P = VI = I²R = V²/R · H = I²Rt · the kilowatt-hour · the fuse
Push current through anything with resistance and energy turns into heat. In a kettle that is the whole point; in a cable it is pure waste; in a fuse it is the safety feature. One formula covers all three, and the square in it is why doubling the current costs four times as much.
This page covers electric power in its three forms, Joule's heating law, the kilowatt-hour you are billed in, and why fuses, filaments and transmission lines all follow from the same sum. By the end you'll be able to:
Every board asks for a power calculation and a cost calculation, usually in the same question. It is also the physics behind every safety rule in a house:
Searched as: joule's law of heating, electric power formula, i squared rt, kilowatt hour, appliance rating, why fuses melt.
Predict the power, lock it, then change the supply, the element and the time. Real water, real specific heat, and it stops at boiling. 🟢 real Joule-law engine
Guess, then lock — the bench keeps its answer until you commit.
Set the supply, the element and the time — then watch where the energy goes.
P = VI, P = I²R and P = V²/R are the same fact three ways. The skill is picking the one that matches what you were given. Six situations — your answers build the table underneath.
You know the voltage and the current. Which form?
Read what is given, then pick the form that needs exactly those.
Choosing the right form first turns most of these questions into one line of arithmetic.
The supply is fixed at 240 V. Choose an element resistance that gives the target power — 3 attempts.
Pick a resistance, then switch on. P = V²/R decides it.
Each label breaks the physics somewhere. Tap the broken idea.
P = VI gives 240 × 2 = 480 W, not 2000 W. A 2000 W appliance on 240 V must draw about 8.3 A.
The fuse must be the weakest point. A 30 A fuse lets the thin flex carry ten times its safe current, so the cable becomes the fuse — by catching fire.
H = I²Rt has the current squared, so doubling it gives four times the heat, not twice. That square is the whole reason cables are rated by current.
Thinking beats memorising. Sort every claim about power and heat into three boxes.
One change at a time, to a resistor on a supply. Say what happens to the heating. The square is the only thing that makes this hard.
Decide what is being held fixed before you answer — that is what decides which formula applies.
Pick an appliance, then choose the smallest standard fuse that will carry it. Too big protects nothing; too small blows every time you switch on.
Standard fuses: 1 A, 3 A, 5 A, 13 A. Work out the current first.
Read the scene. The same formula explains all three.
Both run on the same 240 V. The kettle draws about 10 A against the lamp's 0.25 A, and the heat in the flex itself goes as the square of that.
A good connection has almost no resistance and stays cool. Warmth means I²R is happening where it should not, usually at a loose screw — and it gets worse as it goes.
The lead and the lamp are in series, so they share the supply. A long thin lead can take several volts — and those volts become heat in the lead rather than light in the lamp.
In plain terms: charge pushed through a resistance loses energy to the material, and that energy appears as heat. How fast it appears is the power.
Each coulomb of charge that crosses a potential difference V gives up V joules. Push I coulombs a second and the rate is P = VI watts. That form needs nothing else to be true — it works for a motor, a phone, an LED. The other two forms, I²R and V²/R, come from substituting Ohm's law, so they apply only where a resistance is what you have.
In a fixed cable the resistance is set, so heating follows I². Twice the current is four times the heat; ten times the current is a hundred times. This one fact decides cable thickness, fuse ratings, why power is transmitted at high voltage, and why a loose connection gets hotter the worse it becomes.
Your meter counts energy, not power: kilowatts multiplied by hours. A 2 kW heater for 30 minutes and a 1 kW heater for an hour cost exactly the same. One kilowatt-hour is 3.6 million joules, which is why the unit exists at all — nobody wants a bill written in joules.
A filament lamp's resistance rises steeply as it heats, so the switch-on surge is several times the running current — which is why bulbs nearly always fail at the moment you switch them on. An LED is not a resistor at all, so P = VI still holds while V²/R is meaningless for it. And a heater is 100% efficient at making heat, which is the one appliance where nothing is wasted, because waste is the product.
Exams reward the method. Work it one step at a time — read the thought, predict the line, then reveal it.
Three small labs, each one native to this law.
Six wrong ideas that survive because they sound sensible. Tap a card to bust the myth.
James Joule, running his family's brewery by day, passed currents through wires in water and measured the temperature rise. He found the heat went as the square of the current, and published it at twenty-two.
His claim that heat and work were the same currency was dismissed for years, partly because his temperature measurements were finer than his audience thought possible.
