Section 02
The Idea, Step by Step
Think of electric charge like water moving through pipes. A battery is a pump: it pushes the charge around the loop. Voltage is how hard it pushes, current is how much charge flows past each second, and resistance is how much the pipe squeezes that flow. Open things up (less resistance) and more flows; pinch the pipe (more resistance) and less gets through. That single picture is what a circuit is.
From "push" and "squeeze" to one rule
Three named quantities carry the whole story: voltage $V$ (the push, in volts), current $I$ (the flow, in amperes), and resistance $R$ (the squeeze, in ohms). For an ordinary resistor they lock together in one tidy relationship — Ohm's law — so the current is simply the push divided by the squeeze, $I = V/R$.
The one rule to start from
$$V = IR \qquad\Longrightarrow\qquad I = \frac{V}{R}$$
Worked number: put a $12\text{ V}$ battery across a $100\ \Omega$ resistor and the current is $I = 12/100 = 0.12$ A. The energy it delivers each second is the power $P = IV = 0.12 \times 12 = 1.44$ W, almost all of it turning into heat in the resistor. Double the voltage or halve the resistance and both the current and the power climb.
Chaining resistors, then adding time
Real circuits combine resistors two ways. In series the same current threads through every part, so the resistances simply add: $R_\text{series} = R_1 + R_2 + \cdots$. In parallel each resistor feels the full battery voltage and the current splits between branches, so $\tfrac{1}{R_\text{parallel}} = \tfrac{1}{R_1} + \tfrac{1}{R_2} + \cdots$ — the total always comes out smaller than the smallest single branch, because every extra path gives charge another way through. Add a capacitor and time enters the story: charging through a resistor, the capacitor voltage rises as $V_C(t) = V_0\!\left(1 - e^{-t/\tau}\right)$ with time constant $\tau = RC$. After one $\tau$ it has reached about 63%; after $5\tau$ it is essentially full. The sliders map straight onto this: $V$ is the battery, $R$ and $R_2$ are the two resistors, and $C$ sets how slowly the RC circuit fills.
Try this in the sim above
(1) In Ohm's Law mode, halve $R$ and watch both the current and the power roughly double. (2) Build the same two resistors in Series, then switch to Parallel and compare the total resistance — parallel is always the smaller number. (3) In RC Circuit mode, raise $C$ and watch the charging curve stretch out as the time constant $\tau = RC$ grows.
Section 03
Equations & Derivation
Ohm's Law & Power
$$V = IR,\quad P = IV = I^2 R = \frac{V^2}{R}$$
Series & Parallel Resistors
$$R_{\text{series}} = R_1+R_2+\cdots,\quad \frac{1}{R_{\text{parallel}}} = \frac{1}{R_1}+\frac{1}{R_2}+\cdots$$
RC Circuit — Charging
$$V_C(t) = V_0\!\left(1-e^{-t/\tau}\right),\quad I(t) = \frac{V_0}{R}e^{-t/\tau},\quad \tau = RC$$
Resistivity
$$R = \frac{\rho L}{A},\quad \rho(T) = \rho_0[1+\alpha(T-T_0)]$$
Symbol Definitions
| Symbol | Quantity | SI Unit |
|---|
| $R$ | Resistance | Ω |
| $V$ | Potential difference (voltage) | V |
| $I$ | Current (charge/time) | A |
| $P$ | Power dissipated | W |
| $\rho$ | Resistivity | Ω m |
| $\tau=RC$ | Time constant of RC circuit | s |
| $C$ | Capacitance | F |
1
Ohm's Law. $V=IR$ for ohmic conductors (where $R$ is constant). Many devices are non-ohmic (diodes, transistors). Current $I=dq/dt$ — charge flow per unit time. Conventional current: $+$ to $−$ outside battery.
2
Series circuits. Same current through all components. Voltages add: $V_{\text{total}}=V_1+V_2+\cdots$. Resistances add: $R_{\text{total}}=\sum R_i$. If one component breaks, the whole circuit fails.
3
Parallel circuits. Same voltage across all components. Currents add: $I_{\text{total}}=I_1+I_2+\cdots$. $1/R_{\text{total}}=\sum 1/R_i$. Total resistance is always less than the smallest branch resistance. If one component fails, others continue.
4
RC circuit. Charging capacitor: $V_C(t)=V_0(1-e^{-t/\tau})$, $\tau=RC$. At $t=\tau$: $V_C=0.632V_0$. At $t=5\tau$: $V_C\approx99.3\%V_0$ (fully charged). Energy stored: $U=\frac{1}{2}CV^2$.
Ref: Halliday, Resnick & Walker 10th Ed., Ch. 26–27; Serway & Jewett 8th Ed., Ch. 18–19; Horowitz & Hill — The Art of Electronics (3rd Ed.).
Section 05
Common Misconceptions
❌ Ohm's Law is a fundamental law of physics.
✅ Ohm's Law ($V=IR$ with constant $R$) is an empirical approximation, not a fundamental law. It holds for many conductors (resistors, copper wire) at constant temperature but fails for diodes (non-linear), transistors, thermistors (R changes with T), and superconductors ($R=0$). The actual fundamental law is the Drude model.
📖 HRW 10th Ed., §26-3.
❌ Parallel circuits are safer because they have lower resistance.
✅ Lower resistance means more current from the same voltage source ($I=V/R$), which means more power dissipated ($P=V^2/R$). Household wiring uses parallel circuits for convenience (independent operation) but high current demands require appropriate wire gauges to prevent overheating.
📖 HRW 10th Ed., §27-3.
❌ Current is consumed as it flows through a circuit.
✅ Current (charge/time) is conserved by Kirchhoff's Current Law. The same current exits a resistor as enters it. What is consumed is energy — the potential energy of charges (voltage) drops as it powers the resistor, converted to heat or light.
📖 HRW 10th Ed., §26-4.
Misconception research: McDermott — Tutorials in Introductory Physics; Shipstone (1985), Physics Education.