Current, Resistance & Direct-Current Circuits

Here's the historical accident that trips everyone up: when Ben Franklin guessed at the sign of charge carriers in 1752, he guessed wrong. He called glass-rubbed-with-silk "positive" and amber-rubbed-with-fur "negative." By the time we discovered electrons (1897), the conventions were too entrenched to flip.

So today: in a metal wire, the actual moving charges are negative electrons drifting one way. But by convention, we draw current arrows pointing the opposite direction, as if positive charges were flowing. The math works out identically either way — (negative charge) × (opposite velocity) = (positive charge) × (original velocity).

Drift speed is embarrassingly slow. In a typical copper wire, individual electrons drift at about 10⁻⁴ m/s — slower than a garden snail. Yet when you flip a light switch, the light comes on instantly. How? Because the electric field that pushes the electrons travels at nearly the speed of light through the wire. Every electron starts moving at roughly the same instant, even though each one individually crawls.

This will shock you: V = IR is not a universal law of nature like Newton's laws or Maxwell's equations. It's an empirical observation that many materials happen to follow. "Ohmic" materials are those whose V-I relationship is a straight line; "non-ohmic" materials (diodes, transistors, neon tubes, living tissue) have curved V-I relationships.

Why most metals are ohmic: inside a metal, free electrons collide with vibrating atoms. Double the voltage (= electric field), double the force pushing them, double the drift speed, double the current. The proportionality arises because the scattering rate is roughly constant.

Non-ohmic examples you use daily:

• Diode: current flows easily forward, not at all backward — a one-way valve. Used in every power supply.
• LED: doesn't turn on until you exceed ~2 V, then current rockets up exponentially.
• Transistor: a third terminal controls how much current flows. The basis of every digital computer.
• Incandescent bulb: mildly non-ohmic because resistance rises with temperature.

Proportional reasoning trick (from the book): if you double a wire's length AND double its diameter, what happens to R? Length doubles R, but diameter doubling quadruples area (A = πr²), which quarters R. Net: R × 2 × 1/4 = R/2. Half the resistance.

An 18-gauge copper wire has R = 1.02 Ω at 20 °C. What's its resistance at 0 °C and 100 °C?

At 0 °C: R = 1.02 × [1 + 0.0039 × (0 − 20)] = 1.02 × 0.922 = 0.94 Ω
At 100 °C: R = 1.02 × [1 + 0.0039 × (100 − 20)] = 1.02 × 1.312 = 1.34 Ω

The 40% swing between ice and boiling matters enormously in engineering. Precision resistors need to be made from alloys (like manganin or constantan) whose α is close to zero, so they don't drift with temperature.

Why superconductivity is such a big deal: ~7% of all electrical energy worldwide is lost as heat in power transmission lines. If we had room-temperature superconductors, we could eliminate that loss entirely, cutting global electricity consumption by 7% overnight. That's why labs around the world are still hunting for materials with higher T_c — the economic prize is enormous.

Setup: a fresh battery has ε = 1.5 V and r ≈ 0. An old battery has ε = 1.5 V (unchanged!) but r = 1000 Ω. When supplying I = 1 mA:

V_ab (new) = 1.5 − (10⁻³)(0) = 1.5 V
V_ab (old) = 1.5 − (10⁻³)(1000) = 1.5 − 1.0 = 0.5 V

The old battery's terminal voltage is only 0.5 V, even though its "emf" is still 1.5 V. This is why flashlights dim gradually instead of suddenly cutting out — the internal resistance creeps up as the chemistry degrades.

How to test a battery properly: a voltmeter with no load will read the open-circuit emf (1.5 V) even on a dead battery. You must measure terminal voltage under load (i.e., with a known current drawn) to see the real state. Automotive "load testers" do exactly this for car batteries.

Why car batteries weaken in cold weather: internal resistance of a lead-acid battery rises as temperature drops. A car that starts fine at 20 °C might not have enough V_ab to crank the starter at −20 °C, even though the "emf" reading is unchanged.

In physics, whenever you can compute the same quantity two different ways, it's worth doing so — getting the same answer is a powerful sanity check. Here, V_ab can be computed either by looking at the resistor (Ohm's Law: V = IR) or by looking at the battery (EMF minus internal drop: V = ε − Ir). Both yield 8 V because energy is conserved.

This is Kirchhoff's loop rule sneaking in early. Around the entire loop, ε = IR + Ir. Rearranging: ε − Ir = IR, which says "the battery's terminal voltage equals the voltage drop across the external resistor." That's a statement of conservation — energy gained at the battery = energy dissipated in the resistors.

