Series and parallel circuits — OCR A-Level Physics
Test yourself on Series and parallel circuits with OCR A-Level practice questions.
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Series and parallel circuits explained
Kirchhoff’s second law states that in any closed loop of a circuit, the sum of the electromotive forces equals the sum of the potential differences.
Read the full explanation
It is a statement of conservation of energy: charge returning to its starting point must have the same total energy, so energy gained from sources equals energy lost in components. In a series circuit with a 6.0 V battery and two resistors of 2.0 Ω and 4.0 Ω carrying 1.0 A, the p.d.s are 2.0 V and 4.0 V, which sum to 6.0 V. In multiple-choice questions, identify the loop, assign signs consistently and check that the algebraic sum of e.m.f.s and p.d.s around the loop is zero.
(b) Kirchhoff’s first and second laws applied to electrical circuits
Kirchhoff’s first law states that the sum of currents entering a junction equals the sum leaving it, because charge is conserved. At a junction where 2.0 A enters and splits into I₁ and I₂, I₁ + I₂ = 2.0 A. Kirchhoff’s second law states that around any closed loop, the sum of e.m.f.s equals the sum of p.d.s, because energy is conserved per unit charge. In a series circuit with a 6.0 V battery and two resistors, the p.d.s add to 6.0 V. These laws apply to any circuit, including series and parallel combinations, and are used to analyse unknown currents and voltages. They are fundamental to circuit analysis and are assessed in multiple-choice questions requiring calculation or identification of correct statements.
(c) total resistance of two or more resistors in series; R = R₁ + R₂ + ...
For resistors connected in series, the total resistance is the sum of individual resistances: R = R₁ + R₂ + ... . This is because the same current flows through each resistor, and the total potential difference is the sum of the potential differences across each resistor. For example, a 10 Ω and a 20 Ω resistor in series give a total resistance of 30 Ω. This rule applies to any number of resistors in series. In multiple-choice questions, you may be asked to calculate total resistance or identify the correct expression. Remember that series connections increase total resistance, and the formula is straightforward addition.
(d) total resistance of two or more resistors in parallel; 1/R = 1/R₁ + 1/R₂ + ...
For resistors connected in parallel, the reciprocal of the total resistance equals the sum of the reciprocals of individual resistances: 1/R = 1/R₁ + 1/R₂ + ... . This is because the potential difference across each resistor is the same, and the total current is the sum of the currents through each resistor. For example, two 10 Ω resistors in parallel give 1/R = 1/10 + 1/10 = 2/10, so R = 5 Ω. The total resistance is always less than the smallest individual resistance. In multiple-choice questions, you may need to calculate total resistance or identify the correct formula. Remember to take the reciprocal of the sum to find R.
(e) analysis of circuits with components, including both series and parallel
Analysing circuits with components in both series and parallel requires applying Kirchhoff’s laws and the rules for total resistance. For a circuit with a series part and a parallel part, first simplify the parallel section to a single equivalent resistance using 1/R = 1/R₁ + 1/R₂ + ... , then add series resistances. Use Ohm’s law (V = IR) to find currents and voltages. For example, a 6.0 V battery connected to a 4.0 Ω resistor in series with a parallel pair of 6.0 Ω and 3.0 Ω resistors: the parallel pair has equivalent resistance 2.0 Ω, total resistance 6.0 Ω, total current 1.0 A, and the p.d. across the parallel pair is 2.0 V. Multiple-choice questions may ask for a current, voltage, or equivalent resistance.
(f) analysis of circuits with more than one source of e.m.f.
When a circuit contains more than one source of e.m.f., you cannot treat the current as driven by a single p.d. Instead, apply Kirchhoff's laws: at any junction, the sum of currents entering equals the sum leaving; around any closed loop, the algebraic sum of potential differences is zero. Assign a current direction to each branch, write loop equations using the e.m.f.s and the p.d.s across resistors, and solve the simultaneous equations. For example, two batteries of 6.0 V and 4.0 V in parallel with resistors require you to define loop directions and signs consistently. The method works for any number of sources and resistors, including opposing e.m.f.s.
Your focus
- State Kirchhoff’s second law and relate it to conservation of energy.
