KAHIBARO
Table of Contents
Power in Electric Circuits
Electric power tells us how fast electrical energy is transferred or transformed. When a circuit element such as a resistor, lamp, or motor operates, energy is converted every second into heat, light, motion, or other forms. Power measures this rate.
If an amount of energy $E$ is transferred in a time interval $t$, the average power is
$$
P = \frac{E}{t}
$$
The SI unit of power is the watt, written as $\text{W}$, where
$$
1\ \text{W} = 1\ \text{J/s}
$$
This means a device with power $1\ \text{W}$ converts or transfers $1$ joule of energy every second.
Important idea:
Electric power is the rate of energy transfer.
$$
P = \frac{E}{t}
$$
Unit:
$$
1\ \text{W} = 1\ \text{J/s}
$$
Power from Current and Voltage
In electric circuits, power is closely related to current and voltage. Voltage is energy transferred per unit charge, and current is charge flowing per unit time. Combining these ideas gives the basic electrical power formula:
$$
P = VI
$$
where $P$ is power, $V$ is voltage, and $I$ is current.
This formula means that if a charge moves through a potential difference $V$, and current $I$ flows, then energy is transferred at a rate equal to the product of voltage and current.
For example, if a device operates at $12\ \text{V}$ and draws $2\ \text{A}$, then its power is
$$
P = VI = (12)(2) = 24\ \text{W}
$$
So the device uses electrical energy at a rate of $24$ joules per second.
Core electrical power formula:
$$
P = VI
$$
This formula applies to any circuit element when $V$ is the potential difference across it and $I$ is the current through it.
Power in a Resistor
If the circuit element obeys Ohm's law, then the power formula can be rewritten in useful forms. Since
$$
V = IR
$$
we can substitute into $P = VI$.
Using $V = IR$ gives
$$
P = I(IR) = I^2 R
$$
Using $I = \frac{V}{R}$ gives
$$
P = V\left(\frac{V}{R}\right) = \frac{V^2}{R}
$$
So for a resistor, three equivalent expressions for power are:
$$
P = VI
$$
$$
P = I^2 R
$$
$$
P = \frac{V^2}{R}
$$
Each form is useful depending on which quantities are known.
For a resistor:
$$
P = VI = I^2R = \frac{V^2}{R}
$$
Use only the form that matches the known quantities.
Physical Meaning of Power Dissipation
In many circuits, especially those containing resistors, electrical energy is transformed into thermal energy. This process is often called power dissipation. A hot toaster wire, a glowing bulb filament, and a warm phone charger are all examples of electrical power being converted mainly into heat.
A larger current through a resistor usually means greater heating, because
$$
P = I^2R
$$
The square of the current is important. If the current doubles, the power dissipated becomes four times as large, if the resistance stays the same.
Energy Used by Electrical Devices
Since power is energy per time, the total energy used by an electrical device is
$$
E = Pt
$$
If power is constant, this formula is very simple to use. For example, a $100\ \text{W}$ lamp running for $10\ \text{s}$ uses
$$
E = Pt = (100)(10) = 1000\ \text{J}
$$
Electric companies often measure energy in kilowatt-hours instead of joules. A kilowatt-hour is the energy used by a $1000\ \text{W}$ device running for one hour.
$$
1\ \text{kWh} = 1000\ \text{W} \times 3600\ \text{s} = 3.6 \times 10^6\ \text{J}
$$
This is an energy unit, not a power unit.
Do not confuse power and energy.
Power measures rate of energy transfer:
$$
P = \frac{E}{t}
$$
Energy used over time:
$$
E = Pt
$$
Also,
$$
1\ \text{kWh} = 3.6 \times 10^6\ \text{J}
$$
Power and Sign Conventions
Power can be understood in two common ways. A device may absorb electrical energy, like a resistor or motor, or it may supply electrical energy, like a battery or generator.
If current enters the higher-potential side of an element, the element usually absorbs power. If the element pushes charge and supplies energy to the rest of the circuit, it delivers power.
For beginners, it is often enough to remember that resistors consume power, while sources can provide power.
Typical Power Values
Different devices operate at very different power levels.
| Device | Typical Power |
|---|---|
| Small LED | $0.1\ \text{W}$ to $1\ \text{W}$ |
| Phone charger | $5\ \text{W}$ to $20\ \text{W}$ |
| Light bulb | $5\ \text{W}$ to $100\ \text{W}$ |
| Laptop | $30\ \text{W}$ to $100\ \text{W}$ |
| Electric kettle | $1000\ \text{W}$ to $3000\ \text{W}$ |
These values help build intuition. A kettle heats water quickly because its power is much larger than that of a small lamp.
Simple Example
Suppose a resistor of resistance $R = 6\ \Omega$ carries a current of $I = 3\ \text{A}$.
Using
$$
P = I^2R
$$
we get
$$
P = (3)^2(6) = 9 \times 6 = 54\ \text{W}
$$
So the resistor converts electrical energy into heat at a rate of $54\ \text{J/s}$.
We can also find the voltage across the resistor:
$$
V = IR = (3)(6) = 18\ \text{V}
$$
Then
$$
P = VI = (18)(3) = 54\ \text{W}
$$
The result is the same, as expected.
Visualizing Power in a Circuit Element
In this diagram, current flows through the resistor and there is a voltage drop across it. The resistor absorbs electrical energy and converts it mainly into heat.
Summary Formulas
| Situation | Formula |
|---|---|
| General definition of power | $P = \frac{E}{t}$ |
| Electrical power | $P = VI$ |
| Power in a resistor using current | $P = I^2R$ |
| Power in a resistor using voltage | $P = \frac{V^2}{R}$ |
| Energy used at constant power | $E = Pt$ |
Essential formulas for electric power:
$$
P = \frac{E}{t}, \qquad P = VI
$$
For resistors:
$$
P = I^2R, \qquad P = \frac{V^2}{R}
$$
For constant power over time:
$$
E = Pt
$$
Final Insight
Electric power connects current, voltage, and energy. It tells us how quickly a circuit element uses or supplies energy. In practical circuits, power is one of the most important quantities because it determines heating, brightness, battery usage, and the operating cost of electrical devices.