Common student trap: confusing "emf" (the intrinsic push of the battery) with "terminal voltage" (what you actually measure at the battery's terminals while current flows). These are equal only if r = 0 (ideal battery) or I = 0 (open circuit). Under any normal load on a real battery, V_terminal < ε.

You have three forms of the same equation. Which should you use?

• P = VI — use when you know voltage and current directly. Works for ANY circuit element (not just resistors).
• P = I²R — use when you know current and resistance. Best for elements in series, where current is the same everywhere.
• P = V²/R — use when you know voltage and resistance. Best for elements in parallel, where voltage is the same across each branch.

Power dissipation scales QUADRATICALLY. Double the current and you quadruple the heat in a fixed resistor. This is why power lines run at extremely high voltage — to keep current (and thus I²R loss) small. The national grid ships electricity at 500,000 V for exactly this reason.

LED vs. incandescent — the application box from the book: an incandescent bulb dissipates most of its energy as heat (making the filament glow as a side effect). An LED converts electrical energy directly into light with very little heat loss — which is why a 10 W LED can produce the same light as a 60 W incandescent. Same luminosity, 1/6 the power bill.

Series resistors create a voltage divider. If you have two resistors R₁ and R₂ in series across a voltage V, the voltage across R₂ is:

V_out = V · R₂ / (R₁ + R₂)

So a 9 V battery with two equal resistors gives exactly 4.5 V across each. Want 3 V? Make R₂ twice R₁. Want 1 V? Make R₂ = R₁/8.

Real-world uses: every microphone input on a computer uses a voltage divider to bring line-level signals down to mic-level. Every analog volume control on old stereos is a voltage divider (a "potentiometer" is just a variable resistor acting as a divider). Every temperature sensor that uses a thermistor reads the voltage division between the thermistor and a reference resistor.

Christmas light failure mode: old-style Christmas lights were wired in series, so if one bulb burns out, the whole string goes dark. The problem is that with the burned bulb acting as an "infinite resistance," no current flows at all. Modern Christmas lights solve this with tiny "shunt" wires that short out burned bulbs to preserve the circuit.

Every outlet in your house is wired in parallel with every other outlet. This has two huge advantages:

• Every outlet sees 120 V — the same as the wall socket, regardless of what else is plugged in.
• Turning one device off doesn't affect others — each branch has its own current path.

If houses were wired in series, turning off the kitchen lights would also turn off your fridge, and each device would only receive a fraction of the line voltage depending on how many other devices were on. Chaos.

The "current hog" principle: in parallel, the resistor with the lowest R draws the most current (I = V/R). So if you short-circuit (R → 0) any branch, ALL the current floods through that short, overwhelming the others and potentially tripping a breaker or starting a fire. This is why circuit breakers exist — they sense when parallel current exceeds a safe limit and open the circuit.

The "product over sum" trick: for just two parallel resistors, R_eq = R₁R₂/(R₁+R₂). For three or more, you must use the full 1/R_eq = 1/R₁ + 1/R₂ + 1/R₃ form. Students often forget that "product over sum" is a two-resistor shortcut only.

Series-parallel reduction works for most textbook problems, but it has limits. Consider a bridge circuit — four resistors arranged in a diamond with a fifth resistor across the middle. None of the resistors in the outer diamond are purely in series or parallel with each other, because of the cross-link. You cannot reduce a bridge with the step-by-step method.

When you hit a bridge or any "non-planar" network, you must use Kirchhoff's rules (Slides 14–17). Write junction equations at every node, loop equations around every independent loop, and solve the resulting linear system.

Rule of thumb: if you stare at a circuit for 30 seconds and can't identify a clean series or parallel pair, don't force it — jump straight to Kirchhoff.

Imagine you have two lightbulbs: a 100 W bulb and a 40 W bulb, both rated for 120 V. Which glows brighter when wired in series across 120 V?

Counterintuitive answer: the 40 W bulb is brighter! Here's why: a bulb's "wattage rating" is at its rated voltage (120 V). The 100 W bulb has R = V²/P = 144 Ω. The 40 W bulb has R = 144²/40 = 360 Ω. In series, they carry the same current. With P = I²R, the larger resistance (40 W bulb) dissipates more power — and glows brighter — despite being the "smaller" bulb on the label.

In parallel across 120 V, the rating works normally — the 100 W bulb is brighter because it's designed to dissipate 100 W at 120 V while the 40 W bulb dissipates 40 W.

Understanding this distinction is often the tiebreaker on exam multiple-choice questions about power in resistor networks.

