A current is nothing mystical — it's charge, moving, counted per second. Everything below is that one idea unfolding: what pushes it (voltage), what resists it (resistance), how resistors gang up in series or parallel, what a battery's own guts cost you, and the two accounting rules (Kirchhoff's laws) that let you solve any circuit at all. Seven acts, one thread.
Picture a cross-section of a wire like a tollbooth. Charge (in coulombs) drives through it; current is simply how much charge passes per second: I = Q / t. Watch the odometer below — it's genuinely integrating the current live, the same way a real ammeter's reading corresponds to charge actually crossing that line.
Something has to push that charge through. That push is voltage — and how hard the wire fights back is resistance.
Ohm's Law ties the three together: V = IR. Squeeze the resistor and the same push moves less current through the gap. Push harder and more gets through — the gap doesn't change, only the flow does.
But R isn't arbitrary — it comes from what the wire is made of, and its shape.
Every material has its own resistivity ρ — how stubborn it is, independent of shape. Stretch a wire longer and there's simply more material to fight through; make it fatter and current has more room to spread out. R = ρℓ / A.
Now take that one resistor idea and chain a few of them together, end to end.
One path, no branching — the same current has to squeeze through every resistor in turn, so their resistances simply stack: R' = R₁ + R₂ + R₃. The same current flows everywhere; each resistor just takes its own bite out of the total voltage.
Wire those same three resistors side by side instead of end to end, and the story flips completely.
Now there are multiple paths, all connected across the same two points — so every branch feels the exact same voltage, and each just draws its own current: 1/R' = 1/R₁ + 1/R₂ + 1/R₃. The lowest-resistance branch always carries the most current — same idea as water finding the widest pipe.
So far every battery has been a perfect, lossless pusher. Real ones aren't.
A real battery has its own internal resistance r. The current has to push through that too, so the voltage you actually get at the terminals is always a little less than the battery's full emf: V = V_B − Ir, and the circuit current is I = V_B / (R + r).
Simple loops solve with Ohm's Law alone. Tangled ones — with multiple batteries and junctions — need a stricter set of bookkeeping rules.
Charge can't pile up anywhere, and energy has to balance all the way around any loop. That's it — that's both laws. Together they can solve circuits Ohm's Law alone can't touch.
At any junction, whatever current flows in must flow back out — nothing accumulates. ΣI_in = ΣI_out. Set all four currents yourself — the equal sign turns green the instant they balance, and red the moment they don't.
Around any closed loop, the emfs and the IR drops must balance exactly: ΣV_B = ΣIR. This is the book's own two-battery example, solved live — drag any value and all three currents resolve instantly, including flipping direction (and turning red) if a battery ends up being charged rather than discharged.
Both laws, one wire, always balanced — however tangled the circuit gets.