4  Ohm’s Law and Power

4.1 Start Simple

Imagine choosing a resistor for an indicator LED on a microcontroller pin. Ohm’s Law turns the supply voltage, LED drop, and safe current into a part value, while the power equation checks whether the part can survive. Start with the load you can see, then calculate current, voltage drop, and heat before wiring the circuit.

Phoebe’s Field Notes: Why “Constant Resistance” Is Only Approximately True

Phoebe’s Why

Ohm’s Law treats resistance as a fixed number, but it is really the outcome of electrons scattering off a vibrating lattice, and warmer metal vibrates harder. Raise the temperature and each free electron collides with the lattice more often, shortening the average time it drifts freely between hits – so for the same electric field, less current gets through. That is why the Under-the-Hood section on this page is right to ask “does the resistance change with temperature?” the answer is always yes, by an amount that depends entirely on what the resistor is made of. A precision resistor is deliberately alloyed to make that drift small; a length of bare wire is not, which is exactly why the 5 cm hookup wire in a coin-cell sensor is a worse assumption to warm up than the 260 ohm LED resistor a few paragraphs above.

The Derivation

Resistivity depends on carrier density \(n\), charge \(q\), effective mass \(m\), and the mean time between lattice collisions \(\tau\):

\[\rho = \frac{m}{nq^2\tau}\]

Above a metal’s Debye temperature, phonon population – and so the collision rate – grows in proportion to absolute temperature, which makes \(\tau \propto 1/T\) and therefore \(\rho \propto T\). Linearizing that around a reference temperature \(T_0\):

\[\rho(T) = \rho_0\left[1+\alpha(T-T_0)\right]\]

Geometry barely moves with temperature compared with \(\rho\), so \(R=\rho L/A\) inherits the same law:

\[R(T) = R_0\left[1+\alpha(T-T_0)\right]\]

\(\alpha\) is the temperature coefficient of resistance (TCR), and it is a property of the material, not of Ohm’s Law itself.

Worked Numbers: This Chapter’s LED Resistor Across an Industrial Range

• The chapter’s own 260 ohm LED resistor, as a catalog metal-film part (\(\alpha \approx 100\) ppm/°C \(=1.00\times10^{-4}\)/°C), taken from 25°C to the ends of a -40 to +85°C industrial range: at 85°C, \(\Delta R = 260\times1.00\times10^{-4}\times60 = 1.56\ \Omega\), so \(R=261.56\ \Omega\) (+0.600%); at -40°C, \(\Delta R = 260\times1.00\times10^{-4}\times(-65) = -1.69\ \Omega\), so \(R=258.31\ \Omega\) (-0.650%) – well under 1% across the full range, which is why the Beginner Checklist’s “use real component values” advice does not need a temperature caveat for this part class.
• The same 260 ohm value as bare copper wire instead (\(\alpha \approx 3930\) ppm/°C \(=3.93\times10^{-3}\)/°C, a 39.3x larger coefficient): the identical 60°C rise now moves the resistance by \(100\times3.93\times10^{-3}\times60 = 23.6\)%, not 0.6% – a plain length of wire used as a current-setting element would need the temperature caveat that a purpose-built metal-film resistor does not.
• Reading this back onto Under the Hood’s question: that section asks whether resistance “is stable, or changes with temperature,” without giving a number. A catalog-typical 100 ppm/°C part answers it quantitatively: about 0.6% drift across a full -40 to +85°C industrial swing is small next to the LED current margin already built into choosing a “nearby higher standard value,” so this specific resistor case does not need a temperature correction – but the 23.6% swing on bare wire shows the assumption fails hard for a different component class, which is why the escalation checklist calls out “thermal behavior” as its own line rather than folding it into the basic V=IR formula.

Overview: Ohm's Law Is A Safety Check

Ohm's Law connects voltage, current, and resistance: V = I x R. In IoT hardware, the formula is not just a classroom equation. It is how you check whether a GPIO pin, LED resistor, pull-up, divider, sensor rail, heater, relay coil, or fuse choice is plausible before powering the circuit.

The key habit is to ask what the circuit is trying to prove. If you know supply voltage and resistance, calculate current. If you know current and resistance, calculate voltage drop. If you know voltage and current, calculate power. Then compare the result with the limits of the component, board trace, wire, regulator, battery, or power supply.