Edison's lighting circuits needed a deliberate weak point, and the fuse was patented as one. The same formula that melted the fuse made the filament glow.
Because loss goes as I²R, sending power at high voltage and low current wins over distance. That arithmetic settled the current wars, and it is why transformers are everywhere.
Energy is measured in joules, and your bill is written in kilowatt-hours because 3.6 million of them is a more useful number for a household than one.
About 2.4 kW through a low-resistance element: ten amps, a thick flex, and half a litre boiling in under a minute.
A filament makes light by being white hot, so most of its energy leaves as heat. An LED makes light directly — same room, a tenth of the watts.
Your meter counts kilowatt-hours, not watts. What it charges for is power multiplied by how long you left it running.
One sentence each: what makes the heat, and whether anybody wanted it.
Drag the slider. Same formula, six machines, from a tenth of a watt to fifty megawatts.
Seven orders of magnitude, one formula, and the same square in all of them.
Each one is a build and a measurement — the measuring is what turns a demo into a project.
Build: a low-voltage immersion element in a measured mass of water with a thermometer.
Measure: temperature every 30 s; the graph should be a straight line while nothing boils.
Build: the same element at several supply voltages, each run for the same time.
Measure: plot temperature rise against current, then against current squared. Only one is a straight line.
Build: read your electricity meter, run one known appliance for an hour, read it again.
Measure: compare the difference with the rating plate. They should agree within a few per cent.
Build: two bulbs of similar brightness, one LED and one filament, each in still air.
Measure: the temperature near each after ten minutes, and their rated powers. The difference is the wasted heat.
Build: a lamp at the end of several metres of thin wire, then the same lamp on a short thick lead.
Measure: the voltage at the lamp in each case; the missing volts are being spent heating the wire.
Build: a single strand of thin fuse wire in a low-voltage circuit with a variable supply.
Measure: the current at which it melts, for two thicknesses. Never attempt this on mains.
Straight answers, each one complete on its own:
Power is the rate: how many joules a device uses every second, measured in watts. Energy is the total: power multiplied by how long it ran, measured in joules or in kilowatt-hours. A rating plate states power; a bill charges for energy.
Use the one that matches what you were given. P = VI needs voltage and current and works for any device. P = I²R needs current and resistance, and P = V²/R needs voltage and resistance — both of those come from substituting Ohm's law, so they only apply to something that obeys it.
Because the current appears squared in H = I²Rt. Twice as much charge passes each second, and each unit of charge is pushed through with twice the potential difference across the same resistance, so both factors double and the product is four times as large.
One kilowatt-hour: a kilowatt of power running for one hour, which is 3.6 million joules. A 2 kW heater uses one unit in half an hour; a 10 W charger would take four days. Your meter counts these and your bill multiplies them by a rate.
A fuse protects the cable behind it by being the weakest point. Fit one rated far above the cable's safe current and a fault will heat the cable itself to the point of fire before the fuse ever melts. The fuse should be the smallest rating that carries the appliance comfortably.
It uses more energy every second, which is not the same thing. A filament bulb turns most of its watts into heat, so a 60 W filament and a 9 W LED can give similar light. Compare lumens for brightness and watts for what it costs to run.
Because the loss in the line is I²R, and for a given power, raising the voltage lowers the current. The same 100 kW sent at 25 kV instead of 250 V carries a hundredth of the current and wastes ten thousandth of the heat in the cables.
A cold filament has a much lower resistance than a hot one, so for the first fraction of a second it draws several times its running current. That surge, and the sudden thermal shock, is when an already thinned filament finally breaks.
Spot these before the exam does.
Seven question formats, the way LLOS.ai serves them — multiple choice, multiple-correct, fill-in-the-blank, match, sequence, read-think-connect, and write-your-own. Every question has layered hints: a quick nudge, the reasoning, then a deeper connection — so a wrong answer opens a door, never a dead end. 🟢 received from a board-tagged question bank · seed toward 2,000
Pick your board — the set re-tunes to its wording and emphasis. Competitive draws the JEE / NEET / Olympiad lane.
Four appliances, no numbers yet. Tap them in order, most expensive hour first. Then read the arithmetic — and see what fooled you.
Five LED lamps, all on, lighting a whole room brightly.
A large plasma set, on in the corner, nobody watching.
Plugged in all hour, charging a phone the whole time.
Heating a tank of water in the cupboard, quietly, with no light or sound at all.
Tap the four in the order you believe, most expensive first.
The same sum that boils your kettle is the one that melts a fuse and warms a cable nobody wanted warmed. Only the intention differs.