3  Voltage, Current, and Resistance

3.1 Start Simple

Imagine a small sensor node powered by a coin cell. Voltage is the push available between two points, current is the flow through the chosen path, and resistance is what keeps that flow under control. Start with one source, one load, one return path, and one measurement that proves the circuit is behaving as intended.

Phoebe’s Field Notes: Where V = IR Actually Comes From

Phoebe’s Why

V = IR looks like a rule someone chose, but it falls out of how charge actually moves through a conductor. Drop an electric field along a wire and every free electron feels a steady push, accelerating between collisions with the vibrating lattice – but each collision resets its extra speed to zero. The result is not a runaway electron; it settles into a small, steady average drift velocity proportional to the field, the same way a marble dropped into a pinball machine settles into a steady average fall between bounces. Multiply that drift by how many charge carriers share the wire and how much charge each one carries, and the current comes out proportional to the field – that proportionality is the whole physical content hiding behind the resistor symbol. It also explains something the “3.3 V signal” framing hides: the electric field snaps into place at nearly the speed of light the instant a switch closes, but the actual electrons carrying current through a coin-cell sensor node’s power wire crawl – so slowly that one electron takes the better part of an hour to cross a 5 cm trace.

The Derivation

A field along a wire of length \(L\) carrying voltage \(V\):

\[E = \frac{V}{L}\]

Drift velocity under mobility \(\mu\):

\[v_d = \mu E\]

Current density from free-carrier density \(n\) and charge \(q\):

\[J = nqv_d = nq\mu E = \sigma E\]

where \(\sigma = nq\mu\) is conductivity. Current through cross-section \(A\):

\[I = JA = \sigma A\,\frac{V}{L}\]

so

\[V = I\,\frac{L}{\sigma A} = IR, \qquad R = \frac{\rho L}{A},\ \ \rho = \frac{1}{\sigma}\]

Worked Numbers: This Chapter’s Coin-Cell Wire

• Catalog-typical copper: \(n = 8.50\times10^{28}\ \mathrm{m}^{-3}\), \(\mu = 4.30\times10^{-3}\ \mathrm{m}^2/(\mathrm{V\,s})\) \(\Rightarrow \sigma = nq\mu = 5.86\times10^{7}\) S/m \(\Rightarrow \rho = 1/\sigma = 1.71\times10^{-8}\ \Omega\cdot\mathrm{m}\) – within about 1.7% of the accepted \(1.68\times10^{-8}\ \Omega\cdot\mathrm{m}\), the usual spread across catalog mobility figures
• 5.00 cm of 24 AWG wire (\(d = 0.511\) mm) linking the coin cell to the node: \(A = \pi(d/2)^2 = 2.05\times10^{-7}\ \mathrm{m}^2\); using the accepted \(\rho = 1.68\times10^{-8}\ \Omega\cdot\mathrm{m}\), \(R = \rho L/A = 4.10\times10^{-3}\ \Omega\)
• At an illustrative 50.0 mA active-mode draw (catalog-typical MCU-plus-radio burst current, not stated by the chapter): \(J = I/A = 2.44\times10^{5}\ \mathrm{A/m^2}\), so \(v_d = J/(nq) = 1.79\times10^{-5}\) m/s \(\approx 17.9\ \mu\mathrm{m/s}\)
• Time for one electron to drift the 5 cm trace: \(L/v_d = 2790\) s \(\approx 46.5\) minutes – compare the electric field itself, which reaches the far end in about \(L/c = 167\) ps, close to the speed of light
• Voltage drop over that wire at 50.0 mA: \(V = IR = 0.0500\times4.10\times10^{-3} = 2.05\times10^{-4}\) V \(\approx 0.205\) mV, about \(0.0062\%\) of a 3.3 V rail – small enough to ignore for a short trace, but \(R = \rho L/A\) grows linearly with any longer cable run or thinner gauge

Overview: Electricity Needs a Complete Loop

Electricity is the movement of charge through a circuit. In IoT hardware, that movement powers processors, sensors, radios, indicators, and actuators. A useful beginner model starts with a source, a load, conductors, and a return path. If the path is incomplete, current does not flow through the load in the intended way.

The four quantities that appear in almost every circuit review are voltage, current, resistance, and power. Voltage is the potential difference between two points. Current is charge flow through a path. Resistance limits current. Power is energy use per time and is the reason parts heat up and batteries drain.

The source does not create energy from nothing; it transforms another stored or delivered form into electrical potential. A battery begins with chemical energy, a generator begins with mechanical motion, a solar cell begins with radiant energy, and a power station may begin with non-renewable fuel, renewable flow, or nuclear heat. The upstream source matters for sustainability and capacity, but the circuit still has to prove the same voltage, current, resistance, and power claims.

Read the hierarchy as a review order: name the voltage difference, identify what limits or redirects current, confirm the intended current path, then check power because that is where heat and battery drain appear.

That order keeps beginner debugging grounded. First ask what voltage difference is actually available across the load, not just whether one wire is labeled 3.3 V. Next ask what resistance or impedance is in the path: a resistor, sensor element, wire fault, transistor channel, or accidental short can all change the current. Then ask where current really flows. Finally, calculate or measure power in the parts that drop voltage while carrying current.

The quantities are connected, but they are not interchangeable. A circuit can show a supply voltage while no useful current flows through the load. A low-resistance path can pull too much current even when the voltage looks normal. A small current can still matter if the device sleeps for months on a battery. Useful electricity notes therefore name the two voltage nodes, the current path, the current limit, and the state where the measurement was taken.

If you only need the intuition, keep this sentence: voltage is measured across, current flows through, resistance limits, power heats and drains, and ground is the shared reference.