Voltage Vera

“Before you blame the code, follow the current — most ‘bugs’ are a loose ground or a wrong resistor.”

In this chapter Vera turns each formula into a meter check: predict the number, then probe the built circuit to confirm it.

Use the figure as a formula selector: choose the Ohm's Law form that matches the values you know, then run the matching power check and loop or node sanity check before trusting the circuit.

That selection step matters because each formula answers a different review question. I = V / R estimates the current a path will demand. R = V / I chooses a current-limiting value. P = V x I translates the electrical result into heat and battery cost. Kirchhoff-style loop and node checks catch missing return paths, impossible voltage drops, and current that has no safe place to flow.

If you only remember two formulas, remember I = V / R for current and P = V x I for power. Those two checks catch many unsafe IoT circuit assumptions.

The Core Variables

Voltage

Voltage is the electrical push between two points. Always name the two points being compared, such as 3.3 V rail to ground or GPIO output to LED forward drop.

Current

Current is the flow through a path. It is what heats resistors, stresses GPIO pins, drains batteries, and decides whether protection is needed.

Resistance

Resistance limits current or divides voltage in a resistive path. Use real component values and include the part of the circuit where the voltage actually appears.

Power

Power is energy per second. It is the bridge from a correct electrical calculation to heat, battery life, regulator stress, and part ratings.

Beginner Checklist

• Use V = I x R only for paths that are approximately resistive.
• For LEDs and diodes, subtract the forward voltage before calculating the series resistor current.
• For motors, relays, and coils, expect startup, stall, or switching behavior that a simple DC resistance does not capture.
• After calculating current, compare it with the component's allowed current and the source's ability to supply it.
• After calculating power, compare it with the resistor, regulator, transistor, enclosure, and thermal margin.

Overview Knowledge Check

Practitioner: Turn Formulas Into A Circuit Review

A practical Ohm's Law review records the known values, the equation, the result, and the design decision that follows from the result. The calculation is only useful when it is tied to the real circuit path and to a component limit.

For many IoT circuits, the sequence is simple: identify the source voltage, identify what part of that voltage appears across the resistive element, calculate current, calculate power, and compare both values with ratings. Then measure the built circuit, because breadboard wiring, tolerances, batteries, regulators, and non-resistive loads can shift the result.

Review Ledger

Circuit Case

Known Values

Calculation

Decision

LED indicator

Supply voltage, LED forward voltage, target current, GPIO or driver limit.

R = (Vsupply - Vled) / Iled, then P = Iled^2 x R.

Choose a standard resistor value, keep current below the pin and LED limits, and verify brightness.

Pull-up resistor

Bus voltage, sink-current limit, bus capacitance, required rise time.

Check low-state current with I = V / R; check whether the signal rises fast enough.

Choose a value that is not too strong for the pin and not too weak for the bus timing.

Voltage divider

Input voltage, safe output voltage, input impedance, acceptable divider current.

Vout = Vin x R2 / (R1 + R2); check divider current and loading.

Scale the signal safely without wasting too much current or making the node too noise-sensitive.

Load or heater

Supply voltage, load resistance or rated power, wire path, fuse, switch, and source limit.

I = V / R or I = P / V; then check P = V x I.

Confirm source capacity, protection, wire margin, switch rating, and heat dissipation.

Worked Review: GPIO LED

1. State the path. A 3.3 V GPIO drives a red LED and a series resistor to ground.
2. Estimate resistor voltage. If the LED forward voltage is about 2.0 V, the resistor sees about 1.3 V.
3. Choose current. A visible indicator often needs less than the LED's maximum current; the GPIO limit is also part of the decision.
4. Calculate resistance. For 5 mA, R = 1.3 V / 0.005 A = 260 ohms; a nearby higher standard value reduces current.
5. Check power. Resistor power is small in this case, but the calculation should still be recorded.
6. Measure. Verify voltage and current on the actual board before reusing the design.

Vera’s Probe Points

• Measure: voltage across the series resistor, then the LED loop current.
• Expect: about 1.3 V across the resistor; 5 mA when it is 260 ohms.
• Wrong?: breadboard wiring, tolerances, or a shifted supply — the reasons this review ends by measuring the real board.