- Apply the law to find unknown e.m.f.s or potential differences in a loop.
- Use consistent sign conventions when analysing a closed loop.
Show all 18 objectives
- State Kirchhoff’s first and second laws.
- Apply Kirchhoff’s laws to simple series and parallel circuits to find unknown currents and voltages.
- Identify correct statements about Kirchhoff’s laws in multiple-choice questions.
- State the formula for total resistance of resistors in series.
- Calculate the total resistance of two or more resistors in series.
- Recognise series connections in circuit diagrams.
- State the formula for total resistance of resistors in parallel.
- Calculate the total resistance of two or more resistors in parallel.
- Recognise parallel connections in circuit diagrams.
- Analyse circuits containing both series and parallel components.
- Calculate equivalent resistance, current, and potential difference in mixed circuits.
- Apply Kirchhoff’s laws and Ohm’s law to solve circuit problems.
- Apply Kirchhoff's junction rule to circuits with multiple e.m.f.s.
- Apply Kirchhoff's loop rule to circuits with multiple e.m.f.s.
- Solve simultaneous equations to find unknown currents or potential differences in such circuits.
Series and parallel circuits exam tips
Marking Points
- States Kirchhoff’s second law as the sum of e.m.f.s equals the sum of p.d.s around a closed loop.
- Links the law to conservation of energy for charge moving around a complete circuit.
- Applies the law to series circuits where p.d.s sum to the supply e.m.f.
- Uses the law to find an unknown p.d. or e.m.f. in a loop.
- Recognises that the algebraic sum of potential changes around a closed loop is zero.
- Kirchhoff’s first law: sum of currents entering a junction equals sum leaving, based on conservation of charge.
- Kirchhoff’s second law: around any closed loop, sum of e.m.f.s equals sum of p.d.s, based on conservation of energy.
- Application to junctions: at a node, total current in equals total current out, e.g. I₁ = I₂ + I₃.
- Application to loops: in a series circuit, the sum of potential differences across components equals the supply e.m.f.
- Use in circuit analysis: these laws allow calculation of unknown currents and voltages in series and parallel networks.
- Total resistance in series is the sum of individual resistances: R = R₁ + R₂ + ... .
- The same current flows through each resistor in series.
- The total potential difference is the sum of the potential differences across each resistor.
- Adding resistors in series increases the total resistance.
- Application: calculate total resistance for two or more resistors in series.
- Total resistance in parallel is given by 1/R = 1/R₁ + 1/R₂ + ... .
- The potential difference across each resistor in parallel is the same.
- The total current is the sum of the currents through each resistor.
- The total resistance is always less than the smallest individual resistance.
- Application: calculate total resistance for two or more resistors in parallel.
- Identify series and parallel sections in a circuit diagram.
- Calculate equivalent resistance of parallel resistors using 1/R = 1/R₁ + 1/R₂ + ... .
- Add series resistances to find total resistance.
- Apply Ohm’s law (V = IR) to find current or potential difference.
- Use Kirchhoff’s laws to check current and voltage distributions.
- State Kirchhoff's junction rule: the sum of currents entering a junction equals the sum leaving.
- State Kirchhoff's loop rule: around any closed loop, the algebraic sum of e.m.f.s equals the sum of p.d.s, or the total p.d. is zero.
- Assign a current direction to each branch and apply consistent sign conventions for e.m.f.s and p.d.s.
- Write simultaneous equations for each independent loop and junction, then solve for unknown currents or p.d.s.
- Check the solution: a negative current indicates the actual direction is opposite to the assumed direction.
Examiner Tips
- 💡Mark the chosen loop direction on a diagram before writing the equation.
- 💡List e.m.f.s and p.d.s with their signs, then sum them to zero.
- 💡Check that the total p.d. across series components equals the supply e.m.f. as a quick verification.
- 💡For multiple-choice questions, check each option against the laws: first law involves currents at a junction, second law involves voltages around a loop.
- 💡When calculating, ensure you use consistent signs for currents and voltages; define a direction and stick to it.
- 💡Remember that in a series circuit, current is the same everywhere, so Kirchhoff’s first law is trivially satisfied at any point.
- 💡In multiple-choice questions, if you see resistors in a line, they are in series; simply add their resistances.
