Ohm's Law: V = IR and Power Dissipation P = IV

Every current-limiting resistor, every gate drive network, every trace impedance calculation, and every power budget in a datasheet traces back to two equations: V = I·R and P = I·V. They look trivial enough to skip past, but a surprising fraction of field failures — burnt resistors, overheating regulators, undersized traces — come from an engineer applying them without checking the assumptions behind them or the numbers that fall out. This article covers the law itself, where it breaks down, and how to turn it into a power budget you can trust.

The Law Itself

Ohm's Law states that for an ideal resistive element, voltage across it is directly proportional to the current through it:

V = I·R

where V is in volts, I in amps, R in ohms. Rearranged, it also gives I = V/R and R = V/I. It's an empirical relationship, not a fundamental law of physics — it holds for a specific class of materials and devices called ohmic (or linear resistive) elements, over a defined operating range. Metal-film resistors, PCB copper traces, and wire are ohmic across enormous current and voltage ranges. Diodes, transistors, incandescent filaments, and thermistors are not — their V-I relationship is nonlinear, and applying V = IR to them with a fixed R is only valid as a local, small-signal approximation around an operating point.

Sign convention: current flows from higher to lower potential through a passive resistive element (passive sign convention). Get the polarity backwards in a hand calculation and every downstream power number will still compute correctly in magnitude, but current direction in a schematic will be wrong — worth checking against KVL/KCL before trusting the result.

Why "Ohmic" Isn't Automatic

Resistance itself isn't perfectly constant in real components:

• Temperature coefficient (TCR): resistors have a temperature coefficient, typically ±50 to ±200 ppm/°C for standard film resistors. A resistor carrying significant current self-heats, which changes R, which changes I — a feedback loop that matters in high-power or high-precision designs.
• Tolerance: a "10 kΩ" resistor might be 9.8–10.2 kΩ (±2%) or worse. Worst-case analysis must carry the tolerance through, not just the nominal value.
• Voltage/frequency dependence: at high frequency, skin effect and parasitic inductance/capacitance make the effective impedance deviate from the DC resistance value (see skin effect and non-ideal component topics). At DC and low frequency for a well-behaved resistor, V = IR is essentially exact.

For this article, all examples assume ideal, constant-value resistors at a fixed temperature — standard practice for first-pass hand analysis before adding tolerance/thermal margin.

Power Dissipation: P = IV

Power delivered to (or dissipated by) any two-terminal element is the product of voltage across it and current through it:

P = I·V

For a resistor obeying Ohm's Law, substituting V = IR or I = V/R gives two equivalent forms:

P = I²·R
P = V²/R

All three forms are algebraically identical — pick whichever one uses the two quantities you actually know. This is the single most common shortcut error in hand calculations: using P = V²/R with the source voltage when the resistor in question only sees part of that voltage (e.g., one resistor in a divider), or using P = I²R with total circuit current when the resistor only carries a branch current. Always confirm which V or I applies to the specific element, not the circuit as a whole.

Worked Example: Current-Limiting Resistor for an LED

A 5 V supply drives an LED with a forward voltage Vf = 2.1 V and a target forward current of 15 mA. Find the series resistor value and its power dissipation, and check standard resistor availability.

Step 1 — Apply KVL around the loop:

Vsupply = Vf + I·R
5 V = 2.1 V + (0.015 A)·R

Step 2 — Solve for R:

R = (5 − 2.1) / 0.015 = 2.9 / 0.015 = 193.3 Ω

Nearest standard E24 value: 200 Ω (choosing up, to bias current slightly below target rather than over-driving the LED).

Step 3 — Recompute actual current with the standard value:

I = (5 − 2.1) / 200 = 2.9 / 200 = 14.5 mA

Close to the 15 mA target — acceptable.

Step 4 — Power dissipation in the resistor:

P = I²·R = (0.0145)² × 200 = 0.00021 × 200 = 0.042 W

Cross-check with the alternate form using the voltage actually across the resistor (V_R = I·R = 0.0145 × 200 = 2.9 V, matching the KVL drop exactly):

P = V_R²/R = (2.9)² / 200 = 8.41 / 200 = 0.042 W

Both forms agree at 42 mW. A standard 1/8 W (0.125 W) resistor gives roughly 3× margin above the dissipated power — a reasonable derating for reliability; running a resistor at or near its full rated power in a poorly ventilated enclosure will elevate its temperature well above ambient and accelerate drift or failure.

Step 5 — Sanity check the whole loop: total power delivered by the source is P_total = Vsupply × I = 5 × 0.0145 = 72.5 mW. Power in the resistor (42 mW) plus power in the LED (Vf × I = 2.1 × 0.0145 = 30.5 mW) sums to 72.5 mW — matches the source output exactly, confirming conservation of energy (Tellegen's theorem in miniature) and validating the calculation.

Power Budgeting in Practice

A few practical consequences worth internalizing:

• Power scales with the square of current, not linearly. Doubling current through a fixed resistor quadruples its dissipation — a critical fact when a fault condition doubles or triples expected current draw.
• Resistor power rating is not a hard ceiling in the way voltage rating on a capacitor is — but continuous operation near the rated wattage substantially shortens life and increases resistance drift due to self-heating. Derating to 50% of rated power for continuous operation is a common conservative design rule.
• The same power law applies to any dissipative element, not just resistors: P = IV is fully general (it's the definition of electrical power), while P = I²R and P = V²/R only apply where V = IR holds, i.e., linear resistive elements.
• Trace resistance matters at high current. A PCB trace with even a few tens of milliohms of resistance carrying several amps can dissipate hundreds of milliwatts, enough to cause a measurable temperature rise — this is the basis of copper trace width tables used in PCB layout.

Where the Simple Law Breaks Down

• Non-ohmic devices: diodes, LEDs, transistor junctions follow exponential (Shockley diode equation) or more complex V-I relationships. V = IR is only used locally as a small-signal resistance (r = dV/dI at an operating point), not as a fixed global relationship.
• AC circuits with reactance: with capacitors and inductors in the circuit, voltage and current are no longer in phase, and the simple scalar R must be replaced with complex impedance Z (covered in the impedance and phasor topics) — power calculations then require real, reactive, and apparent power distinctions.
• Superconductors and other exotic materials: R → 0, and the ohmic model doesn't meaningfully describe the physics — not a practical concern in standard embedded design, but worth knowing as a boundary case.

Key Takeaways

• V = I·R holds exactly for ideal ohmic elements (resistors, wires, traces) at fixed temperature and frequency; it is only a local approximation for nonlinear devices like diodes and transistors.
• Power dissipation has three interchangeable forms — P = IV, P = I²R, P = V²/R — use whichever matches the two quantities you actually know for that specific element, not the whole circuit.
• Power scales with the square of current or voltage, so fault conditions that raise current even modestly can cause disproportionate heating.
• Always cross-check a power calculation with a second form or with energy conservation (sum of dissipated power should equal source output power) before trusting the result.
• Resistor tolerance, temperature coefficient, and power derating (commonly 50% of rated wattage for continuous duty) are real-world corrections layered on top of the ideal law — necessary for any design meant to survive outside a simulator.