. The equation is a constraint that V, I, and R must satisfy simultaneously in any moment — closer to V == I * R (an invariant) than to V = I * R (an assignment). In a normal circuit you control V (you pick a battery or supply) and R (you pick a resistor), and I is what emerges to satisfy the constraint. You can't actually "hold current steady and crank the resistance up to make voltage rise" — there's no knob in the physical world that does that. (There are special components called current sources that fix current and let voltage emerge instead, but they're rare in beginner circuits — we'll meet them later.)

The slider below reflects this: you set V and R, you read off the resulting I.

When voltage rises and resistance is held constant, current rises proportionally.

Ohm's Law has a quiet but dramatic consequence at the edges. Look at what I = V / R does as R gets very small:

Principle — as resistance approaches zero, ideal-model current rises without bound. A short circuit creates a low-resistance path that bypasses the intended load. Real sources and wiring limit the current, but overheating and fire remain possible.^[2]

Real supplies have internal resistance and other limits, so current cannot reach infinity. For a worked example, assume a 9 V source has 1 Ω of internal resistance and the shorting wire has negligible resistance. Then I ≈ 9 / 1 = 9 A. The assumed 1 Ω is not a specification for every 9 V battery. Never short a real battery to test this model.

Try it below. Slide the resistance down a logarithmic scale and watch the current curve climb. Then toggle the model supply's internal resistance on and off to compare the ideal limit (I → ∞) with the modeled limit (V / R_internal).

(Aside: not every voltage drop comes from resistance. Transformers and switching regulators step voltage down without dissipating energy as heat — they trade voltage for current via energy storage in magnetic fields. We'll meet them in the AC fundamentals lesson.)

Real-world note — overcurrent protection. A short circuit can drive much more current than the intended load and overheat conductors. A correctly selected circuit breaker or fuse disconnects a circuit under specified overcurrent conditions; its operation depends on the fault current, device characteristics, and installation. It does not make contact with an energized conductor safe, and this slider cannot predict whether or when a breaker will trip. We'll study Japanese 配線用遮断器 (branch-circuit breakers) in the wiring lessons.^[2]

Ohm's Law is one of the few equations in electronics you'll genuinely use every day:

• Sizing a current-limiting resistor for an LED. A typical 5 mm red LED needs ~20 mA at ~2 V. From a 5 V supply, you need a series resistor of $(5 - 2) / 0.02 = 150\ \Omega$.
• Reading a multimeter. When you measure resistance, the meter applies a known voltage and computes $I = V/R$. Current measurement does the inverse — it puts a small known resistor in line and reads the voltage across it.
• Sensor voltage dividers. Photoresistors, thermistors, and other analog sensors are wired with a fixed resistor so that a varying $R$ becomes a varying $V$ that a microcontroller's ADC can read directly.

If you've built anything with an Arduino or Raspberry Pi, every "what resistor should I use?" question lands on Ohm's Law.