- 💡Check units: ensure all resistances are in ohms before adding.
- 💡If a question asks for total resistance of a series combination, look for the sum; do not use the parallel formula.
- 💡In multiple-choice questions, if resistors are connected between the same two points, they are in parallel; use the reciprocal formula.
- 💡For two resistors in parallel, you can use the product-over-sum shortcut: R = (R₁ × R₂)/(R₁ + R₂).
- 💡Always check that your calculated total resistance is less than the smallest individual resistance.
- 💡In multiple-choice questions, redraw the circuit to clearly identify series and parallel sections.
- 💡Calculate equivalent resistances step by step, and keep track of units.
- 💡Use Ohm’s law and Kirchhoff’s laws to find unknown values; check that your answers are physically reasonable.
- 💡Draw the circuit clearly and label all currents with assumed directions before writing equations.
- 💡Use a consistent sign convention: for example, traversing a loop in the chosen direction, treat e.m.f.s that drive current in that direction as positive and those opposing as negative.
- 💡Solve the simultaneous equations carefully, then interpret any negative result as a current flowing opposite to your assumed direction.
Common Mistakes
- Confusing Kirchhoff’s second law with the first law: the correction is that the second law concerns energy around a loop, while the first law concerns charge at a junction.
- Ignoring the direction of the loop when assigning signs: the correction is to choose a direction and treat e.m.f.s and p.d.s consistently.
- Assuming the law applies only to series circuits: the correction is that it applies to any closed loop, including those in parallel networks.
- Misunderstanding: Kirchhoff’s first law applies to any point in a circuit, not just junctions. Correction: it applies specifically to junctions where current can split or combine.
- Misunderstanding: Kirchhoff’s second law applies only to simple series circuits. Correction: it applies to any closed loop, including those with parallel branches.
- Misunderstanding: Kirchhoff’s laws are independent of conservation principles. Correction: they are direct consequences of conservation of charge and energy.
- Misunderstanding: The sum of currents entering a junction is always zero. Correction: the algebraic sum is zero if directions are considered, but the sum of magnitudes entering equals that leaving.
- Misunderstanding: total resistance in series is less than the smallest resistance. Correction: it is always greater than the largest individual resistance.
- Misunderstanding: the formula R = R₁ + R₂ + ... applies to parallel circuits. Correction: it applies only to series circuits; parallel uses reciprocals.
- Misunderstanding: current is different through each resistor in series. Correction: current is the same through all components in series.
- Misunderstanding: voltage across each resistor is the same in series. Correction: voltage divides according to resistance; only current is common.
- Misunderstanding: total resistance in parallel is the sum of individual resistances. Correction: it is the reciprocal of the sum of reciprocals.
- Misunderstanding: total resistance in parallel is greater than the largest resistance. Correction: it is always less than the smallest individual resistance.
- Misunderstanding: forgetting to take the reciprocal of the sum to find R. Correction: after summing reciprocals, invert the result to get R.
- Misunderstanding: voltage is different across each resistor in parallel. Correction: voltage is the same across all parallel branches.
- Misunderstanding: treat the entire circuit as series or parallel without simplifying sections. Correction: break the circuit into series and parallel parts and simplify step by step.
- Misunderstanding: apply the series resistance formula to parallel resistors. Correction: use the reciprocal formula for parallel sections.
- Misunderstanding: assume current is the same through all components in a mixed circuit. Correction: current is the same only in series sections; it divides in parallel branches.
- Misunderstanding: forget to include internal resistance of the battery if given. Correction: include it as a series resistance if the question provides it.
- Assuming that the larger e.m.f. always determines the current direction in every branch; correction: the actual direction depends on the full circuit, and a branch may carry current driven by the smaller source if the larger source is opposed by other p.d.s.
- Adding e.m.f.s arithmetically without considering their polarities; correction: e.m.f.s in opposing senses in a loop must be subtracted, not added.
- Ignoring internal resistance of the sources; correction: each source has internal resistance, which must be included as a resistor in series with its e.m.f. when analysing the circuit.
- Applying the loop rule to a non-closed path; correction: Kirchhoff's loop rule applies only to closed loops.