20.4 Electric Power and Energy

Electric Power and Energy

Electric power and energy describe how electrical circuits transfer and consume energy. Power tells you how fast energy is being used, while energy tracks the total amount consumed over time. Together, they let you calculate everything from how much heat a resistor gives off to how much your electricity bill will be.

Electric Power and Energy

Power Dissipation in Electric Circuits

Power ($P$) is the rate at which energy is transferred or converted. It's measured in watts ($W$), where $1 \text{ W} = 1 \text{ J/s}$.

When current flows through a resistor, electrical energy converts into heat (or light, motion, etc.). You can calculate the power dissipated using three equivalent formulas, each useful depending on what quantities you know:

• $P = IV$ — Use this when you know the current through and voltage across the component. For example, a light bulb drawing 0.5 A at 120 V dissipates 60 W.
• $P = I^2R$ — Use this when you know current and resistance but not voltage. Notice that power depends on the square of current, so doubling the current quadruples the power.
• $P = \frac{V^2}{R}$ — Use this when you know voltage and resistance but not current. Again, power depends on voltage squared.

These three formulas aren't independent. You can derive any one from the other two by substituting Ohm's Law ($V = IR$). They all describe the same physics.

For a power source like a battery, $P = IV$ gives the total power supplied to the circuit. Conservation of energy requires that the total power supplied by all sources equals the total power dissipated by all resistors. This is sometimes called power balance.

Electricity Costs of Household Devices

Utility companies don't charge you for power directly. They charge for energy, which is power multiplied by time. The standard billing unit is the kilowatt-hour (kWh):

$1 \text{ kWh} = 1000 \text{ W} \times 3600 \text{ s} = 3.6 \times 10^6 \text{ J}$

To estimate the cost of running a device:

1. Find the device's power rating in watts (check the label). Example: a 60 W light bulb.
2. Estimate how many hours per day you use it. Example: 4 hours/day.
3. Convert power to kilowatts and multiply by time to get daily energy consumption: $0.060 \text{ kW} \times 4 \text{ h} = 0.24 \text{ kWh/day}$.
4. Multiply by your electricity rate to get the daily cost. At $0.12/kWh: $0.24 \times 0.12 = \$0.029 \text{ per day}$, or about $0.87 per month.

This same method scales to any appliance. A 1500 W space heater running 8 hours a day at $0.12/kWh costs $1.5 \times 8 \times 0.12 = \$1.44 \text{ per day}$, which adds up fast.

Relationships in DC Circuit Components

Ohm's Law ($V = IR$) connects voltage, current, and resistance. It tells you that:

• Increasing voltage across a resistor increases the current through it (if resistance stays constant).
• Increasing resistance decreases the current (if voltage stays constant).

The three power formulas reveal how power depends on circuit quantities:

• From $P = IV$: power is proportional to both current and voltage. Doubling either one (while holding the other fixed) doubles the power.
• From $P = I^2R$: power grows with the square of current. Doubling current through a fixed resistance quadruples the power dissipated. This is why wires carrying large currents need to be thick (low $R$) to avoid overheating.
• From $P = \frac{V^2}{R}$: power grows with the square of voltage and is inversely proportional to resistance. Doubling the voltage across a fixed resistance quadruples the power.

The voltage difference between two points in a circuit is what drives current through that part of the circuit. No voltage difference means no current flow.

Additional Electrical Concepts

• Electrical conductivity measures how easily a material carries current. High conductivity means low resistance, and vice versa. Metals like copper have high conductivity, which is why they're used for wiring.
• Electromotive force (emf) is the energy per unit charge that a source (like a battery) supplies to a circuit. Despite the name, it's not actually a force; it's measured in volts.
• Alternating current (AC) periodically reverses direction, unlike the direct current (DC) discussed in most of this unit. Household outlets supply AC (typically 120 V at 60 Hz in the U.S.). The power formulas above still apply to AC circuits, but with modified definitions of voltage and current (called rms values), which you'll encounter in later units.