Explain the advantages of connecting lamps in parallel in a lighting circuit

4.3.2 Series and Parallel Circuits

Learning objectives (AO1 – AO3)

• Recall the definitions of voltage, current, resistance, power and e.m.f.
• State the qualitative relationships for current, voltage and resistance in series and in parallel (AO1).
• Derive and use the quantitative equations for series‑ and parallel‑combined resistance, e.m.f. and power (AO2).
• Apply Kirchhoff’s junction and loop rules to solve unfamiliar circuit problems (AO3).
• Predict the behaviour of a lighting circuit when lamps are wired in series or in parallel.
• Identify the practical advantages of parallel wiring for domestic lighting.

Reminder – standard circuit symbols (4.3.1)

Example schematic (cell, switch, lamp, ammeter and voltmeter) – students must be able to draw and interpret such diagrams.

Key definitions

• Voltage (V) – electrical potential difference between two points. Unit: volt (V).
• Current (I) – rate of flow of charge through a conductor. Unit: ampere (A).
• Resistance (R) – opposition to the flow of current. Unit: ohm (Ω).
• Power (P) – rate at which electrical energy is converted to other forms.

  \[P = VI = I^{2}R = \frac{V^{2}}{R}\]

  Unit: watt (W).

• Electromotive force (e.m.f.) – energy supplied per unit charge by a source. Measured in volts, symbol \(\mathcal{E}\) or \(E\).

Qualitative statements required by the syllabus (4.3.2)

• In a series circuit the same current flows through every component; the voltage divides in proportion to the resistances.
• In a parallel circuit the voltage across each branch is the same as the source voltage; the total current is the sum of the branch currents.
• The equivalent resistance of a series arrangement is larger than any individual resistance; for a parallel arrangement it is smaller than the smallest individual resistance.
• When ideal e.m.f. sources are connected in series their voltages add; when identical ideal sources are connected in parallel the terminal voltage equals the e.m.f. of each source.

Series circuits

• Current: \[I{\text{total}} = I{1}=I{2}= \dots = I{n}\]
• Voltage across each element: \[V{k}=I{\text{total}}R{k}\qquad\text{and}\qquad V{\text{total}}=\sum{k=1}^{n}V{k}\]
• Equivalent resistance: \[R{\text{eq}} = \sum{k=1}^{n}R_{k}\]

  (for \(n\) identical resistors, \(R_{\text{eq}} = nR\)).

• Combined e.m.f.: \[E{\text{eq}} = \sum{k=1}^{n}E_{k}\]
• Potential‑divider use: In a series chain a known resistor can be used to obtain a required fraction of the source voltage:

  \[V{x}=V{\text{source}}\frac{R{x}}{R{\text{total}}}\]

Parallel circuits

• Voltage across each branch: \[V{1}=V{2}= \dots =V{n}=V{\text{source}}\]
• Current in each branch: \[I{k}= \frac{V{\text{source}}}{R{k}}\qquad\text{and}\qquad I{\text{total}} = \sum{k=1}^{n} I{k}\]
• Equivalent resistance: \[\frac{1}{R{\text{eq}}}= \sum{k=1}^{n}\frac{1}{R_{k}}\]

  (for \(n\) identical resistors, \(R_{\text{eq}} = \frac{R}{n}\)).

• Combined e.m.f.: When identical ideal sources are placed in parallel, the terminal voltage equals the e.m.f. of each source.

Kirchhoff’s rules (useful for unfamiliar contexts)

• Junction (current) rule: The algebraic sum of currents entering a junction equals the sum leaving it.

  \[\sum I{\text{in}} = \sum I{\text{out}}\]

• Loop (voltage) rule: The algebraic sum of the potential differences around any closed loop is zero.

  \[\sum V = 0\]

  (take rises as positive, drops as negative).

Example: A 12 V battery supplies a series resistor \(R{1}=4\;\Omega\) that branches into two parallel resistors \(R{2}=6\;\Omega\) and \(R_{3}=12\;\Omega\). Using the junction rule at the node and the loop rule for each branch gives the same currents as obtained from the series/parallel formulas, confirming the rules.

