Ohm's Law Explained: Formula, Triangle, and Practical Examples
A 12-volt car battery. A 4-ohm subwoofer. Connect them directly and current rips through at 3 amps, dissipating 36 watts of heat in the voice coil. That one sentence uses Ohm's law twice, and it is the kind of back-of-the-envelope check every electrical engineer runs a dozen times a day. Master the relationship between voltage, current, and resistance and most circuit questions collapse into arithmetic.
Georg Simon Ohm worked it out in 1826 and published it in 1827, in a treatise called Die galvanische Kette, mathematisch bearbeitet (roughly: "The galvanic circuit investigated mathematically"). The German scientific establishment hated it. One critic called the work "a web of naked fancies." A government minister declared that "a professor who preached such heresies was unworthy to teach science." Fifteen years later the Royal Society awarded Ohm the Copley Medal, and today his name is on the SI unit of resistance.
The Formula and the Triangle Mnemonic
The law is three symbols long: V = I · R. Voltage equals current times resistance. Volts equal amps times ohms. Rearrange it and you get the two companions that matter just as much:
- I = V / R (solve for current when you know voltage and resistance)
- R = V / I (solve for resistance when you know voltage and current)
The "Ohm's law triangle" is a visual mnemonic for those three forms. Draw a triangle with V on top, I and R on the bottom. Cover the variable you want, and the remaining two are arranged in the right equation: cover V and you see I next to R (multiplication); cover I and you see V over R (division); cover R and you see V over I. It is cheesy, it has nothing to do with the physics, and it works. Most introductory textbooks still print it for that reason.
Use the Ohm's Law calculator when you want any two values entered and the third solved automatically, including the power dissipation that falls out of it. But the arithmetic is easy enough to do in your head once the triangle sticks.
A note on units. One volt pushing one amp through one ohm is the definition. Double the voltage across a fixed resistor and current doubles. Double the resistance at fixed voltage and current halves. That linear proportionality is the entire content of the law, and it is the thing that fails for the "non-ohmic" devices covered later.
Worked Example: Sizing a Current-Limiting Resistor for an LED
An LED is a diode. Connect it directly to a battery and it will pass whatever current the source can deliver until it burns out. Every LED circuit needs a current-limiting resistor in series, and Ohm's law sizes it.
Take a standard red LED running from a 5V rail. The datasheet gives two numbers you care about: forward voltage (call it 2.0 V for a typical red) and forward current (20 mA is the usual design target, though many indicator LEDs look plenty bright at 5 to 10 mA). The LED drops 2.0 V regardless of what you do. The remaining 5 - 2 = 3 V has to fall across the resistor.
A common pick is 330 Ω because it is everywhere in parts drawers. With 3 V across 330 Ω:
- I = V / R = 3 / 330 = 0.00909 A = 9.09 mA
That lands well under the 20 mA limit, giving a comfortably bright indicator with room for supply variation. Power dissipated in the resistor is:
- P = V² / R = 9 / 330 = 0.0273 W = 27.3 mW
A quarter-watt (250 mW) resistor handles that with an order of magnitude to spare. If you wanted closer to 20 mA, solve the other direction: R = 3 V / 0.020 A = 150 Ω. The LED Resistor calculator runs this exact computation with a cleaner interface for batch work, and the Resistor Color Code calculator decodes the bands on the real part once you know the value.
One caveat: forward voltage varies by color. Red and yellow sit around 1.8 to 2.2 V, green around 2.1 to 2.4 V, blue and white around 3.0 to 3.4 V. Swap a blue LED into a circuit sized for red and current drops, dimming the output. Always pull the number from the datasheet of the exact part.
Power Dissipation in Three Forms
Ohm's law gets you two of the three variables in a resistive circuit. The power equation gets you the third output that matters: heat. Power in a DC resistive circuit is P = V · I, watts equal volts times amps. Joule's first law, published in 1841, established that this power shows up as heat in the resistor, and that the heat scales as the square of the current.
Substitute Ohm's law into P = VI and two more forms appear:
- P = V · I (when you know both V and I)
- P = I² · R (when you know the current and the resistance)
- P = V² / R (when you know the voltage across a known resistor)
All three are the same equation wearing different clothes. The right one is whichever lets you skip a step. Designing a pull-up resistor on a 3.3 V line with a 10 kΩ part? P = V²/R = 10.89 / 10,000 = 1.09 mW, so a 1/16 W resistor is fine. Sizing a current-sense shunt at 2 A through 0.01 Ω? P = I²R = 4 × 0.01 = 0.04 W, so a 1/8 W part works.
Go back to the 12 V battery into 4 Ω speaker from the opening. I = 12/4 = 3 A. Power is 12 × 3 = 36 W, equivalently 3² × 4 = 36 W, equivalently 144 / 4 = 36 W. Three roads, one destination. The Electrical Power calculator handles all three forms if you would rather point and click than multiply.
The squared term in P = I²R is why wire gauge matters. Double the current through a fixed wire and heating quadruples.
When Ohm's Law Does Not Apply
Ohm's law is empirical, not fundamental. It holds beautifully for metals and most carbon resistors across many orders of magnitude in voltage. It fails hard for a long list of common components.
