Week 1 — From Fields to Circuits: Charge, Current, Voltage, Ohm’s Law, and the Bench

Overview

Every circuit in this course is, underneath, a configuration of electric and magnetic fields obeying Maxwell’s equations. We almost never solve Maxwell’s equations to analyze a circuit — we use the lumped-element model, where a wire is an equipotential, a resistor is a single number, and current is the same at both ends of a component. This week is about earning that simplification: understanding what voltage and current are at the field level, why the lumped approximation is valid for the slow, small circuits we build, and exactly when it would break. If you only ever learn the rules (“voltage is across, current is through”) without the field picture, you will be helpless the first time a circuit is fast enough or big enough that the rules stop holding.

The second half of the week is pure bench fluency. A staff-level engineer is defined partly by trusting their instruments and knowing their limits. By the end you should be able to measure resistance, DC voltage, and current with the Fluke 117 without thinking, wire a clean circuit on the breadboard, and — critically — predict every measurement before you take it. The discipline we establish now (predict by hand → simulate in Qucs-S → measure → reconcile the three numbers) is the spine of all ten weeks.

This week underpins everything. Week 2’s network analysis assumes the lumped model and Ohm’s law are second nature. Week 3’s capacitors and inductors are the field-energy story continued. The measurement workflow established here is reused unchanged through the transient, AC, filter, and DSP weeks.

Readings

Griffiths Ch. 2 (electrostatics) and Ch. 5 intro (currents): Extract the field meaning of the circuit variables. Voltage between two points is the line integral of \(\mathbf{E}\), i.e. a potential difference \(V = -\int \mathbf{E}\cdot d\boldsymbol{\ell}\); current is the flux of current density, \(I = \int \mathbf{J}\cdot d\mathbf{a}\). You are not solving boundary-value problems here — you are extracting why a “voltage” is a single well-defined number between two nodes.
CAD Ch. 1: Circuit terminology, the four base quantities (charge, current, voltage, power), the passive sign convention, and ideal sources. Extract: the bookkeeping conventions that make later analysis unambiguous.
PEI Ch. 2 (theory): The engineer’s working version of the same ideas — Ohm’s law, power dissipation, resistor color codes, real resistor tolerances and power ratings. Extract: what a real resistor is, not just the ideal one.
ME Experiments 1–4: Tactile intuition — short a battery through your tongue / a resistor, feel that current needs a complete loop, see an LED light. Extract: the visceral sense that current flows in loops and components have polarity.

The deliberate contrast: Griffiths tells you what the quantities are, CAD tells you how to bookkeep them, PEI tells you what real parts do, and ME makes them physical.

Key Concepts

Charge, current, and current density

Charge \(q\) is conserved and quantized in units of \(e = 1.602\times10^{-19}\) C. Current is the rate of charge transport through a surface:

\[ i(t) = \frac{dq}{dt}. \]

At the field level the local quantity is current density \(\mathbf{J}\) (A/m²), and the current through a surface \(S\) is the flux \(I = \int_S \mathbf{J}\cdot d\mathbf{a}\). In a wire of cross-section \(A\) carrying uniform \(\mathbf{J}\), \(I = JA\). One ampere is one coulomb per second — about \(6.24\times10^{18}\) electrons per second past a point. Electrons drift slowly (mm/s) even when the signal (a change in \(\mathbf{E}\)) propagates near the speed of light; the lumped model cares about the latter.

Voltage as a potential difference

In electrostatics \(\mathbf{E} = -\nabla V\), so the work per unit charge to move between points \(a\) and \(b\) is path-independent:

\[ V_{ab} = V_a - V_b = -\int_b^a \mathbf{E}\cdot d\boldsymbol{\ell}. \]

Path-independence is exactly what lets us label each node with a single number. It holds when \(\nabla\times\mathbf{E}=0\), i.e. when \(\partial\mathbf{B}/\partial t\) is negligible over the loop (Faraday’s law: \(\oint \mathbf{E}\cdot d\boldsymbol{\ell} = -d\Phi_B/dt\)). For our slow circuits this is true to extraordinary precision — which is why “the voltage at a node” is meaningful.

The lumped-element approximation and when it holds

The lumped model makes three assumptions, each a limiting case of Maxwell:

No net charge accumulates inside a component (\(\partial q/\partial t \approx 0\) inside) → current in equals current out → KCL.
Negligible magnetic flux through any loop (\(d\Phi_B/dt \approx 0\) outside inductors) → voltage is path-independent → KVL.
Signals change slowly compared to light-transit time across the circuit.

The quantitative test is the electrical size. A signal of frequency \(f\) has wavelength \(\lambda = c/f\) (or, for a sharp edge of rise time \(t_r\), a characteristic length \(\sim c\,t_r\)). The lumped model is valid when the physical circuit dimension \(\ell\) satisfies

\[ \ell \ll \lambda. \]

A rule of thumb is \(\ell < \lambda/10\). At our top scope bandwidth of 100 MHz, \(\lambda = 3\,\text{m}\), so \(\lambda/10 = 30\) cm — a breadboard is comfortably lumped. At 1 GHz it would be 3 cm and the breadboard becomes a distributed (transmission-line) system. This single inequality is the boundary between this course and RF engineering.