Comparison of series and parallel formulas

Worked example – three identical 60 W, 240 V lamps

For each lamp:

\[

R = \frac{V^{2}}{P}= \frac{240^{2}}{60}=9.60\times10^{2}\;\Omega\;(3\;\text{significant figures})

\]

1. Lamps in series

• Equivalent resistance: \(R_{\text{eq}} = 3R = 2.88\times10^{3}\;\Omega\).
• Total current: \[

  I = \frac{V{\text{source}}}{R{\text{eq}}}= \frac{240}{2.88\times10^{3}} = 8.3\times10^{-2}\;\text{A}

  \] (2 s.f.).

• Voltage across each lamp: \(V_{\text{lamp}} = IR = 0.083\times960 \approx 80\;\text{V}\).
• Power per lamp: \[

  P_{\text{lamp}} = I^{2}R = (0.083)^{2}\times960 \approx 6.6\;\text{W}

  \] – very dim.

2. Lamps in parallel

• Equivalent resistance: \[

  \frac{1}{R_{\text{eq}}}= \frac{1}{960}+\frac{1}{960}+\frac{1}{960}= \frac{3}{960}

  \;\;\Rightarrow\;\; R_{\text{eq}} = 3.20\times10^{2}\;\Omega

  \] (2 s.f.).

• Total current: \[

  I_{\text{total}} = \frac{240}{320}=7.5\times10^{-1}\;\text{A}

  \] (2 s.f.).

• Current through each lamp: \[

  I_{\text{lamp}} = \frac{240}{960}=2.5\times10^{-1}\;\text{A}

  \] (2 s.f.).

• Power per lamp: \[

  P_{\text{lamp}} = V I = 240 \times 0.25 = 60\;\text{W}

  \] – full brightness.

Note on significant figures: All final answers are given to the same number of significant figures as the data supplied (here 2 s.f.).

Internal resistance of the supply: The calculations above assume an ideal source (\(r_{\text{int}}=0\)). In real mains circuits a small internal resistance exists, causing a tiny voltage drop when large currents flow; this is rarely required for IGCSE calculations but may appear in “unfamiliar context” questions.

Power loss in wiring

• For a conductor of resistance \(r\) carrying current \(I\), the heating loss is \(P_{\text{loss}} = I^{2}r\).
• In a parallel lighting circuit the total current is split among several branches, so each branch carries a smaller current and the associated \(I^{2}r\) loss in the branch conductors is reduced.
• This is one reason why domestic circuits use relatively thin (and cheaper) wiring for each lighting point.

Advantages of connecting lamps in parallel (lighting circuits)

• Uniform brightness: Every lamp receives the full mains voltage, so each operates at its rated power.
• Independent operation: Failure (open circuit) of one lamp does not affect the others.
• Control flexibility: Switches, dimmers or timers can be placed on individual branches without altering the operation of the remaining lamps.
• Reduced heating in conductors: The total current is divided, minimising I²R losses and allowing the use of thinner, cheaper wiring.
• Safety: Lower current in any single conductor reduces the risk of overheating and fire.

Practical skills linked to this topic (AO2 & AO3)

• Design & drawing: Produce a clear schematic of a lighting circuit showing cell, switch, lamp, ammeter and voltmeter (AO1).
• Construction: Build series and parallel arrangements of identical lamps (or resistors) on a breadboard.
• Measurement: Use a voltmeter to record the voltage across each lamp and an ammeter to record the total current. Record values in a table.
• Data analysis: Calculate the theoretical currents and voltages using the formulas, then compute percentage errors and discuss sources of discrepancy (wire resistance, internal resistance of the source, meter loading) (AO2).
• Experiment planning (AO3): Write a brief experimental plan before the activity, stating hypothesis, variables, safety precautions and method.
• Investigation of lamp failure: Remove one lamp from a parallel circuit, observe the effect on the remaining lamps, and explain the observation using the qualitative statements and Kirchhoff’s junction rule.

Summary

• Series: same current, voltage divides, \(R_{\text{eq}}\) larger, e.m.f. adds.
• Parallel: same voltage, current divides, \(R_{\text{eq}}\) smaller, terminal voltage equals each source (identical e.m.f.).
• Kirchhoff’s rules allow analysis of any combination of series and parallel elements.
• Parallel wiring is preferred for domestic lighting because it gives uniform brightness, independent operation, flexible control and reduced heating losses.
• Accurate prediction of circuit behaviour using the series‑ and parallel‑combination equations, together with careful measurement and evaluation, fulfills the requirements of the Cambridge IGCSE 0625 syllabus.