4cPower and Energy in Circuits
Master electric power (P = IV = I²R = V²/R), energy in joules and kilowatt-hours, household electricity, and how circuit breakers keep your home safe.
Every electrical device you use — from your phone charger to a power plant — is governed by the same power equations. Understanding P = IV lets you calculate electricity bills, design safe circuits, and understand why high-voltage power transmission saves enormous amounts of energy. These are the formulas behind every watt on your electric bill.
Why does a 100 W light bulb get hotter than a 60 W bulb — and how does your electric meter know exactly how much to charge you each month?
Lesson Overview
Electric power is the rate at which electrical energy is converted to other forms (heat, light, mechanical energy). The fundamental formula is P = IV (watts). Combined with Ohm's law (V = IR), this gives two equivalent forms: P = I²R and P = V²/R. Electrical energy consumed is E = Pt (joules). For billing purposes, energy is measured in kilowatt-hours (kWh): 1 kWh = 3.6×10⁶ J. Household circuits in North America operate at 120 V (standard outlets) or 240 V (large appliances). Circuit breakers and fuses protect wiring by interrupting current when it exceeds a safe level, preventing overheating and fires. Electricity cost = energy (kWh) × rate ($/kWh).
Key Equations
Worked Examples
A toaster operates at 120 V and draws a current of 8.0 A. (a) What is its power rating? (b) How much energy does it use in 3.0 minutes?
A 60 Ω resistor is connected to a 12 V battery. Find the power dissipated using all three power formulas.
A household uses the following appliances for 8 hours per day: a 1500 W hair dryer (0.25 h/day), a 200 W TV (4 h/day), and ten 60 W light bulbs (8 h/day). If electricity costs $0.12/kWh, what is the daily cost?
A circuit breaker is rated at 15 A on a 120 V circuit. What is the maximum power that can be drawn before the breaker trips? What is the minimum resistance of a device that can be safely connected?
A 240 V electric dryer is rated at 5400 W. (a) What current does it draw? (b) What is its resistance? (c) How much does it cost to run for 45 minutes at $0.15/kWh?
Guided Problems
A 100 W light bulb and a 60 W light bulb are both designed for 120 V. Which has the higher resistance? Calculate both resistances.
Hint: Use P = V²/R → R = V²/P. A higher power rating at the same voltage means lower resistance (more current flows).
An electric kettle is rated 1200 W at 120 V. How long does it take to heat water, consuming 0.050 kWh of energy? Express your answer in minutes.
Hint: E = Pt → t = E/P. Convert 0.050 kWh to joules (or convert P to kW) and solve for t.
Three resistors (10 Ω, 20 Ω, 30 Ω) are connected in parallel across a 60 V source. Find the power dissipated in each resistor and the total power delivered by the source.
Hint: In parallel, each resistor has the full 60 V across it. Use P = V²/R for each. Total power = sum of individual powers (or P_total = V × I_total).
A fuse rated at 5 A is used in a 12 V circuit. A device with resistance 1.5 Ω is connected. Will the fuse blow? Explain.
Hint: Find the current: I = V/R = 12/1.5. Compare to the fuse rating of 5 A.
A family's monthly electricity bill shows they used 720 kWh. The rate is $0.13/kWh. What is their bill? What is their average power consumption in watts?
Hint: Cost = energy × rate. Average power: P = E/t where t = 30 days × 24 h/day = 720 h.
Key Vocabulary
Electric Power
The rate at which electrical energy is converted to other forms of energy. P = IV = I²R = V²/R. Measured in watts (W); 1 W = 1 J/s.
Example: A 60 W light bulb converts 60 joules of electrical energy to light and heat every second.
Kilowatt-hour (kWh)
A unit of electrical energy equal to the energy consumed by a 1 kW device running for 1 hour. 1 kWh = 3.6×10⁶ J. Used by electric utilities for billing.
Example: Running a 1000 W (1 kW) microwave for 1 hour uses 1 kWh of energy.
Circuit Breaker
A resettable safety device that automatically interrupts current flow when it exceeds a rated value, protecting wiring from overheating. Modern replacement for fuses.
Example: A 20 A circuit breaker trips if a short circuit causes 30 A to flow, preventing a fire.
Fuse
A one-time safety device containing a thin wire that melts and breaks the circuit when current exceeds a rated value. Must be replaced after it blows.
Example: A 3 A fuse in a lamp protects it from current surges — if 5 A flows, the fuse wire melts and breaks the circuit.
Workbook Check — Interactive Quiz
Interactive Practice — 5 Questions
A device operates at 120 V and draws 2.0 A. Its power consumption is:
Which formula correctly gives power in terms of current and resistance only?
How many joules of energy are in 1 kilowatt-hour?
A 40 Ω resistor dissipates 90 W. What voltage is across it?
A circuit breaker rated at 20 A protects a 120 V circuit. What is the maximum safe power load?
Independent Practice
A 1500 W space heater runs for 6.0 hours. (a) How much energy does it use in joules? (b) How many kWh is that? (c) At $0.11/kWh, what does it cost?
A 9.0 V battery powers a small motor with resistance 18 Ω. Find (a) the current, (b) the power dissipated, and (c) the energy used in 5.0 minutes.
Two resistors, 30 Ω and 60 Ω, are connected in series across a 90 V source. Find the power dissipated in each resistor and verify that total power equals P = V²/R_eq.
Explain why a circuit breaker is rated in amperes, not watts. Why does the same 15 A breaker protect both a 120 V and a 240 V circuit differently?
★ A household has a 200 A service at 240 V (the total power available). List five major appliances with their typical wattages and show that their simultaneous use would not exceed the service limit. Then calculate the monthly cost if each runs an average of 2 hours per day at $0.14/kWh.
ChallengeCommon Mistakes
Using P = IV but forgetting that I and V must be for the same component
In a series circuit, I is the same everywhere but V differs across each resistor. In parallel, V is the same but I differs. Always match I and V to the same element.
Confusing watts (power) with watt-hours or kilowatt-hours (energy)
Power (W) is the rate of energy use. Energy (J or kWh) = power × time. A 100 W bulb uses 100 W of power and 0.1 kWh of energy per hour.
Thinking a higher-wattage appliance always has higher resistance
At the same voltage, higher power means lower resistance: R = V²/P. A 100 W bulb has lower resistance than a 60 W bulb (both at 120 V).
Forgetting to convert minutes or hours to seconds when calculating energy in joules
E = Pt requires t in seconds when P is in watts and E is in joules. Convert: 1 min = 60 s, 1 h = 3600 s.
Math Tips
Three power formulas: P = IV, P = I²R, P = V²/R. Choose based on what you know: if you have I and R but not V, use P = I²R.
Energy in kWh: E(kWh) = P(kW) × t(h). Convert watts to kilowatts by dividing by 1000. This is the unit on your electric bill.
Cost calculation: Cost = kWh × $/kWh. Always check units — power in kW, time in hours, rate in $/kWh.
Circuit breaker check: I = P/V. If I > breaker rating, the breaker trips. Maximum safe power = breaker rating × voltage.