Power and energy

YOU WILL BE ABLE TO

Define power as p = dw/dt and derive p = v·i.
Compute power absorbed or delivered by an element.
Compute energy from power over time; convert to kilowatt-hours.
State and use conservation of power in a circuit.

Voltage is energy per charge; current is charge per time. Multiply them and the charge cancels, leaving energy per time — power. This little chain rule is the most-used equation in electronics.

p = \frac{dw}{dt} = \frac{dw}{dq}\,\frac{dq}{dt} = vi

power, watts

One watt (W) is one joule per second. Power tells you how fast energy is being converted right now — chemical to electrical in a battery, electrical to light and heat in an LED, electrical to heat in a resistor. Every component on a real board has a power rating, and exceeding it is how you let the magic smoke out.

Energy is power accumulated over time. For constant power, w = p·t. Electricity meters measure energy in kilowatt-hours: 1 kWh = 1000 W × 3600 s = 3.6 MJ. When your hydro bill charges per kWh, it is charging per 3.6 million joules.

w = \int p \, dt \; ; \quad 1\ \text{kWh} = 3.6\ \text{MJ}

energy

Energy is conserved, so in any circuit, at every instant, the total power absorbed by all elements sums to zero: whatever the sources deliver, the rest of the circuit absorbs. This is a ruthless error-checker. When you finish analysing a circuit, add up every element's power; if the sum isn't zero, you made an arithmetic mistake somewhere, guaranteed.

\sum p = 0

conservation of power

WORKED EXAMPLE

LED power

An LED drops 2.0 V while carrying 10 mA.
p = v·i = 2.0 V × 0.010 A = 0.020 W = 20 mW.

The LED absorbs 20 mW.

WORKED EXAMPLE

The hydro bill

A 1500 W space heater runs 6 hours per day for 30 days.
Energy = 1.5 kW × 6 h × 30 = 270 kWh.
At $0.10/kWh (roughly Ontario's off-peak rate), cost = 270 × $0.10 = $27.00.

270 kWh, about $27

WORKED EXAMPLE

Power balance check

A battery delivers 5 W. Resistor 1 absorbs 3 W, resistor 2 absorbs 2 W.
Write delivery as negative absorption: battery absorbs −5 W.
Sum: −5 + 3 + 2 = 0. ✓ The books balance.

Σp = 0 — the analysis is consistent.

PRACTICE

A USB port supplies 5 V at up to 500 mA. What is its maximum power?

A resistor absorbs 0.25 W while carrying 50 mA. What voltage is across it?

A Raspberry Pi Pico draws about 25 mA at 3.3 V when idle. How much energy does it use in 24 hours, in joules and in watt-hours?

In a circuit, element A absorbs 12 W, element B absorbs −20 W, and element C absorbs 5 W. What does element D absorb, and which elements are sources of energy?