Electric circuits: voltage, current, and resistance
A cell lifts charge to a higher potential, and the charge flows back down through the circuit. A voltage is always measured between two points: V = V_red − V_black.
Cells in a row add their voltages, a flipped cell takes its voltage away, and cells side by side give the voltage of one. A battery gives every coulomb it moves V joules: W = Q·V. Stack, flip, line up, power a lamp, and count cells.
A voltmeter reads the potential at its red probe minus the potential at its black probe: V = V_red − V_black. Swap the probes, try an open switch and a plain wire, find the voltage at a junction, and set two cells against each other.
A current is the charge that passes a cut across the wire each second: I = Q ÷ t, and one ampere is one coulomb a second, about 6 × 10¹⁸ electrons. Count the charge at the gate, turn minutes and hours into seconds, and time a slow current.
Electrons jitter at about a million metres a second but drift along a wire like snails: I = n·e·A·v_d gives under a tenth of a millimetre a second for 1 A in copper. Yet a lamp lights at once, because the signal runs along the wire at 2 × 10⁸ m/s.
A wire's resistance grows with its length and falls with its cross-section: R = ρ·L ÷ A. Twice the length doubles R, twice the diameter gives a quarter, and stretching a wire to twice its length makes R four times larger. Unroll, stretch, and cut a coil to size.
A metal resists more as it heats: R = R₀·(1 + α·(T − T₀)). A lamp filament at 2500 °C has twelve times its cold resistance, so a lamp takes a surge of current when switched on, and a platinum wire makes a thermometer. Heat, glow, and read the temperature.
For a fixed resistor, the current grows in proportion to the voltage: V = I·R, a straight line through the origin on an I–V graph. Find a current, a resistance, or a voltage from the other two, read a line, and pick the resistor that keeps an LED at 20 mA.
I = V ÷ R sets every current here, and a fuse melts the moment the current reaches its rating. Find a kettle's current, the smallest resistance a fuse survives, why a fuse does not protect a person, the most voltage a resistor can take, and the huge current of a short circuit.
In a series circuit there is only one path, so the same current flows through every bulb and the resistances add: R = R₁ + R₂ + …. Find the current, the total resistance, one bulb's share of the voltage, why a fourth bulb dims them all, and how many fairy lights the mains needs.
Two resistors in series split the supply in proportion to their resistances: V_out = V·R₂ ÷ (R₁ + R₂). Find an output voltage, design a divider for 5 V, read a light sensor and a thermistor, and find the voltage partway down a chain of three.
Parallel branches each get the full voltage: the currents add and the combined resistance falls, 1/R = 1/R₁ + 1/R₂ + …. Find a combined resistance and a main current, see why the other lamps stay lit when one is unplugged, choose a resistor, and count the lamps a breaker allows.
Current is never used up at a junction: what flows in flows out, I = I₁ + I₂ + …. Branches in parallel share one voltage, so the smaller resistance takes the larger share. Find a missing current, divide a current, add the branches, see why a new branch leaves the old one alone, and design a shunt.
Power is the rate a bulb turns electrical energy into light and heat: P = V·I = I²·R = V² ÷ R. Its rating only fixes its resistance. Find a power, a bulb's resistance, what two bulbs in series give, why the 60 W bulb wins when paired with a 100 W one, and what half the voltage does.
The same power can be sent as a large current at low voltage or a small current at high voltage, and the line's heating grows with the current squared: P_loss = I²·R. Find the loss at two voltages, design a line that loses 1%, heat a line, and size a heater at the house.
The energy a kettle supplies is its power times the time, E = P·t, and the water needs m·c·ΔT to warm up. Race a kettle against your predicted time, find the energy it uses, see what heat losses do, choose a power, and see why a kettle abroad takes four times as long.
The kilowatt-hour on an energy bill is not a power but an energy: 1 kW for 1 h, or 3600 kJ. Find a lamp's energy, what standby costs in a month, a power bank in kilojoules, what an LED saves over 1000 hours, and how much energy a phone battery holds.