Here's a mnemonic that actually works: "Series Shares Current; Parallel Pairs Pressure." Alliteration-based but unambiguous. In series, all elements share (= have the same) the current. In parallel, all elements share (= have the same) the voltage (pressure).

From these two "shared" quantities you can derive everything else:

• In series, if I is shared, then V_i = I · R_i is proportional to R. Bigger R → bigger V drop.
• In parallel, if V is shared, then I_i = V / R_i is inversely proportional to R. Bigger R → smaller I.

Then power follows: P = VI. In series, the big-R element has a big V (and fixed I), so big P. In parallel, the small-R element has a big I (and fixed V), so big P. All four rules in the matrix above drop out of one memorized fact: "series shares I, parallel shares V."

The two rules feel like arbitrary recipes, but they're not — they're direct consequences of fundamental physics.

Junction rule = charge conservation. No charge can accumulate at a wire junction (there's nowhere for it to go in a steady-state DC circuit). So whatever flows in must flow out, instantaneously. Water in a pipe T-junction works the same way: flow rate in = flow rate out.

Loop rule = energy conservation. Electric potential is a well-defined number at every point in the circuit. If you walk around a loop and measure voltage changes, the total must be zero when you return — otherwise potential would be multi-valued, which is physically impossible. The gravity analogy: if you hike around a mountain and come back to where you started, your net change in altitude is zero.

Because these rules are rooted in conservation, they work for ANY circuit — DC, AC, linear, non-linear, with capacitors, with inductors, with diodes — anything. They are among the most universal equations in all of physics.

Many students get confused because "walking with the current through a resistor" sounds like you'd gain energy, but it's actually a drop. The reason: you're walking "downhill" through the resistor — current flows from high potential to low potential through a resistor. So moving with the flow means you're moving from + to −, losing potential.

It's exactly like walking along a river. The water flows from high ground to low ground. If you walk downstream (with the current), you're going downhill — losing altitude (potential). If you walk upstream (against the current), you're going uphill — gaining altitude. Same idea.

The battery is the "uphill" element: current inside a battery is driven from − to + by chemical energy (against the natural electric-field direction). So walking from − to + through a battery means walking with the "uphill pump" — gaining potential energy, hence +ε. Walking from + to − means working backwards through the pump (drop).

Common mistake: drawing current going the "wrong way" then getting confused about the sign. Fix: always trust your initial current direction guess. If the math gives a negative number, the real current simply flows the other way. You don't redraw anything — you just reinterpret the sign.

1. Label every branch current with an arbitrary direction. If you guess wrong, you'll get a negative answer and just reinterpret — no need to redo.
2. Apply KCL at every node to reduce the number of unknowns. For an N-loop circuit, you get N−1 independent junction equations.
3. Pick N independent loops and apply KVL to each. "Independent" means each loop contains at least one element the others don't. For planar circuits, the "windows" (mesh loops) work perfectly.
4. Walk each loop in ONE chosen direction (CW or CCW doesn't matter, as long as you're consistent). Use the sign rules from Slide 14.
5. Solve the resulting linear system. For 2 unknowns, substitution. For 3+, elimination or matrix methods.

The universal verification: plug your answers back into EVERY loop equation. If all of them sum to zero, you're done.

This is the single most common mistake on circuit lab work, and it can destroy instruments.

Voltmeter in series: a voltmeter has a huge internal resistance (typically 10 MΩ). Putting it in series with the circuit means that resistance is now part of the current path — the circuit's total resistance becomes effectively infinite, so essentially no current flows. Your circuit "turns off" and the voltmeter reads the full supply voltage (or close to it) regardless of what's in the circuit. Not dangerous — just useless.

Ammeter in parallel: an ammeter has a tiny internal resistance (milliohms). Putting it in parallel with a resistor creates a near-short — a huge current rushes through the ammeter, potentially blowing its internal fuse or burning out the measurement coil. This is how students destroy lab ammeters. Always think: "A = series, V = parallel" before connecting anything.

Why ideal ammeters have R = 0: if R_A were nonzero, inserting the ammeter would add to the circuit's total resistance, changing the very current you're trying to measure. An ideal ammeter is a perfect conductor that reads current without disturbing it.

Why ideal voltmeters have R = ∞: the voltmeter must not draw any current from the circuit being measured. If it did, the current through the resistor you're measuring would change, shifting the voltage you observe. A modern digital voltmeter typically has R ≈ 10 MΩ — close enough to ∞ for most purposes.