The First Four Ideas

Voltage

A difference in electrical potential. It is measured between two points and gives context to every signal level.

Current

Charge flow through a closed path. It must be limited so wires, pins, regulators, and parts stay within their ratings.

Resistance

Opposition to current. It is used in current limiting, pull-ups, dividers, sensing elements, and protection networks.

Power

Energy per time. Power checks connect electrical design to heat, battery life, regulator capacity, and safety margin.

Beginner Checks

• Confirm the circuit has a source, a load, and a return path.
• Measure voltage across the load, not at a single point without a reference.
• Measure current in the path that actually feeds the load.
• Check component ratings for voltage, current, and power before assuming the circuit is safe.
• Use a common ground when two parts must compare signal voltages.

Overview Knowledge Check

Practitioner: Review the Circuit as Evidence

A practical electricity review turns a schematic or breadboard into a set of claims that can be checked. The review asks where current flows, what voltage each part sees, which element limits current, where heat may appear, and how measurements prove the design matches the intended operating state.

For IoT work, this review should include both steady behavior and transitions. A board may be safe while idle but fail when a radio transmits, a motor starts, a sensor heater turns on, or a GPIO pin changes mode. The electrical record should name the states that were checked.

How to Review a Simple IoT Circuit

1. Trace the loop. Mark the source, load, current-limiting element, switch or transistor, and return path.
2. Check voltage references. Confirm ground, supply rails, logic thresholds, and signal references are compatible.
3. Estimate current. Use Ohm's Law where it applies, then check current against pin, regulator, sensor, and load ratings.
4. Check power. Identify where voltage drop and current become heat, and compare against component ratings.
5. Measure the real state. Record voltage, current, and behavior with the actual wiring, firmware state, and load.

Circuit Review Ledger

Item

Question

Useful Evidence

Common Failure

Loop

Where does current leave the source, pass through the load, and return?

Annotated schematic, continuity check, and measured voltage across the load.

A missing return path or loose ground makes the circuit look unpredictable.

Reference

Do the communicating parts share a valid voltage reference?

Ground check, rail measurement, logic-level check, and pull-up or pull-down value.

A signal is read without a shared ground or stable bias.

Current

What limits current, and are pins, wires, regulators, and loads within ratings?

Ohm's Law calculation, current measurement, and component datasheet limits.

A GPIO pin drives a load directly without a current limit or driver stage.

Power

Where does heat appear, and can the part or package dissipate it?

Power calculation, thermal observation, regulator margin, and measured current state.

The circuit works briefly but overheats, resets, or drains the source too quickly.

Practitioner Knowledge Check

Under the Hood: References and Ratings Define Trust

Under the hood, electricity in IoT circuits is about boundaries. Voltage only has meaning between two nodes. Current only flows through a complete path. Resistance, capacitance, inductance, semiconductor behavior, and wiring all shape the actual current path. Ground is not magic; it is the reference node that parts use to compare signal voltages.

Ratings are also boundaries. A part can tolerate only certain voltages, currents, power dissipation, and temperature conditions. Exceeding a rating may cause immediate failure, gradual damage, misleading readings, or intermittent resets. That is why electrical evidence should include both calculations and measurements under real operating states.

The deeper rule is that circuit claims must balance. Around a closed loop, the voltage rises from sources and the voltage drops across loads must account for one another. At a node, current entering and current leaving must have a path. If those balances do not match the sketch, the sketch is missing a path, a reference, or a load state. This is why a breadboard that looks connected can still fail when a ground rail is split, a jumper lands one row off, or a meter lead measures against the wrong reference.

Measurements also change the evidence depending on where the instrument is placed. A voltmeter is placed across two nodes and usually draws very little current, so it can show a plausible voltage on a weak or floating node. An ammeter is placed in series with the path and changes the circuit if it is put across a source by mistake. A current-limited supply, bench meter, and temperature check together give a safer picture than any single number.

Boundary Checks

Node Boundary

Voltage measurements need named nodes: supply to ground, signal to ground, or voltage across a specific component.

Path Boundary

Current follows available paths, including unintended paths through protection diodes, pull-ups, shields, or shared grounds.

Rating Boundary

Part ratings set allowed voltage, current, power, temperature, polarity, and transient behavior.

State Boundary

A measurement applies only to the firmware state, load, wiring, source, and environment where it was taken.

Failure Patterns

• Floating input: a signal has no stable pull-up, pull-down, or driver, so it reads random values.
• Ground mismatch: two devices exchange signals without a valid common reference.
• Hidden load: an LED, sensor heater, motor start, or radio burst pulls more current than the source can supply.
• Optimistic measurement: voltage and current are measured only at idle, not during startup, transmit, switching, or fault states.

Under-the-Hood Knowledge Check

3.2 Summary

Electricity in IoT circuits is about controlled charge flow through a complete path. Voltage is a difference between nodes, current flows through loads, resistance limits current, and power connects electrical behavior to heat and energy use. Reliable circuit work combines a clear schematic, rating checks, and measurements taken in the real operating state.

3.3 Key Takeaway

Start every electricity review by tracing the loop, naming the reference, checking current limits, and measuring the real circuit state.

3.4 See Also

Electricity Fundamentals

Review the broader electricity evidence model for IoT circuits.

Ohm’s Law

Use voltage, current, resistance, and power equations for resistive circuit checks.

Electricity Common Pitfalls

Avoid common wiring, grounding, unit, current-limit, and power-rating mistakes.

Electricity Applications

Apply these checks to pull-ups, dividers, loads, protection, and sensor circuits.