F2.2 · Analyse series, parallel, and mixed circuits with Ohm’s and Kirchhoff’s laws

Ontario Grade 11 Physics

Electricity and Magnetism

Using Ohm’s law and Kirchhoff’s laws

What you will learn

• Identify series and parallel parts in a circuit.
• Use Ohm’s law to find voltage, current, or resistance.
• Apply Kirchhoff’s current and voltage laws to mixed circuits.
• Check circuit answers using units and the circuit’s physical layout.

1. Prerequisites and circuit patterns

• Series: one path; equal current through each component.
• Parallel: shared endpoints; equal voltage across each branch.
• Equivalent resistance is measured in ohms.

2. Ohm’s law and Kirchhoff’s laws

• Ohm’s law connects voltage, current, and resistance.
• At each junction, entering current equals leaving current.
• Around a complete loop, voltage rises and drops balance.

3. A reliable analysis method

• Determine connections from endpoints, not from how a drawing looks.
• Simplify groups before finding branch values.
• Use Kirchhoff’s laws to verify the circuit as a whole.

4. Units and reasonableness checks

• Include units and a sensible number of significant figures.
• Check voltage balance around loops and current balance at junctions.
• Explain the direction if a calculated current is negative.

Series and parallel patterns

Worked example

Two resistors in series

1. Set up
   The system is the source and both resistors. Conventional current is chosen from the positive terminal through the 2.0 Ω and 4.0 Ω resistors toward the negative terminal. The unknowns are the total resistance, current, and two drops.
2. Combine resistance
   There is one path, so the series resistances add. This gives the resistance seen by the source.
   $$R_{\mathrm{total}}=2.0\,\Omega+4.0\,\Omega=6.0\,\Omega$$
3. Find current
   Use Ohm’s law for the whole circuit. The source voltage divided by total resistance gives the current, which is the same through both series resistors.
   $$I=\frac{12\,\mathrm{V}}{6.0\,\Omega}=2.0\,\mathrm{A}$$
4. Find voltage drops
   Apply Ohm’s law to each resistor. Current flows from the source’s positive terminal through each resistor, so each listed value is a voltage drop in that direction.
   $$V_1=(2.0\,\mathrm{A})(2.0\,\Omega)=4.0\,\mathrm{V},\quad V_2=(2.0\,\mathrm{A})(4.0\,\Omega)=8.0\,\mathrm{V}$$

Worked example

Two resistors in parallel

1. Set up
   The system is the source and two branches connected across the same pair of points. Choose conventional current from the positive terminal toward the junction, then into each branch. Both resistors have the source voltage across them.
2. Find equivalent resistance
   For parallel resistors, add the reciprocals and then take the reciprocal of the result. The equivalent resistance is less than either branch resistance.
   $$\frac{1}{R_{\mathrm{eq}}}=\frac{1}{6.0\,\Omega}+\frac{1}{3.0\,\Omega}=\frac{1}{2.0\,\Omega},\quad R_{\mathrm{eq}}=2.0\,\Omega$$
3. Find total current
   Use the source voltage and equivalent resistance in Ohm’s law. This is the current entering the junction before it divides between branches.
   $$I_{\mathrm{total}}=\frac{12\,\mathrm{V}}{2.0\,\Omega}=6.0\,\mathrm{A}$$
4. Find branch currents
   Each branch has 12 V across it. Apply Ohm’s law separately to each branch; the lower resistance carries the greater current.
   $$I_{6.0\,\Omega}=\frac{12\,\mathrm{V}}{6.0\,\Omega}=2.0\,\mathrm{A},\quad I_{3.0\,\Omega}=\frac{12\,\mathrm{V}}{3.0\,\Omega}=4.0\,\mathrm{A}$$

Worked example

A mixed circuit

1. Set up
   The system includes the source, a series resistor, and a parallel pair. Choose conventional current from the positive terminal through the 2.0 Ω resistor, then into the parallel junction. The unknowns are total current, voltage drops, and branch currents.
2. Simplify the parallel pair
   Find the pair’s equivalent resistance using the parallel rule. Then add the 2.0 Ω series resistance to get total circuit resistance.
   $$R_{\mathrm{parallel}}=\left(\frac{1}{6.0\,\Omega}+\frac{1}{3.0\,\Omega}\right)^{-1}=2.0\,\Omega,\quad R_{\mathrm{total}}=2.0\,\Omega+2.0\,\Omega=4.0\,\Omega$$
3. Find total current and series drop
   Use Ohm’s law with the source voltage and total resistance. The same total current passes through the series resistor before reaching the junction.
   $$I_{\mathrm{total}}=\frac{18\,\mathrm{V}}{4.0\,\Omega}=4.5\,\mathrm{A},\quad V_{2.0\,\Omega}=(4.5\,\mathrm{A})(2.0\,\Omega)=9.0\,\mathrm{V}$$
4. Find branch values
   The remaining source voltage is across the parallel pair. Both branches share that voltage. Apply Ohm’s law to each branch and check that their currents recombine to the incoming current.
   $$V_{\mathrm{parallel}}=18\,\mathrm{V}-9.0\,\mathrm{V}=9.0\,\mathrm{V},\quad I_{6.0\,\Omega}=1.5\,\mathrm{A},\quad I_{3.0\,\Omega}=3.0\,\mathrm{A}$$

Common mistakes and how to avoid them

Lesson summary

• Use Ohm’s law to relate voltage, current, and resistance.
• In series, current is shared and resistances add.
• In parallel, voltage is shared and branch currents add at a junction.
• In mixed circuits, simplify groups and use Kirchhoff’s laws to check loops and junctions.

Check your understanding

Key terms

Conventional current

The circuit-current direction defined as the direction positive charge would move.

Junction

A point where a circuit path splits or joins.

Equivalent resistance

One resistance that has the same overall effect as a group of resistors.

Voltage drop

A decrease in electrical potential across a circuit component in the chosen direction of travel.

Loop

A closed path through a circuit that returns to its starting point.

Continue through SPH3U

View the complete SPH3U Ontario Grade 11 Physics curriculum and lessons

• F2.1 · Use terminology for current, voltage, resistance, power, and transformers
• F2.3 · Design and explain mixed direct-current circuits
• F1.1 · Analyse social and economic impacts of electromagnetic technologies
• F1.2 · Assess electrical generation efficiency and sustainability
• F2.4 · Investigate properties of magnetic fields
• F2.5 · Investigate magnetic fields around conductors and solenoids

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About this lesson and its review

Published by DoAssignment. This reviewed lesson follows Ontario Grade 11 Physics (SPH3U), expectation F2.2. It is a study resource, not an official curriculum publication.

Before publication, content is checked for structure, mathematical or chemical notation, calculations, course boundaries, and readability. Errors can still occur, so corrections are welcomed.

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