KAHIBARO
Table of Contents
Energy in Electric Circuits
Electrical energy is the energy transferred when electric charges move through a circuit under the influence of an electric potential difference. In simple terms, a battery or power supply gives charges energy, and circuit elements can use that energy in different ways.
When a charge $q$ moves through a potential difference $V$, the electrical energy transferred is
$$
E = qV
$$
This relation is one of the most important ideas in basic circuit physics. It tells us that the transferred energy depends on how much charge moves and how much potential difference pushes it.
Important formula:
$$
E = qV
$$
where $E$ is electrical energy, $q$ is charge, and $V$ is potential difference.
Meaning of the Formula
The unit of charge is the coulomb, $\mathrm{C}$, and the unit of potential difference is the volt, $\mathrm{V}$. Since
$$
1 \ \mathrm{V} = 1 \ \mathrm{J/C}
$$
it follows that
$$
E = qV \Rightarrow \mathrm{C} \cdot \mathrm{J/C} = \mathrm{J}
$$
So electrical energy is measured in joules, $\mathrm{J}$.
If a charge of $2 \ \mathrm{C}$ moves through a potential difference of $6 \ \mathrm{V}$, then the energy transferred is
$$
E = qV = (2)(6) = 12 \ \mathrm{J}
$$
This means $12$ joules of energy are transferred.
Energy Supplied and Energy Used
In a circuit, energy is usually supplied by a source such as a battery. That energy is then transferred to other components. A lamp may convert electrical energy into light and heat. A motor may convert it into motion. A resistor usually converts it mainly into thermal energy.
Electrical energy is not lost, it is transformed from one form into another. This is an application of conservation of energy.
Key idea:
A source supplies electrical energy, and circuit components transform that energy into other forms such as heat, light, or mechanical energy.
Energy and Current
Current tells us how much charge passes per second. Since
$$
I = \frac{q}{t}
$$
we can write the charge as
$$
q = It
$$
Substituting into the energy formula gives
$$
E = qV = (It)V
$$
so
$$
E = VIt
$$
This is very useful in circuits, because voltage, current, and time are often easier to measure than total charge.
Useful circuit energy formula:
$$
E = VIt
$$
This gives the electrical energy transferred in time $t$.
Electrical Energy in Resistors
When current flows through a resistor, electrical energy is usually converted into thermal energy. This is often called heating. Devices such as electric heaters, toasters, and filament bulbs rely on this effect.
Using Ohm's law, energy expressions can also be rewritten. Since
$$
V = IR
$$
we can combine this with
$$
E = VIt
$$
to get other equivalent forms:
$$
E = I^2Rt
$$
and, using $I = \frac{V}{R}$,
$$
E = \frac{V^2}{R} t
$$
These forms are especially useful when resistance is known.
Equivalent formulas for electrical energy in a resistor:
$$
E = VIt
$$
$$
E = I^2Rt
$$
$$
E = \frac{V^2}{R} t
$$
Comparing the Forms
Different forms of the energy equation are useful in different situations.
| Known quantities | Best formula to use |
|---|---|
| Charge and voltage | $E = qV$ |
| Voltage, current, and time | $E = VIt$ |
| Current, resistance, and time | $E = I^2Rt$ |
| Voltage, resistance, and time | $E = \frac{V^2}{R}t$ |
All these formulas describe the same transferred electrical energy, but each is convenient for a different problem.
Everyday Units of Electrical Energy
In many practical situations, especially for homes, electrical energy is not given in joules but in kilowatt-hours, abbreviated as $\mathrm{kWh}$. This is a unit of energy, not power.
One kilowatt-hour means using a power of $1 \ \mathrm{kW}$ for $1$ hour. Since
$$
1 \ \mathrm{kW} = 1000 \ \mathrm{W}
$$
and
$$
1 \ \mathrm{hour} = 3600 \ \mathrm{s}
$$
then
$$
1 \ \mathrm{kWh} = 1000 \times 3600 = 3.6 \times 10^6 \ \mathrm{J}
$$
So
$$
1 \ \mathrm{kWh} = 3.6 \times 10^6 \ \mathrm{J}
$$
This unit is commonly used on electricity bills.
Important conversion:
$$
1 \ \mathrm{kWh} = 3.6 \times 10^6 \ \mathrm{J}
$$
A Simple Example
Suppose a device operates at a potential difference of $12 \ \mathrm{V}$ and a current of $2 \ \mathrm{A}$ for $5 \ \mathrm{s}$. The electrical energy transferred is
$$
E = VIt = (12)(2)(5) = 120 \ \mathrm{J}
$$
So the device transfers $120$ joules of electrical energy in that time.
Visualizing Energy Transfer in a Circuit
A simple way to think about a circuit is that the source gives energy to the charges, and the charges carry that energy to the components where it is transformed.
Final Idea
Electrical energy describes how much energy is transferred in an electric circuit. The basic relation is $E = qV$, and in working circuits it is often written as $E = VIt$. In resistive devices, this energy is commonly changed into heat, and in practical electricity use it is often measured in kilowatt-hours.