Worked Review: Simple Lamp Load

A classroom lamp circuit uses the same ledger. If a 12 V source drives a lamp path that behaves like a 3 ohm load, the current estimate is I = V / R = 12 V / 3 ohms = 4 A. The next review step is not just writing "4 A"; it is checking whether the battery or supply, switch, wire, connector, and lamp rating can safely carry that current. The matching power check is P = V x I = 12 V x 4 A = 48 W, which explains why a small formula result can become a heat, fuse, or source-capacity decision.

Vera’s Probe Points

• Measure: current in the lamp loop, with the meter in series.
• Expect: 12 V / 3 ohms = 4 A, and a 48 W power check.
• Wrong?: a sagging source or wire resistance in the loop — recheck supply, switch, connector, and lamp ratings.

Practitioner Knowledge Check

Under The Hood: Where Ohm's Law Stops Being Enough

Ohm's Law is exact for an ideal resistor at a fixed condition. IoT hardware is full of parts that are only partly resistive, change with temperature, or behave differently during startup and switching. The equation remains valuable because it gives a first-order ledger, but the ledger must identify where a deeper model or measurement is required.

The common trap is treating the calculated value as the final truth. A motor may draw far more current during startup or stall than its running current suggests. A diode has a nonlinear voltage-current curve. A battery voltage may sag under pulse load. A regulator may dissipate heat that limits current below the headline rating. A long wire may add enough resistance to change the load voltage.

For non-ideal parts, use Ohm's Law as a local approximation and then ask which assumption was hidden. Is the resistance stable, or does it change with temperature and current? Is the supply stiff, or does its internal resistance lower the voltage during a radio burst or motor start? Is the load passive, or does it contain semiconductor junctions, coils, capacitors, firmware-controlled modes, or protection paths?

A strong review therefore writes the condition next to the equation: steady state at a measured supply voltage, startup at current limit, LED at an expected forward voltage, divider under ADC loading, or resistor after warm-up. The math gives the expected value, and the measurement proves whether the real circuit stayed inside that model. When the two disagree, the next task is not to force the formula, but to identify the missing physical behavior.

Formula Set

Ohm's Law

V = I x R, I = V / R, and R = V / I. Use them for resistive paths and first-pass checks.

Power

P = V x I, P = I^2 x R, and P = V^2 / R. Use them to check heat, source load, and battery drain.

Series Paths

Series resistances add, current is common, and voltage divides according to resistance when the load does not disturb the divider.

Parallel Paths

Parallel branches share voltage, currents add, and total resistance is lower than the smallest branch resistance.

Escalation Checks

• Nonlinear parts: LEDs, diodes, transistors, and many sensors need their voltage-current behavior checked, not just a resistor equation.
• Dynamic loads: motors, relays, radios, and capacitors can create startup, inrush, stall, or pulse currents.
• Thermal behavior: resistance, regulator dropout, and safe operating margin can change as parts warm up.
• Source limits: batteries and regulators have internal limits that can sag voltage under load.
• Measurement placement: current must be measured in series with the path, while voltage is measured across two points.
• Protection: fuses, current limiting, flyback paths, and driver selection must match the real fault modes.

Under-the-Hood Knowledge Check

4.2 Summary

• Ohm’s Law links voltage, current, and resistance, and it is most useful when tied to a named circuit path.
• Power equations turn a current or voltage calculation into heat, battery drain, source capacity, and rating checks.
• LED resistors, pull-ups, voltage dividers, heaters, and simple loads all benefit from a small review ledger before power is applied.
• Nonlinear and dynamic parts need more than the simple equation: check startup, stall, pulse load, forward voltage, source sag, and thermal behavior.
• A defensible circuit decision records the formula, known values, result, rating comparison, and measurement evidence.

Key Takeaway

Use Ohm’s Law as a review workflow, not a memorized triangle: name the path, calculate current and power, compare with limits, and measure the built circuit.

4.3 See Also

Electricity Fundamentals

Review voltage, current, resistance, power, grounding, and measurement habits before applying formulas.

Electricity Applications

Apply circuit calculations to pull-ups, dividers, loads, protection, and troubleshooting decisions.

Electricity Common Pitfalls

Avoid unit mistakes, unsafe current paths, missing power checks, and misleading measurements.

Semiconductors, Doping, and Diodes

Move beyond resistive assumptions into nonlinear devices used throughout IoT electronics.