19.4 Electric Power

Electric Power in Circuits

Electric power describes how quickly electrical energy gets converted into other forms like heat, light, or motion. It's measured in watts and connects voltage, current, and resistance through a few key formulas. Understanding power is essential for analyzing how much energy devices use and how circuits distribute that energy.

Whether components are wired in series or parallel, total power always equals the sum of individual component powers. Some elements (like resistors) consume power irreversibly, while reactive elements (like capacitors and inductors) store energy temporarily and can release it back into the circuit.

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Calculation of Electric Power

Power ($P$) is the rate at which electrical energy is converted into other forms. It's measured in watts ($W$), where $1 \, W = 1 \, J/s$.

There are three equivalent formulas for calculating power, and which one you reach for depends on what quantities you know:

• $P = VI$ — Use when you know voltage and current.
  • A 12 V battery supplying 2 A of current provides $12 \, V \times 2 \, A = 24 \, W$ of power.
• $P = \frac{V^2}{R}$ — Use when you know voltage and resistance (but not current).
  • A 120 V appliance with 60 Ω resistance consumes $\frac{120^2}{60} = 240 \, W$.
• $P = I^2 R$ — Use when you know current and resistance (but not voltage).
  • A 5 Ω resistor carrying 3 A dissipates $3^2 \times 5 = 45 \, W$.

All three formulas come from combining $P = VI$ with Ohm's law ($V = IR$). If you forget one, you can always derive it from the other two.

Power Dissipation in Resistor Configurations

Regardless of how resistors are connected, the total power dissipated in a circuit equals the sum of the power dissipated by each resistor:

$P_{total} = P_1 + P_2 + ... + P_n$

What changes between series and parallel is how voltage and current distribute across the resistors, which affects how you calculate each individual power.

Series circuits: The current is the same through every resistor, but voltage divides across them proportionally to resistance. Use the voltage drop across each resistor to find its power.

• Two resistors in series (10 Ω and 20 Ω) connected to a 30 V source:
  1. Find total resistance: $R_{total} = 10 + 20 = 30 \, \Omega$
  2. Find current: $I = \frac{30 \, V}{30 \, \Omega} = 1 \, A$
  3. Find voltage drops: $V_1 = 1 \times 10 = 10 \, V$, $V_2 = 1 \times 20 = 20 \, V$
  4. Find power for each: $P_1 = \frac{10^2}{10} = 10 \, W$, $P_2 = \frac{20^2}{20} = 20 \, W$
  5. Total power: $P_{total} = 10 + 20 = 30 \, W$

Notice that the larger resistor dissipates more power in a series circuit because it has a larger voltage drop.

Parallel circuits: The voltage is the same across every branch, but current divides among them inversely proportional to resistance. Use the current through each resistor to find its power.

• Two resistors in parallel (5 Ω and 10 Ω) connected to a 30 V source:
  1. Each resistor sees the full 30 V.
  2. Find branch currents: $I_1 = \frac{30}{5} = 6 \, A$, $I_2 = \frac{30}{10} = 3 \, A$
  3. Find power for each: $P_1 = 6^2 \times 5 = 180 \, W$, $P_2 = 3^2 \times 10 = 90 \, W$
  4. Total power: $P_{total} = 180 + 90 = 270 \, W$

In parallel, the smaller resistor dissipates more power because it draws more current.

Power Consumption vs. Energy Storage

Resistive elements consume power irreversibly. Electrical energy is converted into heat or light and cannot be recovered by the circuit.

• Resistors, light bulbs, and heating elements all fall into this category.
• A 60 W light bulb converts 60 J of electrical energy per second into light and heat.

Reactive elements store energy temporarily in fields and can release it back into the circuit later. They don't permanently consume energy in the ideal case.

• Capacitors store energy in electric fields: $E = \frac{1}{2}CV^2$
  • A 100 μF capacitor charged to 10 V stores $\frac{1}{2} \times 100 \times 10^{-6} \times 10^2 = 0.005 \, J$
• Inductors store energy in magnetic fields: $E = \frac{1}{2}LI^2$
  • A 50 mH inductor carrying 2 A stores $\frac{1}{2} \times 50 \times 10^{-3} \times 2^2 = 0.1 \, J$

The key distinction: resistors convert energy out of the circuit permanently, while capacitors and inductors act as temporary energy reservoirs. Conservation of energy still holds throughout the system.

Types of Electric Current and Power Factor

• Direct current (DC) flows in one direction. Batteries and most electronics use DC.
• Alternating current (AC) periodically reverses direction. Power grids and household outlets supply AC.

In AC circuits, voltage and current don't always peak at the same time (they can be out of phase), which means not all the apparent power actually does useful work. The power factor captures this:

$\text{Power Factor} = \frac{\text{Real Power}}{\text{Apparent Power}}$

A power factor of 1 means all the power is being used productively. Values less than 1 indicate that some power is cycling back and forth between the source and reactive elements without doing useful work, reducing overall system efficiency.