B.5.4 Power and Resistor Configurations

Electrical Power and Resistors in Circuits

Electrical Power in Resistors

Definition

Power

Power is the rate at which energy is transferred or converted.

In electrical circuits, power is dissipated aspower is dissipated as thermal energy in resistors.

The formula for power in a resistor is:

$P = I V$

Where:

Pis the power (in watts, W)
Iis the current (in amperes, A)
Vis the potential difference (in volts, V)

Using Ohm’s Law ( $V = I R$ ), we can express power in two other forms:

In terms of current and resistance:

$P = I^{2} R$

In terms of voltage and resistance:

$P = \frac{V^{2}}{R}$

Tip

Choose the power formula that matches the information you have.

If you know the current and resistance, use $P = I^{2} R$ .
If you know the voltage and resistance, use $P = \frac{V^{2}}{R}$ .

Example question

Power

A resistor has a resistance of 5 Ω and a current of 2 A flowing through it. Calculate the power dissipated.

Solution

Using $P = I^{2} R$ :

$P = (2 A)^{2} \times 5 Ω = 20 W$

Resistors in Series Circuits

Definition

Series circuit

In a series circuit, resistors are connected end-to-end, forming a single path for current.

Key Characteristics of Series Circuits

Current: The current is the same through all resistors.

$I = I_{1} = I_{2} = I_{3} = \dots$

Voltage: The total voltage across the circuit is the sum of the voltages across each resistor.

$V = V_{1} + V_{2} + V_{3} + \dots$

Resistance: The total resistance is the sum of the individual resistances.

$R_{s} = R_{1} + R_{2} + R_{3} + \dots$

Drawing of a series circuit with 3 resistors.

Tip

In series circuits, adding more resistors increases the total resistance, which decreases the current for a given voltage.

Example question

Series circuit

Three resistors with resistances 2 Ω, 4 Ω, and 6 Ω are connected in series. Calculate the total resistance and the current if the total voltage is 12 V.

Solution

Total Resistance: $R_{s} = 2 Ω + 4 Ω + 6 Ω = 12 Ω$
Current (using Ohm’s Law, $I = \frac{V}{R}$ ): $I = \frac{12 V}{12 Ω} = 1 A$

Resistors in Parallel Circuits

Definition

Parallel circuit

In a parallel circuit, resistors are connected across the same two points, providing multiple paths for current.

Key Characteristics of Parallel Circuits

Voltage: The voltage across each resistor is the same.

$V = V_{1} = V_{2} = V_{3} = \dots$

Current: The total current is the sum of the currents through each resistor.

$I = I_{1} + I_{2} + I_{3} + \dots$

Resistance: The reciprocal of the total resistance is the sum of the reciprocals of the individual resistances.

$\frac{1}{R_{p}} = \frac{1}{R_{1}} + \frac{1}{R_{2}} + \frac{1}{R_{3}} + \dots$

Drawing of a parallel circuit with 3 resistors.

Tip

In parallel circuits, adding more resistors decreases the total resistance, which increases the current for a given voltage.

Example question

Parallel circuit

Three resistors with resistances 2 Ω, 4 Ω, and 6 Ω are connected in parallel. Calculate the total resistance and the current if the total voltage is 12 V.

Solution

Total Resistance: $\frac{1}{R_{p}} = \frac{1}{2} + \frac{1}{4} + \frac{1}{6} = \frac{6}{12} + \frac{3}{12} + \frac{2}{12} = \frac{11}{12}$ $R_{p} = \frac{12}{11} Ω \approx 1.09 Ω$
Total Current (using Ohm’s Law, $I = \frac{V}{R}$ ): $I = \frac{12 V}{1.09 Ω} \approx 11.01 A$

Combining Series and Parallel Circuits

Many circuits contain a combination of series and parallel resistors.combination of series and parallel resistors.
To analyze these circuits:
1. Simplifythe circuit by reducing series and parallel groups step-by-step.
2. Calculatethe total resistance.
3. Use Ohm’s Law to find thetotal current.
4. Work backwardto find the current, voltage, or power for individual resistors.

Common Mistake

A common mistake is to assume that current is the same in parallel resistors.

Remember, current splits in parallel paths, but voltage remains constant across each path.

Example question

Combined circuits

Consider a circuit with three resistors: 2 Ω and 4 Ω in series, and a 6 Ω resistor in parallel with the series combination. The total voltage is 12 V. Calculate the current through each series resistor.

Solution

Simplify the Series Resistors: $R_{s} = 2 Ω + 4 Ω = 6 Ω$
Combine with the Parallel Resistor: $\frac{1}{R_{total}} = \frac{1}{6} + \frac{1}{6} = \frac{2}{6} = \frac{1}{3}$ $R_{total} = 3 Ω$
Calculate the Total Current: $I = \frac{12 V}{3 Ω} = 4 A$
Find the Voltage Across the Series Resistors: $V_{s} = I R_{s} = 4 A \times 6 Ω = 24 V$
Find the Current Through Each Series Resistor (same current in series):
- 2 Ω resistor: $I = 4 A$
- 4 Ω resistor: $I = 4 A$

Why Does This Matter?

Theory of Knowledge

How does the concept of conservation of energy apply to the distribution of power in series and parallel circuits?
Can you think of real-world systems that mimic these principles?

Understanding how resistors combine in circuits is essential for designing and analyzing electrical systems, from simple household wiring to complex electronic devices.

You've read 2/2 free chapters this week.

Upgrade to PLUS or PRO to unlock all notes, for every subject.