Ohm’s Law in Series and Parallel Circuits Revision Notes for Grade 11 NSC Matric Physical Sciences

Ohm's Law in Series and Parallel Circuits

Introduction to series and parallel circuits

When working with electrical circuits, resistors can be connected in two fundamental ways: in series or in parallel. Understanding how current and voltage behave in these configurations is essential for applying Ohm's Law effectively.

In a series circuit, there is only one path for current to flow through all components. In a parallel circuit, there are multiple paths for current to flow, with each component on a separate branch.

When circuits contain more than one resistor, we can calculate the total combined resistance of all resistors. This combined resistance is called the equivalent resistance.

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Understanding the difference between series and parallel connections is fundamental to circuit analysis. The key distinction is the number of current paths available.

Equivalent resistance in series circuits

Definition: Equivalent resistance in a series circuit is the sum of all individual resistances.

For resistors connected in series, the equivalent resistance is:

$R_s = R_1 + R_2 + R_3 + ... + R_n$

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Worked Example: Calculating Series Equivalent Resistance

In the circuit shown above with three resistors in series:

R₁ = 3 Ω
R₂ = 10 Ω
R₃ = 5 Ω

Solution: $R_s = R_1 + R_2 + R_3 = 3\,\Omega + 10\,\Omega + 5\,\Omega = 18\,\Omega$

This means the three resistors together act like a single 18 Ω resistor.

Equivalent resistance in parallel circuits

Definition: Equivalent resistance in a parallel circuit is calculated using the reciprocal formula.

For resistors connected in parallel, the equivalent resistance is:

$\frac{1}{R_p} = \frac{1}{R_1} + \frac{1}{R_2} + \frac{1}{R_3} + ... + \frac{1}{R_n}$

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Worked Example: Calculating Parallel Equivalent Resistance

Using the parallel circuit shown above with three resistors:

R₁ = 10 Ω
R₂ = 2 Ω
R₃ = 1 Ω

Solution: $\frac{1}{R_p} = \frac{1}{10} + \frac{1}{2} + \frac{1}{1} = 0.1 + 0.5 + 1 = 1.6$

Therefore: $R_p = \frac{1}{1.6} = 0.625\,\Omega$

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Notice that the equivalent resistance in parallel is always smaller than the smallest individual resistor.

Applying Ohm's Law in series circuits

Key principles for series circuits

Understanding these fundamental principles is crucial for series circuit analysis:

Current is the same throughout all components: $I = I_1 = I_2 = I_3$
Voltage adds up across all components: $V = V_1 + V_2 + V_3$

The current flows in one continuous path, so it must be identical at every point. However, each resistor drops a portion of the total voltage according to its resistance value.

Worked example: calculating current in a series circuit

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Worked Example: Current in a Series Circuit

Question: Calculate the current flowing through this series circuit with two resistors.

Given:

V = 12 V
R₁ = 2 Ω
R₂ = 4 Ω

Solution:

Step 1: Find the total equivalent resistance

$R_s = R_1 + R_2 = 2\,\Omega + 4\,\Omega = 6\,\Omega$

Step 2: Apply Ohm's Law to find current

Using $V = I \times R$ , we rearrange to get: $I = \frac{V}{R}$
$I = \frac{12\,V}{6\,\Omega} = 2\,A$

The current flowing through the circuit is 2 A.

Worked example: finding unknown resistance in series

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Worked Example: Finding Unknown Resistance in Series

Question: Find the unknown resistance R₂ in this series circuit.

Given:

V = 1.5 V
R₁ = 1 Ω
I = 0.25 A
R₂ = ?

Solution:

Step 1: Use Ohm's Law to find total resistance

$R = \frac{V}{I} = \frac{1.5\,V}{0.25\,A} = 6\,\Omega$

Step 2: Find the unknown resistance

Since $R_s = R_1 + R_2$
$R_2 = R_s - R_1 = 6\,\Omega - 1\,\Omega = 5\,\Omega$

Therefore, R₂ = 5 Ω.

Worked example: calculating voltage drops in series

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Worked Example: Voltage Drops in Series Circuit

Question: For a series circuit with three resistors, calculate the voltage drop across each resistor.

Given:

V = 18 V
I = 2 A
R₁ = 1 Ω
R₂ = 3 Ω
R₃ = ? (unknown)

Solution:

Step 1: Calculate voltage drop across R₁

$V_1 = I \times R_1 = 2\,A \times 1\,\Omega = 2\,V$

Step 2: Calculate voltage drop across R₂

$V_2 = I \times R_2 = 2\,A \times 3\,\Omega = 6\,V$

Step 3: Find voltage drop across R₃

Since $V = V_1 + V_2 + V_3$
$V_3 = V - V_1 - V_2 = 18\,V - 2\,V - 6\,V = 10\,V$

Step 4: Calculate R₃

$R_3 = \frac{V_3}{I} = \frac{10\,V}{2\,A} = 5\,\Omega$

Applying Ohm's Law in parallel circuits

Key principles for parallel circuits

These fundamental principles govern parallel circuit behavior:

Voltage is the same across all components: $V = V_1 = V_2 = V_3$
Current adds up through all branches: $I = I_1 + I_2 + I_3$

In parallel circuits, each resistor is connected directly across the voltage source, so they all experience the same voltage. However, the current divides among the different paths according to each resistor's resistance.