Ohm’s law — and what it really claims

For an ohmic material, \(\mathbf{J} = \sigma\mathbf{E}\) (conductivity \(\sigma\)). Integrating over a uniform conductor of length \(L\) and area \(A\) gives the familiar lumped form:

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

where \(\rho = 1/\sigma\) is resistivity. Ohm’s law is not a law of nature like Maxwell’s equations — it is a constitutive, empirical, linear approximation that many materials obey over a useful range. Diodes and transistors (Week 7) flagrantly violate it; that nonlinearity is the whole point of microelectronics.

Power and the passive sign convention

Instantaneous power delivered to a two-terminal element is

\[ p(t) = v(t)\,i(t). \]

Under the passive sign convention, current is defined entering the \(+\) terminal; then \(p>0\) means the element absorbs power and \(p<0\) means it delivers power. A resistor always absorbs: \(p = vi = i^2R = v^2/R \ge 0\), dissipated as heat. This sign discipline is what keeps energy bookkeeping consistent when we get to sources, capacitors, and amplifiers. Resistor power rating matters in practice: a \(\frac14\) W resistor at 5 V across \(100\,\Omega\) dissipates \(v^2/R = 0.25\) W — right at its limit, and it will run hot.

Real resistors vs ideal resistors

An ideal resistor is one number. A real resistor has: a tolerance (±5% for the common carbon-film parts in the kit, encoded in the 4th color band), a power rating (typically \(\frac14\) W in the kit — exceed it and the value drifts or it burns), a temperature coefficient, and tiny parasitic inductance/capacitance that only matter at high frequency. The color code: each band is a digit/multiplier — e.g. brown-black-red = 1, 0, ×100 = \(1\,\text{k}\Omega\). Always confirm with the meter; reading bands wrong is the most common Week-2 bug.

Theory Exercises

Starting from \(V = -\int \mathbf{E}\cdot d\boldsymbol{\ell}\) and \(\mathbf{J}=\sigma\mathbf{E}\), derive \(R = \rho L / A\) for a uniform cylindrical conductor, stating every assumption.
State the three field-level conditions under which the lumped model is valid, and tie each to either KCL or KVL.
Compute the electrical size \(\ell/\lambda\) for a 15 cm breadboard trace at 1 kHz, 1 MHz, 100 MHz, and 1 GHz. At which frequency does the lumped model first become questionable by the \(\lambda/10\) rule?
A \(\frac14\) W, \(220\,\Omega\) resistor: what is the maximum voltage you may put across it, and the maximum current through it, before exceeding the power rating?
Using the passive sign convention, show that a resistor can never deliver net power. Then give an example of a two-terminal element for which \(p<0\) is possible.
Decode the color bands for the following kit values and give the tolerance: \(1\,\text{k}\Omega\), \(10\,\text{k}\Omega\), \(470\,\Omega\), \(4.7\,\text{k}\Omega\).

Lab / Bench Work

Bring-up (ME-style): Power the breadboard from the MB102 module (verify it outputs 5 V and 3.3 V on the correct rails with the Fluke before connecting anything). Build the canonical first circuit: 5 V → \(330\,\Omega\) → LED → ground. Confirm it lights; reverse the LED and confirm it does not (polarity/diode preview for Week 7).

Ohm’s law verification: Pick five resistor values spanning the kit (e.g. \(100\,\Omega\) to \(10\,\text{k}\Omega\)). For each, measure \(R\) with the Fluke (ohmmeter, component out of circuit), then build a simple source–resistor loop, measure \(V\) across it and \(I\) through it (meter in series — this is the one measurement people wire wrong), and check \(V=IR\) to within the resistor tolerance.

Divider sanity check (sets up Week 2): Build a two-resistor divider, predict the mid-node voltage by hand, draw and simulate it in Qucs-S (ngspice backend), then measure. Record all three numbers.

Measurement Methodology

Resistance: Measure resistors out of circuit — parallel paths corrupt an in-circuit reading. Note the meter’s own resolution and that very low resistances include lead resistance (null it if the meter supports relative mode).
Voltage: The voltmeter is high-impedance and goes in parallel with the element. Its loading is negligible here (\(>10\,\text{M}\Omega\) input vs our k\(\Omega\) resistors) but state that assumption.
Current: The ammeter is low-impedance and goes in series — you must break the loop. The Fluke’s current jacks are fused; using the wrong jack or leaving it in current mode across a voltage source is the classic way to blow the fuse. Build the habit of returning the dial/leads to voltage mode.
Reconcile three numbers for every quantity: hand prediction, Qucs-S, measurement. A >5% discrepancy that is not explained by tolerance is a bug — find it before moving on. This habit is the entire point of the bench half of the course.

Expected baselines: Measured resistor values within ±5% of nominal (kit tolerance). Ohm’s-law residual \(|V - IR|/V\) under a few percent, dominated by resistor tolerance and meter accuracy, not by any modeling error.

Connections

This week is the foundation slab. Week 2 builds systematic network analysis directly on KCL, KVL, and Ohm’s law established here. Week 3 continues the Griffiths field-energy thread into capacitance and inductance. The predict→simulate→measure→reconcile loop is used unchanged every subsequent week; the only thing that grows is the sophistication of the circuit and the instrument (LCR meter in Week 3, scope from Week 4 on).

The lumped-vs-distributed boundary (\(\ell \ll \lambda\)) is worth internalizing now: it is exactly why this course can treat a wire as ideal, and exactly why Course 4’s high-speed digital signals and any future RF work cannot.