RC circuits exhibit exponential behavior because the current charging the capacitor depends on the voltage difference driving it, which in turn depends on how much charge is already there. As Q grows, V_C grows, the driving voltage ε − V_C shrinks, so current tapers off. The rate of growth is proportional to the remaining "distance to go" — that's the mathematical definition of exponential decay.

Mathematically: dQ/dt = (ε − Q/C)/R. This is a first-order linear ODE whose solution is the exponential Q(t) = Cε(1 − e^(−t/RC)).

Real-world RC circuits you use every day:

• Camera flash: a capacitor slowly charges from a battery (large τ = slow to charge), then discharges through a flash tube in microseconds (tiny τ = fast discharge).
• Keyboard debounce: when you press a physical key, the contacts bounce for a few milliseconds. An RC filter smooths this out so the computer only sees one "clean" press.
• Windshield wiper delay: the "intermittent" setting uses an adjustable RC timer. Turn the knob → change R → change the wait time between wipes.
• Defibrillator: a large capacitor charges slowly to 1000+ V, then releases a massive pulse through the patient's chest in ~10 ms.
• Audio tone control: every bass/treble knob on an amplifier is an RC filter. Rolling off highs = adjusting an RC time constant.

Dry skin has a resistance of ~100 kΩ, so touching a 120 V wall outlet with a dry finger draws only I = V/R = 120/100,000 = 1.2 mA — noticeable but not dangerous. Wet skin, however, drops to ~1 kΩ, so the same 120 V now drives 120 mA — firmly in the "ventricular fibrillation" zone. This is why electricity + water is a deadly combination.

Why "can't let go" is so dangerous: above ~10 mA, the muscles in your hand contract involuntarily and grip even tighter, locking you onto the live wire. You literally can't open your hand. This is why electricians are trained to test wires with the back of the hand — if they shock themselves, the muscle contraction flings the hand away instead of clamping onto the wire.

GFCI (ground-fault circuit interrupter) outlets monitor the difference between current going out and coming back on the hot and neutral wires. Normally they should be equal; if they differ by more than 5 mA, it means current is leaking somewhere — probably through a human. GFCIs trip within 25 ms — fast enough to prevent fibrillation. By law, all bathroom and kitchen outlets in the US must be GFCI-protected.

Heart surgery levels: when electrodes are placed directly on the heart, there's no protective skin resistance. Just 25 µA (the book's number) can trigger ventricular fibrillation. For reference, that's the current drawn by a typical LED — truly minuscule. This is why ALL surgical equipment must be "isolated" and grounded, and why the entire operating room is built as a "safe zone" to prevent stray currents.

In the 1880s, Thomas Edison championed direct current (DC) while Nikola Tesla and George Westinghouse pushed alternating current (AC). The deciding factor was exactly the calculation above: AC can be stepped up and down easily using transformers, but DC cannot. Transformers only work with changing currents (Faraday's Law), so if you wanted to step DC from 10 kV to 500 kV, you'd need rotating mechanical devices — clunky and lossy.

AC won because it enabled efficient long-distance transmission. Westinghouse lit the 1893 World's Fair and the Niagara Falls plant, while Edison's DC systems required a power plant every mile or two. By 1900 it was over.

But DC is making a comeback: modern HVDC (high-voltage DC) transmission lines use solid-state electronics to convert between AC and DC. For very long distances (> 600 km) or underwater cables, HVDC is now more efficient than HVAC because it doesn't suffer from capacitive losses. China's Zhundong–Wannan line runs 3,300 km at 1.1 MV — the highest-voltage power line ever built. Tesla would be thrilled.

Chapter 19 is your first exposure to the full machinery of circuit analysis. Every future chapter (magnetism in ch. 20, induction in ch. 21, AC in ch. 22, electromagnetic waves in ch. 23) builds on these fundamentals. If you understand the structure below, you can reconstruct any formula in this chapter from first principles:

• Current carriers feel a force — that's Ohm's Law at the microscopic level
• Batteries do work to move charges uphill — that's EMF
• Energy is conserved around any closed loop — that's Kirchhoff's loop rule
• Charge is conserved at junctions — that's Kirchhoff's junction rule
• Power = V × I everywhere — use the form that matches your known variables

Every problem in this chapter reduces to one of these truths. Master them and the formulas become obvious. Struggle with them and you'll be memorizing patterns forever.

Next chapters build on chapter 19 like this: ch. 20 adds magnetic forces on moving charges (which are currents). Ch. 21 shows that changing magnetic fields induce EMFs — and those EMFs drive currents through exactly the circuit math you learned here. So every "flux change" ultimately becomes a Kirchhoff problem. Nothing is wasted.