Worked example: calculating current in parallel circuit

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Worked Example: Total Current in Parallel Circuit

Question: Calculate the total current in this parallel circuit.

Given:

V = 12 V
R₁ = 2 Ω
R₂ = 4 Ω

Solution:

Step 1: Find equivalent resistance

$\frac{1}{R_p} = \frac{1}{R_1} + \frac{1}{R_2} = \frac{1}{2} + \frac{1}{4} = \frac{2}{4} + \frac{1}{4} = \frac{3}{4}$

Therefore: $R_p = \frac{4}{3} = 1.33\,\Omega$

Step 2: Apply Ohm's Law

$I = \frac{V}{R_p} = \frac{12\,V}{1.33\,\Omega} = 9\,A$

The total current flowing through the circuit is 9 A.

Worked example: finding unknown resistance in parallel

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Worked Example: Finding Unknown Resistance in Parallel

Question: Two resistors are connected in parallel. Find the unknown resistance R₂.

Given:

V = 9 V
R₁ = 3 Ω
I = 4.8 A (total current)
R₂ = ?

Solution:

Step 1: Find equivalent resistance

$R = \frac{V}{I} = \frac{9\,V}{4.8\,A} = 1.875\,\Omega$

Step 2: Use parallel resistance formula

$\frac{1}{R} = \frac{1}{R_1} + \frac{1}{R_2}$

Rearranging: $\frac{1}{R_2} = \frac{1}{R} - \frac{1}{R_1} = \frac{1}{1.875} - \frac{1}{3} = 0.533 - 0.333 = 0.2$

Therefore: R₂ = 5 Ω

Worked example: current distribution in parallel

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Worked Example: Current Distribution in Parallel Circuit

Question: Calculate the current through each resistor in a parallel circuit.

Given:

V = 18 V
R₁ = 4 Ω
R₂ = 12 Ω

Solution:

Step 1: Calculate current through R₁

Since voltage is the same across parallel resistors:
$I_1 = \frac{V}{R_1} = \frac{18\,V}{4\,\Omega} = 4.5\,A$

Step 2: Calculate current through R₂

$I_2 = \frac{V}{R_2} = \frac{18\,V}{12\,\Omega} = 1.5\,A$

Step 3: Verify using total current

Total current: $I = I_1 + I_2 = 4.5\,A + 1.5\,A = 6\,A$

Series and parallel networks

Many circuits combine both series and parallel arrangements. These are called series-parallel networks or combination circuits.

To solve combination circuits, follow these systematic steps:

Identify parallel and series sections
Calculate equivalent resistance for each parallel section
Simplify the circuit to a series arrangement
Apply series circuit rules to find total resistance

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Worked Example: Combination Circuit Analysis

Consider a circuit where two parallel sections are connected in series:

Step 1: Calculate equivalent resistance of each parallel section

For Parallel Circuit 1: $\frac{1}{R_{p1}} = \frac{1}{10} + \frac{1}{10} = \frac{2}{10}$ Therefore: $R_{p1} = 5\,\Omega$

For Parallel Circuit 2: $\frac{1}{R_{p2}} = \frac{1}{10} + \frac{1}{10} = \frac{2}{10}$

Therefore: $R_{p2} = 5\,\Omega$

Step 2: Add series resistances

Total equivalent resistance: $R = R_{p1} + R_{p2} = 5\,\Omega + 5\,\Omega = 10\,\Omega$

Common exam tips and problem-solving strategies

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Series Circuit Strategies

Always start by finding the equivalent resistance
Remember current is constant throughout
Use voltage division to find individual voltage drops
Check that all voltage drops add to the source voltage

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Parallel Circuit Strategies

Calculate equivalent resistance using the reciprocal formula
Remember voltage is constant across all branches
Find individual currents using Ohm's Law for each resistor
Verify that branch currents add to the total current

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Common Mistakes to Avoid

Don't mix up series and parallel resistance formulas
Always draw the circuit diagram before calculating
Double-check your equivalent resistance calculations
Ensure your final answer has correct units (A, V, or Ω)

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Key Points to Remember:

Series circuits have the same current flowing through all components, while voltages add up across resistors
Parallel circuits have the same voltage across all components, while currents add up through different branches
Equivalent resistance in series is the sum of individual resistances: $R_s = R_1 + R_2 + ...$
Equivalent resistance in parallel uses the reciprocal formula: $\frac{1}{R_p} = \frac{1}{R_1} + \frac{1}{R_2} + ...$
Always apply Ohm's Law $(V = I \times R)$ after finding the equivalent resistance to solve for unknown quantities

Ohm’s Law in Series and Parallel Circuits (Grade 11 NSC Matric Physical Sciences): Revision Notes