Circuits: voltage, current, resistance

A phone charger sips a couple of watts; a kettle pulls a thousand. A thin extension cord warns you not to run a heater through it, while the thick cable to your oven stays cool. What is actually flowing in those wires, and why does thickness decide whether a cord stays cool or starts a fire?

In the last lesson, charge that could move was the whole story of a spark. A circuit is what you get when you give that moving charge a loop to go around and a reason to keep going. The charges are already in the wire, like water already filling a pipe. A battery does not supply them - it pushes them.

It is not. Exactly as much current flows out of the bulb as flows into it. Charge is conserved - it cannot pile up or vanish in the wire. What the bulb uses up is not current but energy: each charge arrives with energy and leaves with less, having dumped the difference as light and heat. Keep those two ideas separate and circuits stop being mysterious. Let us name them.

Three quantities, built from a water pipe

The water analogy is imperfect but it gets the roles right, so use it with care. Imagine water in a closed loop of pipe driven by a pump.

current is how much charge flows past a point each second, like the flow rate of the water. It is measured in amperes. voltage is the energy handed to each unit of charge, like the pressure the pump provides. It is a difference between two points, which is the key: voltage does not flow, it is across things, the way pressure is a difference between two ends of a pipe. resistance is how much the pipe fights the flow: a narrow, rough pipe has high resistance and throttles the current.

Play with it before the formula

Below is a battery driving two identical bulbs. Flip between series and parallel, drag the battery voltage and bulb resistance, and watch the current and each bulb's brightness respond. Then thin out the wire gauge and watch the leads glow red as they waste power to heat.

Ohm's law: the three tie together

For a huge class of materials, the current through a component is simply proportional to the voltage across it. Double the push, double the flow. The constant of proportionality is the resistance. This is Ohm's law (a stated empirical relationship, true for ohmic materials like metal wire^[2]):

That one line explains the predict-probe about series and parallel. Two identical bulbs in series share one loop, so the same current threads both, and the battery's voltage splits between them - half each, so each bulb is dim. The same two bulbs in parallel each connect straight across the battery, so each gets the full voltage and burns bright, but now the battery must supply both currents at once.

Why thick wires run cool: power is I squared R

A charge dropping through a voltage $V$ gives up energy, and energy per second is power. Multiplying voltage (energy per charge) by current (charge per second) gives power (energy per second):

The heat wasted in a wire is $I^{2}R$: it depends on the wire's own small resistance and, crucially, on the square of the current. A thick wire has lower resistance, so at the same current it wastes far less power as heat and stays cool. A thin cord has higher resistance; push a heater's large current through it and $I^{2}R$ can get hot enough to melt the insulation^[1]. That is not a safety footnote bolted on afterward - it is the same equation the widget draws as a red glow.

def solve(volts, r_bulb, r_wire, mode):
    if mode == "series":
        r_total = r_wire + 2 * r_bulb      # bulbs add in series
    else:
        r_total = r_wire + r_bulb / 2      # bulbs halve in parallel
    current = volts / r_total              # Ohm's law: I = V / R
    wire_heat = current**2 * r_wire        # P = I^2 R wasted in the leads
    return current, wire_heat

# same battery, same bulbs, thinner wire -> more heat wasted
print(solve(6, 8, 0.5, "parallel"))  # thick wire: tiny wire_heat
print(solve(6, 8, 5.0, "parallel"))  # thin wire: much larger wire_heat

Check yourself