Power and energy in a circuit

A circuit exists to move energy from a source to somewhere it is wanted, and the rate of that transfer is power. Every component either delivers power, as a battery does, or consumes it, as a resistor, lamp or motor does, and the arithmetic of power is short and governs an enormous amount of practical design, from why a phone charger is warm to why a power line runs at hundreds of thousands of volts.

Power is voltage times current

Recall two definitions. Voltage is energy per unit charge, and current is charge per unit time. Multiply them and the charge cancels, leaving energy per unit time, which is power:

with power in watts, symbol W, where one watt is one joule per second. A component with 12 V across it carrying 2 A is handling $12\times 2=24$ W, turning twenty-four joules into some other form every second. The formula is completely general: it holds for a resistor, a motor, a battery, anything, because it is just the two definitions multiplied. Where the energy goes depends on the component. In a resistor it all becomes heat. In a motor most becomes motion. In a battery being charged it becomes stored chemistry. The watt does not care; it only counts the rate.

The three faces of resistor power

For a resistor, Ohm's law lets power be written three equivalent ways, and choosing the right one is half the skill. Starting from $P=VI$ and substituting $V=IR$ or $I=V/R$ gives

All three give the same number for a given resistor; they differ in which two quantities you happen to know. The middle form, $P=I^{2}R$, is the one to burn into memory, because it carries the subject's recurring warning. Power lost as heat in a resistance grows with the square of the current. Double the current through a wire and you quadruple the heat it makes. This is why current, not voltage, is what melts wires and burns out the ESCs of the drone course, and why the fat leads on a high-current build were fat. It is also the argument for high-voltage power transmission: to deliver a fixed power a grid can use high voltage and low current or low voltage and high current, and since the loss in the lines is $I^{2}R$, sending it at high voltage and low current wastes far less in heating the cables. The same power, chosen as low current, arrives nearly intact.

Example. A resistor carries 0.5 A and has 6 V across it. What power does it dissipate, and check it a second way given its resistance.

$P=VI=6\times 0.5=3$ W. Its resistance is $R=V/I=6/0.5=12$ ohms, so $P=I^{2}R=0.25\times 12=3$ W, agreeing. The three forms are a built-in check on your arithmetic.

Now you. A 100 ohm resistor has 10 V across it. Use the form that needs only those two numbers to find the power.

From power to energy

Power is a rate; energy is the total delivered, power multiplied by the time it flows:

in joules when power is in watts and time in seconds. A 3 W resistor left on for 10 s dissipates 30 J. Because the joule is a small unit for household amounts of energy, the electricity meter uses the kilowatt-hour, the energy of one kilowatt flowing for one hour, which is $1000\times 3600=3.6$ million joules. A 2 kW heater run for three hours uses 6 kWh, and that is the number a bill charges for. Energy, not power, is what is bought and paid for, and the distinction between the rate and the total is one that trips people constantly: a powerful device used briefly can cost less than a weak one left on all day.

Example. A 60 W lamp is left on for 5 hours. How much energy does it use, in kilowatt-hours?

$0.060$ kW times 5 hours is $0.30$ kWh. In joules that is $60\times (5\times 3600)=1.08$ million J, which is why the kilowatt-hour exists: 0.30 is a friendlier number than a million.

Now you. A 2000 W kettle runs for 4 minutes. How much energy does it use, in kilowatt-hours?

Efficiency, and where the rest goes

No real device turns all the power it draws into what you want. A motor draws electrical power and delivers mechanical power, but some is lost to heat in its resistance and friction, and the ratio of useful output to total input is its efficiency. A device that draws 100 W and delivers 85 W of useful output is 85 per cent efficient, and the missing 15 W has become heat, which is why hard-working electronics need cooling. Efficiency is where the earlier lessons connect: the $I^{2}R$ loss in every wire and winding is the leak, and a well-designed system is one that keeps current low where it can, spends voltage deliberately, and treats every watt of heat as a watt it failed to use. With power and energy in hand, the circuit can now be more than a single loop, and the next lesson gives the two laws that handle any arrangement of components, however tangled.