Diodes. A silicon diode sits at roughly 0 V of current up to about 0.6 V forward, then conducts almost as a short above 0.7 V. The I-V curve is exponential, not linear. Applying V = IR to a diode gives nonsense. This is exactly why LED circuits need external current-limiting resistors: the LED cannot limit itself.
Incandescent filament bulbs. A tungsten filament at room temperature has maybe one-tenth of its operating resistance. Cold inrush current into a 60 W 120 V bulb can spike to 10× the steady-state value during the first few milliseconds as the filament heats. The steady-state resistance obeys Ohm's law at the operating temperature, but the transient does not.
Thermistors. These are designed to change resistance with temperature. NTC thermistors drop resistance as they warm up. PTC thermistors raise resistance. Plot V against I on a thermistor and the curve is anything but linear, because R is a function of the self-heating the current causes.
Semiconductors in general. Transistors, MOSFETs, and semiconductor junctions all violate Ohm's law by design. Their whole point is non-linear behavior that lets one signal control another.
Materials also break down entirely under strong fields. Air at sea level breaks down at roughly 3 MV/m; once it does, arcing current has almost nothing to do with the pre-breakdown "resistance" of the gap.
Temperature and Real-World Wire Resistance
Resistance changes with temperature. Copper's resistance rises about 0.39% per degree Celsius near room temperature. Aluminum is similar. Tungsten is about 0.45% per degree, which is why the incandescent-bulb inrush effect is so dramatic. A 100-meter run of 14 AWG copper (about 0.83 Ω at 20 °C) rises to roughly 0.88 Ω at 35 °C in a hot attic.
This matters more than it sounds. Electricians rarely sweat 6% variation on a branch circuit, but long runs, data-center buses, and precision current shunts do. The four-terminal Kelvin measurement exists specifically because two-wire resistance measurement cannot distinguish the unknown resistor from the lead wires once their values become comparable.
For sizing purposes, the Voltage Drop calculator and Wire Size calculator compute resistance at standard reference temperatures. If your environment runs hot, derate the ampacity accordingly; NEC tables already bake this in via ambient temperature correction factors.
Maxwell noted the thermal headache in 1876, devising methods to separate Joule heating from the measurement itself. The Peltier effect at contacts and Seebeck voltages from temperature gradients also sneak into low-level measurements, sometimes producing thermal errors comparable in size to the resistance being measured.
AC, Impedance, and the Two-Character Upgrade
Feed a sine wave into a resistor and Ohm's law still holds at every instant: v(t) = i(t) · R. Feed the same sine wave into a capacitor or inductor and voltage and current are no longer in phase, so a single real number R cannot describe the relationship.
The fix is impedance, written Z, a complex-valued generalization of resistance. The form of Ohm's law is unchanged: V = I · Z. The difference is that Z has magnitude and phase. For an ideal resistor, Z = R (phase 0). For an ideal inductor at frequency ω, Z = jωL (current lags voltage by 90°). For an ideal capacitor, Z = 1/(jωC) (current leads voltage by 90°). Combine them in series or parallel with the same rules as resistors, using complex arithmetic.
The Voltage Divider calculator covers the resistive case. Drop capacitors or inductors into the same topology and you have an RC Filter, where the cutoff frequency depends on both components and Ohm's law becomes the frequency-domain tool behind every first-order filter you will ever design.
For DC, impedance reduces back to resistance and everything collapses to V = IR. For 60 Hz power systems, inductive reactance in motors and transformers is usually small enough that simple power calculations still work, though power factor corrections show up once inductive loads get significant. The Power Factor calculator handles that correction for real and apparent power.
Common Mistakes in Applying Ohm's Law
Three failure modes catch even experienced people.
Forgetting which voltage goes into the formula. V in V = IR is the voltage across the resistor in question, not the supply voltage. In the LED example, the supply was 5 V but the voltage across the resistor was 3 V. Using the full 5 V would give 15.2 mA (5/330), which is close enough here not to fry anything but wrong in principle and dangerously wrong in circuits with large supply voltages.
Mixing unit prefixes. Milliamps times kilo-ohms gives volts. Microamps times mega-ohms also gives volts. Amps times ohms gives volts. Mix them in any other combination and the result is off by a factor of 1000 or 1,000,000. Every practicing engineer has blown a part this way at least once.
Treating Ohm's law as Kirchhoff's laws. Ohm's law describes one component at a time. Kirchhoff's voltage law (the sum of voltages around a loop is zero) and current law (the sum of currents at a node is zero) tie the circuit together. You need both. In series, currents are equal and voltages add. In parallel, voltages are equal and currents add. The Resistor Network calculator handles the series and parallel arithmetic once you know which topology applies.
Go back to the opening image. A 12 V car battery, a 4 Ω speaker, 3 A, 36 W. Every circuit you will ever encounter reduces to variations on that equation stack. Learn the three forms of Ohm's law and the three forms of the power equation, know when linearity breaks, and keep track of units. The rest is topology.
Sources: Wikipedia, "Ohm's law"; Wikipedia, "Georg Ohm"; Wikipedia, "Electric power"; NIST SI unit definitions; Horowitz and Hill, The Art of Electronics, 3rd edition.